define analysis dimensional

Units Of Measurement

Dimensional Analysis

We quantify the size and shape of things using Dimensional Analysis. It helps us study the nature of objects mathematically. It involves lengths and angles as well as geometrical properties such as flatness and straightness. The basic concept of dimension is that we can add and subtract only quantities with the same dimensions. Similarly, two physical quantities can be equal only if they have the same dimensions.

Dimensional Analysis Explained

The study of the relationship between physical quantities with the help of dimensions and units of measurement is termed dimensional analysis. Dimensional analysis is essential because it keeps the units the same, helping us perform mathematical calculations smoothly.

Unit Conversion and Dimensional Analysis

Dimensional analysis is also called Factor Label Method or Unit Factor Method because we use conversion factors to get the same units. To help you understand the stated better, let’s say you want to know how many metres make 3 km?

We know that 1000 metres make 1 km,

3 km = 3 × 1000 metres = 3000 metres

Here, the conversion factor is 1000 metres.

Using Dimensional Analysis to Check the Correctness of Physical Equation

Let’s say that you don’t remember whether

time = speed/distance, or
time = distance/speed

We can check this by making sure the dimensions on each side of the equations match.

Reducing both the equations to its fundamental units on each side of the equation, we get

$\begin{array}{l}[T]=\frac{[L][T]^{-1}}{L}=[T]^{-1}\,\,(Wrong) \end{array} $
$\begin{array}{l}[T]=\frac{[L]}{[L][T]^{-1}}=[T]\,\,(Right) \end{array} $

However, it should be kept in mind that dimensional analysis cannot help you determine any dimensionless constants in the equation.

Homogeneity Principle of Dimensional Analysis

Principle of Homogeneity states that dimensions of each of the terms of a dimensional equation on both sides should be the same. This principle is helpful because it helps us convert the units from one form to another. To better understand the principle, let us consider the following example:

Example 1: Check the correctness of physical equation s = ut + ½ at 2 . In the equation, s is the displacement, u is the initial velocity, v is the final velocity, a is the acceleration and t is the time in which change occurs.

We know that L.H.S = s and R.H.S = ut + 1/2at 2

The dimensional formula for the L.H.S can be written as s = [L 1 M 0 T 0 ] ………..(1)

We know that R.H.S is ut + ½ at 2 , simplifying we can write R.H.S as [u][t] + [a] [t] 2

=[L 1 M 0 T 0 ]………..(2)

Hence, by the principle of homogeneity, the given equation is dimensionally correct.

Applications of Dimensional Analysis

Dimensional analysis is a fundamental aspect of measurement and is applied in real-life physics. We make use of dimensional analysis for three prominent reasons:

To check the consistency of a dimensional equation
To derive the relation between physical quantities in physical phenomena
To change units from one system to another

Limitations of Dimensional Analysis

Some limitations of dimensional analysis are:

It doesn’t give information about the dimensional constant.
The formula containing trigonometric function, exponential functions, logarithmic function, etc. cannot be derived.
It gives no information about whether a physical quantity is a scalar or vector.

Additional Solved Problems

1. Check the correctness of the physical equation v 2 = u 2 + 2as 2 . Solution:

The computations made on the L.H.S and R.H.S are as follows:

L.H.S: v 2 = [v 2 ] = [ L 1 M 0 T –1 ] 2 = [ L 1 M 0 T –2 ] ……………(1)

R.H.S: u 2 + 2as 2

Hence, [R.H.S] = [u] 2 + 2[a][s] 2

Hence, by the principle of homogeneity, the equation is not dimensionally correct.

2. Evaluate the homogeneity of the equation when the rate flow of a liquid has a coefficient of viscosity η through a capillary tube of length ‘l’ and radius ‘a’ under pressure head ‘p’ given as $\begin{array}{l}\frac{dV}{dt}=\frac{\pi p a^4}{8l\eta}\end{array} $ Solution: $\begin{array}{l} \frac{ dV }{ dt } = \frac{ \pi p ^ { 4 }}{ 8 l \eta } \end{array} $ $\begin{array}{l} [\textup{L.H.S}] = \frac{ [dV] }{[dt]} = \frac{[M^{0} L^{3} T^ {0}]}{[M^{0} T^{0} T^{1}]} = [M^{0}L^{3}T^{-1}] \textup{ …..(1)} \end{array} $ $\begin{array}{l} [\textup{R.H.S}] = \frac{[p] [a] ^ {4}}{[l] [\eta]} \end{array} $ $\begin{array}{l} \therefore [\textup{R.H.S}] = \frac{ [M ^ { 1 } L ^ { -1 } T ^ { -2 }] [M ^ { 0 } L ^ { 1 } T ^ { 0 }]^ { 4 }} { [M ^ { 0 } L ^ { 1 } T ^ { 0 }] [M ^ { 1 } L ^ { -1 } T ^{ -1 }] } \end{array} $ $\begin{array}{l} = \frac{ [M^{ 1 } L ^ { -1 } T ^ { -2 }] [M ^ { 0 }L ^ { 4 } T ^ { 0 } ] }{ [M^{ 1 }L^ { 0 }T^ { -1 }] } \end{array} $ $\begin{array}{l} = \frac{ [M^ { 1 } L ^ { 3 } T^ { 2 }] }{ [M^ { 1 } L^ { 0 } T ^ { -1 }] } = [M^ { 0 }L ^ { 3 }T^{ -1 }] \textup{ …….(2)} \end{array} $

Hence, by the principle of homogeneity, the given equation is homogenous.

Frequently Asked Questions – FAQs

What is dimensional analysis, state true or false: dimensional analysis can not be used to find dimensionless constants., state the principle of homogeneity of dimensions, why do we use dimensional analysis.

To change units from one system to another.

What are the limitations of dimensional analysis?

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Principles of the Dimensional Analysis

First Online: 03 November 2016

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Nearly all scientists at conjunction with simplifying a differential equation have probably used dimensional analysis. Dimensional analysis (also called the factor-label method or the unit factor method) is an approach to problem that uses the fact that one can multiply any number or expression without changing its value. This is a useful technique. However, the reader should take care to understand that chemistry is not simply a mathematics problem. In every physical problem, the result must match the real world.

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Dimensional Analysis

What is Dimensional Analysis?

To solve the mathematical problems of physical quantities, it is important to have a brief knowledge of units and dimensions. The basic concept of dimensions is that only those quantities can be added or subtracted which have the same dimension. This concept helps us to derive relationships between physical quantities.

Dimensional analysis is the study of the relation between physical quantities based on their units and dimensions. It is used to convert a unit from one form to another. While solving mathematical problems, it is necessary to keep the units the same to solve the problem easily.

Do you know what is the significance of dimensional analysis? Well! In engineering and science, dimensional analysis describes the relationships between different physical quantities based on their fundamental qualities such as length, mass, time, and electric current, and units of measure like miles v/s kilometres, or pounds v/s kilograms.

In other words, in Physics, we study two types of physical quantities, i.e., fundamental and derived. The seven fundamental units include mass, length, amount of substance, time, luminous intensity, and electric current. However, if we combine two or more fundamental units, we get derived quantities.

For examples, we denote [M] for mass, [L] for length, and so on. Similarly, for speed, which is a derived quantity given by distance/time, we denote it with [M]/[L] or [ M L -1 ]. This is how we derive the dimensional formula of various quantities.

The conversion factor used is based on the unit that we desire in the answer. Further, we will derive the dimensional formula of various quantities on this page.

How to Perform Dimensional Analysis?

(Image to be added soon)

Unit Conversion

Dimensional analysis is also called a Unit Factor Method or Factor Label Method because a conversion factor is used to evaluate the units.

For example, suppose we want to know how many meters there are in 4 km.

Normally, we calculate as-

1 km = 1000 meters

4 km = 1000 × 4 = 4000 meters

(Here the conversion factor used is 1000 meters)

Principle of Homogeneity of Dimensional Analysis

This principle depicts that, “the dimensions are the same for every equation that represents physical units. If two sides of an equation don’t have the same dimensions, it cannot represent a physical situation.”

For example, in the equation

[M a L b T c ] = MxLyTz

As per this principle, we have

Example of Dimensional Analysis

For using a conversion factor, it is necessary that the values must represent the same quantity. For example, 60 minutes is the same as 1 hour, 1000 meters is the same as 1 kilometre, or 12 months is the same as 1 year.

Let us try to understand it in this way. Imagine you have 15 pens and you multiply that by 1, now you still have the same number of 15 pens. If you want to find out how many packages of the pen are equal to 15 pens, you need the conversion factor.

Now, suppose you have a packaged set of ink pens in which each package contains 15 pens. Let's consider that you have 6 packages. To calculate the total pens, you have to multiply the number of packages by the number of pens in each package. This comes out to be:

15 × 6 = 90 pens

Some other examples of conversion factors that are used in day to day life are:

A simple example : the time taken by a harmonic oscillator.

A complex example: the energy of vibrating conduction or wire.

A third example : demand versus capacity for a disk that is rotating.

Applications of Dimensional Analysis

Dimensional analysis is an important aspect of measurement, and it has many applications in Physics. Dimensional analysis is used mainly because of five reasons, which are:

To check the correctness of an equation or any other physical relation based on the principle of homogeneity. There should be dimensions on two sides of the equation. The dimensional relation will be correct if the L.H.S and R.H.S of an equation have identical dimensions. If the dimensions on two sides are incorrect, then the relations will also be incorrect.

Dimensional analysis is used to convert the value of a physical quantity from one system of units to another system of units.

It is used to represent the nature of physical quantity.

The expressions of dimensions can be manipulated as algebraic quantities.

Dimensional analysis is used to derive formulas.

Limitations of Dimensional Analysis

Some major limitations of dimensional analysis are:

Dimensional analysis doesn't provide information about the dimensional constant.

Dimensional analysis cannot derive trigonometric, exponential, and logarithmic functions.

It doesn't give information about the scalar or vector identity of a physical quantity.

Example of Dimensional Formula: Derivation for Kinetic Energy

The dimensional formula of any physical entity is the mathematical expression representing the powers to which the fundamental units (mass M, length L, time T) are to be raised to obtain one unit of a derived quantity.

Let us now understand the dimensional formula with an example. Now, we know that kinetic energy is one of the fundamental parts of Physics, hence its formula plays a vital role in many fields of Physics. So, let us derive the dimensional formula of kinetic energy.

The kinetic energy has a dimensional formula of,

[M L 2 T -2 ]

M = Mass of the object

L = Length of the object

T = Time taken

Kinetic energy (K.E) is given by = \[\frac {1} {2}\] [Mass x Velocity 2 ] ---- (I)

The dimensional formula of Mass is = [ M 1 L 0 T 0 ] --- (ii)

We know that,

Velocity = Distance × Time -1

= L x T -1 (dimensional formula)

Velocity has a dimensional formula [ M 0 L 1 T -1 ] ----- (iii)

On substituting equation (ii) and iii) in the above equation (i) we get,

Kinetic energy (K.E) = \[\frac {1} {2}\] [Mass x Velocit y 2 ]

Or, K.E = [ M 1 L 0 T 0 ] [ M 0 L 1 T -1 ] 2 = [ M (0 +1) L (1 + 1) T (-1 + -1) ]

Therefore, on solving, we get the dimensional formula for kinetic as [ M 0 L 2 T -2 ].

From this context, we understand that in dimensional analysis a set of units helps us establish the form of an equation and to check that the answer is free of even minute errors.

Solved Example

1. Find out how many feet are there in 300 centimeters (cm).

Ans. We need to convert cm into feet.

Firstly, we have to convert cm into inches, and then inches into feet, as we can't directly convert cm into feet.

The calculation of two conversion factors is required here:

Then, 300 cm = 300 x \[\frac {1} {30.48}\] feet

= 9.84 feet

FAQs on Dimensional Analysis

1. Find out how Many Feet are in 300 Centimeters (cm).

We need to convert cm into feet.

Then, 300 cm = 300× \[\frac {1} {30.48}\] feet

2. Check the Consistency of the Dimensional Equation of Speed.

Dimensional analysis is used to check the consistency of an equation.

Speed = \[\frac {Distance} {Time}\]

[LT -1 ] = \[\frac {L} {T}\]

[LT -1 ] = [LT -1 ]

This equation is correct dimensionally because it has the same dimensions of speed on both sides. This basic test of dimensional analysis is used to check the consistency of equations, but it doesn't check the correctness of an equation.

By this method, the constants of some physical quantities cannot be determined.

Sine of angle = \[\frac {Length} {Length}\] , and hence it is unit-less.

So, it is a dimensionless quantity.

3. How to Check For Dimensional Consistency

Let us consider one of the equations, let say, constant acceleration,

The equation is given by

s = ut + \[\frac {1} {2}\] at 2

s: displacement = it is a unit of length,

Hence its dimension is L

ut: velocity x time, its dimension is LT -1 x T = L

\[\frac {1} {2}\] at 2 = acceleration x time, its dimension is LT -2 x T 2 = L

All these three terms must have the same dimensions in order to be correct.

As these terms have units of length, the equation is dimensionally correct.

Topics > Matter & Measurements > Dimensional Analysis

Dimensional analysis (also called factor-label method or picket fence) is a method used to convert between units, perform unit conversions, and check the correctness of mathematical equations.

It relies on the principles of dimensional consistency, which states that physical quantities being added, subtracted, multiplied, or divided must have the same dimensions or units.

The process of dimensional analysis involves using conversion factors , which are ratios of equivalent quantities expressed in different units , to convert from one set of units to another.

Here's a step-by-step approach to using dimensional analysis:

Identify the given quantity : Identify the quantity you have, along with its unit.

Determine the desired unit : Determine the unit you want to convert the given quantity to.

Set up conversion factors : Find or derive the appropriate conversion factors that relate the given unit to the desired unit. Conversion factors often come from conversion tables, unit equivalencies, or relationships derived from mathematical formulas.

Construct conversion factor chains: Use multiple conversion factors as needed to create a chain of ratios that cancel out unwanted units and leave you with the desired unit. Each conversion factor should be chosen in a way that the units cancel out appropriately.

Perform the calculation : Multiply the given quantity by the conversion factors, making sure that units cancel out correctly. The final result will be the desired quantity expressed in the desired unit.

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Examples and Practice Problems

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Dimensional Analysis

Dimensional analysis is a method of reducing the number of variables required to describe a given physical situation by making use of the information implied by the units of the physical quantities involved. It is also known as the "theory of similarity" .

Physical Quantities, Units and Dimensions

In observing the physical world, we make use of different physical concepts such as size, distance, time, temperature, etc., and to make quantitative deductions, we must adopt independent and reproducible methods of measuring these physical quantities. "Measurement" in essence means the comparison of the unknown quantity with a known reference, as in the measurement of distance using a ruler, or the balancing of an unknown weight against known weights on a chemical balance. For this purpose, agreement on a known "unit" for each physical quantity is needed; for example, the French Bureau of Standards keeps a block of platinum whose mass is defined to be one kilogram, and known "weights" used for chemical balances are ultimately "calibrated" by comparison with this primary reference—usually by a series of intermediate comparisons.

In fact, it is not necessary to have an independent primary reference for every physical quantity of interest since some physical quantities are defined in terms of others. For example, "velocity" is defined as rate of change of position, or as length travelled in a given time; thus if length is measured in meters and time in seconds, velocity can be measured in meters per second. Units based directly on primary references are called fundamental or basic units, while those defined in terms of other units are called compound or derived units. (See also SI Units .)

Given such a definition of one quantity in terms of others, a choice of basic primary references is thus possible. In the above case for example, astronomers prefer to take the velocity of light and the terrestrial year as primary references, so that length is measured in the derived units "light-years."

where T is the temperature; x, the distance, and the "constant" of proportionality λ is called the Thermal Conductivity . Experiment shows that λ is not, in fact, a constant but varies with both temperature and the nature of the solid; it is, in fact, a new physical variable defined by the physical law, just as velocity was defined above.

There are situations where the new physical variable defined by a physical law does turn out to be constant. For example. Joule's experiments on the conversion of mechanical work (W) into heat (Q) are summarized by the equation:

which defines J as the "mechanical equivalent of heat". Joule found that J was independent of the particular conversion process involved and always had the same value (J = 4.184 × 10 7 erg/calorie ).

Such a constant is called a "universal constant" (other examples are Planck's constant and Boltzmann's constant), and when a law of this kind is available, there is further flexibility in choosing the set of fundamental units:

Keep the original system, with heat and work both based on arbitrary references (e.g., calorie and erg), and accept J as a new physical quantity (with units erg/calorie) which is relevant in any situation where work is converted to heat or vice versa.

Drop one of the original primary references (e.g., calorie) and use the law to define its units in terms of the other quantities involved by making the universal constant unity (i.e., a pure number). In this example, this implies that heat and work are the same kind of physical quantity (both are forms of energy). This is called a coherent choice of units.

It may be thought that a coherent choice is always the preferred choice, but this is not necessarily the case.

To summarize, a given physical system is described in terms of a set of physical variables of various types. To make quantitative observations, units of measurement for each variable are required. For this purpose, a subset of physical variables must be chosen, to which arbitrary reference units are assigned, then definitions or physical laws are used to derive units for the remaining variables.

The basic physical quantities for which arbitrary references are assigned are referred to as "dimensions," and the following notation is used:

Example: Heat transfer to fluid flowing through a pipe. The heat transfer coefficient (α) between the fluid and pipe-wall will possibly depend on fluid properties: density (ρ), viscosity (η), specific heat (c p ), thermal conductivity (λ), and also on the fluid mean velocity (u), the length (l) and diameter (D) of the pipe, and the temperature difference (ΔT) between the wall and the fluid.

It is reasonable to take mass [M], length [L], time [T], heat [H] and temperature [Θ] as basic dimensions.

Then, for example, α = [H/L 2 TΘ] and the indices for all the variables can be set out in the form of a table called the dimensional matrix :

then for a typical physical variable v j :

where m ij , i = 1, 2, ... k, j = 1, 2, ... n are the entries in the dimensional matrix, and it follows that:

This equation formally defines a unit-transformation.

Dimensionless Form of Physical Laws

Clearly the outcome of any set of experiments cannot depend on the particular choice of basic units for the variables, which implies that any mathematical equation representing a valid physical law must be invariant to a unit-transformation. Such an equation is said to be dimensionally homogeneous . Thus if: f(v 1 , v 2 , ... v n ) = 0 represents a physical law, then:

where c 1 , c 2 , ... c n satisfy Eq. (1) for some set of ratios U i1 /U i2 , i = 1, 2, ... k. This condition is a constraint on the form of an equation representing a physical law, which allows it to be expressed in terms of a reduced number of variables.

Example: Consider the simple fluid flow equation:

This can be seen more clearly if the equation is divided by its first term, to give:

Here, the units cancel within each bracketed group of variables, which are thus Dimensionless Groups or pure numbers, and the equation has been reduced to a relationship between five groups, rather than the nine original variables. Different physical situations giving the same values for the five groups are said to be "similar."

More generally, any dimensionally homogeneous equation can be reduced to dimensionless form and "similar" solutions can be exploited.

Complete Sets of Dimensionless Groups

It is clear that the members of a given set of variables can be combined to form dimensionless groups in various different ways. In general, we have

where p is the column vector with elements p j . Now (3) has (n - r) linearly independent solutions p (1) , p (2) , ... p (n-r) , where r is the rank of the matrix M, and any other solution is a linear combination of these solutions. This means that a set of (n - r) independent dimensionless groups can be formed, such that none of these can be formed by combination of the other groups in the set, but any group not in the set can be formed by combination of groups in the set. Such a set of dimensionless groups is called a complete set , and clearly any physical law must be expressible as a relation between members of this set.

In order to find the complete set, a subset of linearly independent columns of M must be found, then the columns permuted so that these are the first r columns and M can be written [M 1 , M 2 ]. Then, using (3):

A more intuitive way to describe this procedure is to select "units" for each basic dimension as a combination of variables describing the system, using the same number of variables as there are basic dimensions. Then using these units for each of the remaining variables generates the required set of dimensionless groups.

Example: Heat transfer to fluid flowing through a pipe. Using the dimensional matrix given earlier, lengths can be measured in pipe-diameters D, and temperatures with ΔT as the unit. For mass, the mass of unit volume of the fluid, ρD 3 can be used, and for heat, the capacity of this volume for unit temperature-rise, ρD 3 c p ΔT. Finally, for time, D/u can be used. For the remaining variables:

whence ∏ 1 = α/ρc p u

Had ΔT been omitted from the table, the rows for H and Θ would have been identical, except for a change of sign, showing that all variables only involve H and Θ as a ratio H/Θ, so units for H and Θ cannot be separately formed. Replacing the separate rows by a row for H/Θ (in fact identical to that for H), a unit can be defined [H/Θ] = ρD 3 c p , and the same four dimensionless groups as before can then be obtained.

In general, more complicated groupings of the original choice of basic dimensions may have to be used to obtain a matrix with linearly independent rows, which thus define a new reduced set of basic dimensions, necessary and just sufficient to define the units for all the remaining variables. Again, this is called a complete set of basic dimensions.

This leads to the famous "∏-Theorem:" "For a physical system described by n physical variables using a complete set of r basic dimensions, the laws governing the system can be expressed as mathematical relations among at most (n - r) dimensionless groups of variables."

Buckingham enunciated this theorem in 1914 without the all-important qualification that the basic dimensions form a complete set, and, of course, was unable to prove it. This remained a challenge until the correct form was enunciated and proved by Langhaar (1951).

Applications of Dimensional Analysis

The most obvious advantage of putting physical laws in dimensionless form is that it reduces the number of independent variables needed to describe the situation. Thus, for example, in planning an experimental investigation of heat transfer to fluid in a pipe, the form of the function: ∏ 1 = φ(∏ 2 , ∏ 3 , ∏ 4 ) can be investigated, rather than α = f(D, L, u, ΔT, ρ, η, λ, c p ). Moreover, to vary ∏ 2 , ∏ 3 and ∏ 4 , the most convenient parameters can be chosen. Thus a rig can be built using only a single diameter pipe, and temperatures measured at several points along the length to obtain the effect of varying ∏ 2 , while varying the flow-rate (for a given fluid) gives the effect of varying ∏ 3 . It is in fact better to use ∏' 4 = ∏ 3 /∏ 4 = c p η/λ, which involves only fluid properties, rather than ∏ 4 , and to investigate variation of ∏' 4 by choosing a range of different fluids. Note again that it is easier to find a range of fluids to cover a range of values for ∏' 4 , rather than separate ranges for the individual properties c p , η, λ, which tend to vary together.

In this example, the analysis itself indicated that one variable (ΔT) was irrelevant—or that other factors were being ignored. It also showed that the effect of varying pipe-diameter can be deduced from experiments on a single pipe.

The latter is a special case of exploiting "similar" solutions, which is perhaps better illustrated by the following example:

To eliminate effects of the shape factors (l 1 /l, ... etc.), a scale model is built, geometrically similar to the prototype aircraft, so that these have the same value for model and prototype.

To vary the other two groups independently, the air velocity u can be altered, but otherwise it is necessary either to build several models of different sizes or vary the air properties—in practice, a single model is used and either air temperature or air pressure is varied.

However, if the wind-tunnel uses only atmospheric air and only one model is available, only a partial solution is possible. If in fact u is well below us, then drag does not depend on u s , and hence only one group (the Reynolds Number ulρ/η) is important. Remember however that the model size (l m ) is much smaller than the prototype (l p ), so the velocity (u m ) required in the wind-tunnel will be higher than the prototype velocity of interest (u p ); hence it is the Mach Number (u m /u sm ) of the model which limits the range of validity of the tests. In general, of course, conditions can be chosen so that:

which yields F p /u p l p η p = F m /u m l m η m . These are then similar solutions, and conditions (4) are often called "similarity conditions."

It is quite common for similarity conditions to be incompatible, making it impossible to model actual conditions in all respects on a different scale.

Finally, the larger the number of basic dimensions for a given set of variables, the smaller the number of dimensionless groups in the complete set, and the simpler the resulting system.

Now if a given physical law is relevant, a noncoherent choice of units requires addition of the universal constant as a relevant physical variable. This is not necessary for a coherent choice, but then there is one less basic dimension and the number of groups is the same in each case.

On the other hand, if a physical law is not relevant the universal constant is not needed, but a coherent choice of units will reduce the number of basic dimensions and create an extra dimensionless group.

This is illustrated by the heat transfer example, where generation of heat by fluid friction was ignored. A coherent choice of heat unit would not have indicated that ΔT was irrelevant, and hence would have generated five groups.

Buckingham, E. (1914) "On Physically Similar Systems: Illustrations of the Use of Dimensional Equations", Phys. Rev., 4 , 345.

Langhaar, H. L. (1951) "Dimensional Analysis and the Theory of Models" , John Wiley, New York. DOI: 10.1016/0016-0032(52)90438-9
Buckingham, E. (1914) "On Physically Similar Systems: Illustrations of the Use of Dimensional Equations", Phys. Rev., 4 , 345. DOI: 10.1103/PhysRev.4.345

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Learning Objectives

To be introduced to the dimensional analysis and how it can be used to aid basic chemistry problem solving.
To use dimensional analysis to identify whether an equation is set up correctly in a numerical calculation
To use dimensional analysis to facilitate the conversion of units.

Dimensional analysis is amongst the most valuable tools physical scientists use. Simply put, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are more accurate or easier to find than others.

A Macroscopic Example: Party Planning

If you have every planned a party, you have used dimensional analysis. The amount of beer and munchies you will need depends on the number of people you expect. For example, if you are planning a Friday night party and expect 30 people you might estimate you need to go out and buy 120 bottles of sodas and 10 large pizza's. How did you arrive at these numbers? The following indicates the type of dimensional analysis solution to party problem:

\[(30 \; \cancel{humans}) \times \left( \dfrac{\text{4 sodas}}{1 \; \cancel{human}} \right) = 120 \; \text{sodas} \label{Eq1} \]

\[(30 \; \cancel{humans}) \times \left( \dfrac{\text{0.333 pizzas}}{1 \; \cancel{human}} \right) = 10 \; \text{pizzas} \label{Eq2} \]

Notice that the units that canceled out are lined out and only the desired units are left (discussed more below). Finally, in going to buy the soda, you perform another dimensional analysis: should you buy the sodas in six-packs or in cases?

\[(120\; { sodas}) \times \left( \dfrac{\text{1 six pack}}{6\; {sodas}} \right) = 20 \; \text{six packs} \label{Eq3} \]

\[(120\; {sodas}) \times \left( \dfrac{\text{1 case }}{24\; {sodas}} \right) = 5 \; \text{cases} \label{Eq4} \]

Realizing that carrying around 20 six packs is a real headache, you get 5 cases of soda instead.

In this party problem, we have used dimensional analysis in two different ways:

In the first application (Equations $\ref{Eq1}$ and Equation $\ref{Eq2}$), dimensional analysis was used to calculate how much soda is needed need. This is based on knowing: (1) how much soda we need for one person and (2) how many people we expect; likewise for the pizza.
In the second application (Equations $\ref{Eq3}$ and $\ref{Eq4}$), dimensional analysis was used to convert units (i.e. from individual sodas to the equivalent amount of six packs or cases)

Using Dimensional Analysis to Convert Units

Consider the conversion in Equation $\ref{Eq3}$:

\[(120\; {sodas}) \times \left( \dfrac{\text{1 six pack}}{6\; {sodas}} \right) = 20 \; \text{six packs} \label{Eq3a} \]

If we ignore the numbers for a moment, and just look at the units (i.e. dimensions ), we have:

\[\text{soda} \times \left(\dfrac{\text{six pack}}{\text{sodas}}\right) \nonumber \]

We can treat the dimensions in a similar fashion as other numerical analyses (i.e. any number divided by itself is 1). Therefore:

\[\text{soda} \times \left(\dfrac{\text{six pack}}{\text{sodas}}\right) = \cancel{\text{soda}} \times \left(\dfrac{\text{six pack}}{\cancel{\text{sodas}}}\right) \nonumber \]

So, the dimensions of the numerical answer will be "six packs".

How can we use dimensional analysis to be sure we have set up our equation correctly? Consider the following alternative way to set up the above unit conversion analysis:

\[ 120 \cancel{\text{soda}} \times \left(\dfrac{\text{6 sodas}}{\cancel{\text{six pack}}}\right) = 720 \; \dfrac{\text{sodas}^2}{\text{1 six pack}} \nonumber \]

While it is correct that there are 6 sodas in one six pack, the above equation yields a value of 720 with units of sodas 2 /six pack .
These rather bizarre units indicate that the equation has been setup incorrectly (and as a consequence you will have a ton of extra soda at the party).

Using Dimensional Analysis in Calculations

In the above case it was relatively straightforward keeping track of units during the calculation. What if the calculation involves powers, etc? For example, the equation relating kinetic energy to mass and velocity is:

\[E_{kinetics} = \dfrac{1}{2} \text{mass} \times \text{velocity}^2 \label{KE} \]

An example of units of mass is kilograms (kg) and velocity might be in meters/second (m/s). What are the dimensions of $E_{kinetic}$?

\[(kg) \times \left( \dfrac{m}{s} \right)^2 = \dfrac{kg \; m^2}{s^2} \nonumber \]

The $\frac{1}{2}$ factor in Equation \ref{KE} is neglected since pure numbers have no units. Since the velocity is squared in Equation \ref{KE}, the dimensions associated with the numerical value of the velocity are also squared. We can double check this by knowing the the Joule ($J$) is a measure of energy, and as a composite unit can be decomposed thusly:

\[1\; J = kg \dfrac{m^2}{s^2} \nonumber \]

Units of Pressure

Pressure ( P ) is a measure of the Force ( F ) per unit area ( A ):

\[ P =\dfrac{F}{A} \nonumber \]

Force, in turn, is a measure of the acceleration ($a$) on a mass ($m$):

\[ F= m \times a \nonumber \]

Thus, pressure ($P$) can be written as:

\[ P= \dfrac{m \times a}{A} \nonumber \]

What are the units of pressure from this relationship? ( Note: acceleration is the change in velocity per unit time )

\[ P =\dfrac{kg \times \frac{\cancel{m}}{s^2}}{m^{\cancel{2}}} \nonumber \]

We can simplify this description of the units of Pressure by dividing numerator and denominator by $m$:

\[ P =\dfrac{\frac{kg}{s^2}}{m}=\dfrac{kg}{m\; s^2} \nonumber \]

In fact, these are the units of a the composite Pascal ( Pa ) unit and is the SI measure of pressure.

Performing Dimensional Analysis

The use of units in a calculation to ensure that we obtain the final proper units is called dimensional analysis . For example, if we observe experimentally that an object’s potential energy is related to its mass, its height from the ground, and to a gravitational force, then when multiplied, the units of mass, height, and the force of gravity must give us units corresponding to those of energy.

Energy is typically measured in joules, calories, or electron volts (eV), defined by the following expressions:

1 J = 1 (kg·m 2 )/s 2 = 1 coulomb·volt
1 cal = 4.184 J
1 eV = 1.602 × 10 −19 J

Performing dimensional analysis begins with finding the appropriate conversion factors . Then, you simply multiply the values together such that the units cancel by having equal units in the numerator and the denominator. To understand this process, let us walk through a few examples.

Example $\PageIndex{1}$

Imagine that a chemist wants to measure out 0.214 mL of benzene, but lacks the equipment to accurately measure such a small volume. The chemist, however, is equipped with an analytical balance capable of measuring to $\pm 0.0001 \;g$. Looking in a reference table, the chemist learns the density of benzene ($\rho=0.8765 \;g/mL$). How many grams of benzene should the chemist use?

\[0.214 \; \cancel{mL} \left( \dfrac{0.8765\; g}{1\;\cancel{mL}}\right)= 0.187571\; g \nonumber \]

Notice that the mL are being divided by mL, an equivalent unit. We can cancel these our, which results with the 0.187571 g. However, this is not our final answer, since this result has too many significant figures and must be rounded down to three significant digits. This is because 0.214 mL has three significant digits and the conversion factor had four significant digits. Since 5 is greater than or equal to 5, we must round the preceding 7 up to 8.

Hence, the chemist should weigh out 0.188 g of benzene to have 0.214 mL of benzene.

Example $\PageIndex{2}$

To illustrate the use of dimensional analysis to solve energy problems, let us calculate the kinetic energy in joules of a 320 g object traveling at 123 cm/s.

To obtain an answer in joules, we must convert grams to kilograms and centimeters to meters. Using Equation \ref{KE}, the calculation may be set up as follows:

\[ \begin{align*} KE &=\dfrac{1}{2}mv^2=\dfrac{1}{2}(g) \left(\dfrac{kg}{g}\right) \left[\left(\dfrac{cm}{s}\right)\left(\dfrac{m}{cm}\right) \right]^2 \\[4pt] &= (\cancel{g})\left(\dfrac{kg}{\cancel{g}}\right) \left(\dfrac{\cancel{m^2}}{s^2}\right) \left(\dfrac{m^2}{\cancel{cm^2}}\right) = \dfrac{kg⋅m^2}{s^2} \\[4pt] &=\dfrac{1}{2}320\; \cancel{g} \left( \dfrac{1\; kg}{1000\;\cancel{g}}\right) \left[\left(\dfrac{123\;\cancel{cm}}{1 \;s}\right) \left(\dfrac{1 \;m}{100\; \cancel{cm}}\right) \right]^2=\dfrac{0.320\; kg}{2}\left[\dfrac{123 m}{s(100)}\right]^2 \\[4pt] &=\dfrac{1}{2} 0.320\; kg \left[ \dfrac{(123)^2 m^2}{s^2(100)^2} \right]= 0.242 \dfrac{kg⋅m^2}{s^2} = 0.242\; J \end{align*} \nonumber \]

Alternatively, the conversions may be carried out in a stepwise manner:

Step 1: convert $g$ to $kg$

\[320\; \cancel{g} \left( \dfrac{1\; kg}{1000\;\cancel{g}}\right) = 0.320 \; kg \nonumber \]

Step 2: convert $cm$ to $m$

\[123\;\cancel{cm} \left(\dfrac{1 \;m}{100\; \cancel{cm}}\right) = 1.23\ m \nonumber \]

Now the natural units for calculating joules is used to get final results

\[ \begin{align*} KE &=\dfrac{1}{2} 0.320\; kg \left(1.23 \;ms\right)^2 \\[4pt] &=\dfrac{1}{2} 0.320\; kg \left(1.513 \dfrac{m^2}{s^2}\right)= 0.242\; \dfrac{kg⋅m^2}{s^2}= 0.242\; J \end{align*} \nonumber \]

Of course, steps 1 and 2 can be done in the opposite order with no effect on the final results. However, this second method involves an additional step.

Example $\PageIndex{3}$

Now suppose you wish to report the number of kilocalories of energy contained in a 7.00 oz piece of chocolate in units of kilojoules per gram.

To obtain an answer in kilojoules, we must convert 7.00 oz to grams and kilocalories to kilojoules. Food reported to contain a value in Calories actually contains that same value in kilocalories. If the chocolate wrapper lists the caloric content as 120 Calories, the chocolate contains 120 kcal of energy. If we choose to use multiple steps to obtain our answer, we can begin with the conversion of kilocalories to kilojoules:

\[120 \cancel{kcal} \left(\dfrac{1000 \;\cancel{cal}}{\cancel{kcal}}\right)\left(\dfrac{4.184 \;\cancel{J}}{1 \cancel{cal}}\right)\left(\dfrac{1 \;kJ}{1000 \cancel{J}}\right)= 502\; kJ \nonumber \]

We next convert the 7.00 oz of chocolate to grams:

\[7.00\;\cancel{oz} \left(\dfrac{28.35\; g}{1\; \cancel{oz}}\right)= 199\; g \nonumber \]

The number of kilojoules per gram is therefore

\[\dfrac{ 502 \;kJ}{199\; g}= 2.52\; kJ/g \nonumber \]

Alternatively, we could solve the problem in one step with all the conversions included:

\[\left(\dfrac{120\; \cancel{kcal}}{7.00\; \cancel{oz}}\right)\left(\dfrac{1000 \;\cancel{cal}}{1 \;\cancel{kcal}}\right)\left(\dfrac{4.184 \;\cancel{J}}{1 \; \cancel{cal}}\right)\left(\dfrac{1 \;kJ}{1000 \;\cancel{J}}\right)\left(\dfrac{1 \;\cancel{oz}}{28.35\; g}\right)= 2.53 \; kJ/g \nonumber \]

The discrepancy between the two answers is attributable to rounding to the correct number of significant figures for each step when carrying out the calculation in a stepwise manner. Recall that all digits in the calculator should be carried forward when carrying out a calculation using multiple steps. In this problem, we first converted kilocalories to kilojoules and then converted ounces to grams.

Converting Between Units: Converting Between Units, YouTube(opens in new window) [youtu.be]

Dimensional analysis is used in numerical calculations, and in converting units. It can help us identify whether an equation is set up correctly (i.e. the resulting units should be as expected). Units are treated similarly to the associated numerical values, i.e., if a variable in an equation is supposed to be squared, then the associated dimensions are squared, etc.

Contributors and Attributions

Mark Tye (Diablo Valley College)

Mike Blaber ( Florida State University )

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Test And Quizzes for Biology, Pre-AP, Or AP Biology For Teachers And Students

Dimensional Analysis: Definition, Examples, And Practice

If you’ve heard the term “dimensional analysis,” you might find it a bit overwhelming. While there’s a lot to “unpack” when learning about dimensional analysis, it’s a lot easier than you might think. Learn more about the basics and a few examples of how to utilize the unique method of conversion.

Dimensional Analysis: Definition, Examples, and Practice

As we mentioned, you may hear dimensional analysis referred to as unit analysis; it is often also known as factor-label method or the unit factor method. A formal definition of dimensional analysis refers to a method of analysis “in which physical quantities are expressed in terms of their fundamental dimensions that is often used.”

Most people might agree that this definition needs to be broken down a bit and simplified. It might be easier to understand this method of analysis if we look at it as a method of solving problems by looking converting one thing to another.

While dimensional analysis may seem like just another equation, one of the unique (and important) parts of the equation is that the unit of measurement always plays a role in the equation (not just the numbers).

We use conversions in everyday life (such as when following a recipe) and in math class or in a biology course. When we think about dimensional analysis, we’re looking at units of measurement, and this could be anything from miles per gallon or pieces of pie per person.

Many people may “freeze up” when they see a dimensional analysis worksheet or hear about it in class, but if you’re struggling with some of the concepts, just remember that it’s about units of measurements and conversion. Dimensional analysis is used in a variety of applications and is frequently used by chemists and other scientists.

The Conversion Factor in Dimensional Analysis

One important thing to consider when using dimensional analysis is the conversion factor. A conversion factor , which is always equal to 1, is a fraction or numerical ratio that can help you express the measurement from one unit to the next.

When using a conversion factor, the values must represent the same quantity. For example, one yard is the same as three feet or seven days is the same as one week. Let’s do a quick example of a conversion factor.

Imagine you have 20 ink pens and you multiply that by 1; you still have the same amount of pens. You might want to find out how many packages of pens that 20 pens equal and to figure this out, you need your conversion factor.

Now, imagine that you found the packaging for a set of ink pens and the label says that there are 10 pens to each package. Your conversion factor ends up being your conversion factor. The equation might look something like this:

20 ink pens x 1 package of pens/10 pens = 2 packages of ink pens. We’ve canceled out the pens (as a unit) and ended up with the package of pens.

While this is a basic scenario, and you probably wouldn’t need to use a conversion factor to figure out how many pens you have, it gives you an idea of what it does and how it works. As you can see, conversion factors work a lot like fractions (working with numerators and denominators)

Even though you’re more likely to work with more complex units of measurement while in chemistry, physics, or other science and math courses, you should have a better understanding of using the conversion factor in relation to the units of measurement.

Steps For Working Through A Problem Using Dimensional Analysis

Like many things, practice makes perfect and dimensional analysis is no exception. Before you tackle a dimensional analysis that your instructor hands to you, here are some tips to consider before you get started.

Read the problem carefully and take your time
Find out what unit should be your answer
Write down your problem in a way that you can understand
Consider a simple math equation and don’t forget the conversion factors
Remember, some of the units should cancel out, resulting in the unit you want
Double-check and retry if you have to
The answer you come up with should make sense to you

To help you understand the basic steps we are using an easy problem that you could probably figure out fairly quickly. The question is: How many seconds are in a day?

First, you need to read the question and determine the unit you want to end up with; in this case, you want to figure out “seconds in a day.” To turn this word problem into a math equation, you might decide to put seconds/day or sec/day.

The next step is to figure out what you already know. You know that there are 60 seconds to one minute and you also know that there are 24 hours in one day; all of these units work together, and you should be able to come up with your final unit of measurement. Again, it’s best to write down everything you know into an equation.

After you’ve done a little math, your starting factor might end up being 60 seconds/1 minute. Next, you will need to work your way into figuring out how many seconds per hour. This equation will be 60 seconds/1 minute x 60 minutes/1 hour. The minutes cancel themselves out, and you have seconds per hour.

Remember, you want to find out seconds per day so you’ll need to add another factor that will cancel out the hours. The equation should be 60 seconds/1 minute x 60 minutes/1 hour x 24 hours/1 day. All units but seconds per day should cancel out and if you’ve done your math correctly 86,400 seconds/1 day.

When doing a dimensional analysis problem, it’s more important to pay attention to the units and make sure you are canceling out the right ones to get the final product. Doing your math correctly important, but it’s easier to double-check than trying to backtrack and figure out how you ended up with the wrong unit.

Our example is relatively simple, and you probably had no problem getting the right answer or using the right units. As you work through your science courses, you will be faced with more difficult units to understand. While dimensional analysis will undoubtedly be more challenging, just keep your eye on the units, and you should be able to get through a problem just fine.

Why Use Dimensional Analysis?

As we’ve demonstrated, dimensional analysis can help you figure out problems that you may encounter in your everyday. While you’re likely to explore dimensional analysis a bit more as you take science courses, it can be particularly helpful for Biology students to learn more.

Some believe that dimensional analysis can help students in Biology have a “better feel for numbers” and help them transition more easily into courses like Organic Chemistry or even Physics (if you haven’t taken those courses yet).

Can you figure out a math equation or a word problem without dimensional analysis? Of course, and many people have their own ways of working through a problem. If you do it correctly, dimensional analysis can actually help you answer a problem more efficiently and accurately.

Ready To Test Your Dimensional Analysis Skills?

If you want to practice dimensional analysis, there are dozens of online dimensional analysis worksheets. While many of them are pretty basic or geared towards specific fields of study like Chemistry, we found a worksheet that has an interesting variety. Test out what we’ve talked about and check your answers when you’re done.

How many minutes are in 1 year?
Traveling at 65 miles/hour, how many minutes will it take to drive 125 miles to San Diego?
Convert 4.65 km to meters
Convert 9,474 mm to centimeters
Traveling at 65 miles/hour, how many feet can you travel in 22 minutes? (1 mile = 5280 feet)

Ready to check out your answers and see more questions? Click here .

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Although you will of course need a complete physical model (represented as a set of mathematical equations) to fully describe a physical system, you can get surprisingly far with a simple method that requires no detailed knowledge at all. This method is known as dimensional analysis , and based on the observation in the previous section that the two sides of any physical equation have to have the same dimension. You can use this principle to qualitatively understand a system, and make predictions on how it will respond quantitatively if you change some parameter. To understand how dimensional analysis works, an example is probably the most effective - we’ll take one that is ubiquitous in classical mechanics: a mass oscillating on a spring (known as the harmonic oscillator), see Figure $\PageIndex{1}$.

Example $\PageIndex{1}$: Dimensional Analysis of the Harmonic Oscillator

Consider the harmonic oscillator consisting of a mass of magnitude m, suspended on a spring with spring constant k. If you pull down the mass a bit and release, it will oscillate with a frequency $\omega$. Can we predict how this frequency will change if we double the mass?

There are two ways to answer this question. One is to consider all the forces acting on the mass, then use Newton’s second law to derive a differential equation (known as the equation of motion) for the mass, solve it, and from the solution determine what happens if we change the mass. The second is to consider the dimensions of the quantities involved. We have a mass, which has dimension of mass ($M$), as it is one of our basic quantities. We have a spring with spring constant k, which has dimensions of force per unit length, or mass per unit time squared:

\[[k]={F \over L}={MLT^{-2} \over L}={M \over T^2} \label{1.2.1} \]

Note the notation [k] for the dimension of k. For the frequency, we have $[\omega]={1 \over T}$. Now we know that the frequency is a function of the spring constant and the mass, and that both sides of that equation must have the same sign. Since there is no mass in the dimension of the frequency, but it exists in the dimension of both the spring constant and the mass, we know that $\omega$ must depend on the ratio of k and m: $\omega \sim {k \over m}$. Now ${[{k \over m}]}={1 \over T^2}$, and from $[\omega]={1 \over T}$, we conclude that we must have

\[\omega \sim \sqrt{k \over m} \label{1.2.2}\]

Equation \ref{1.2.2} allows us to answer our question immediately: if we double the mass, the frequency will decrease by a factor of $\sqrt2$.

Note that in Equation \ref{1.2.2} I did not write an equals sign, but a ‘scales as’ sign ($\sim$, sometimes also written as ). That is because dimensional analysis will not tell us about any numerical factor that may appear in the expression, as those numerical factors have no unit (or, more correctly, have no dimension - they are dimensionless).

You may object that there might be another factor at play: shouldn’t gravity matter? The answer is no, as we can also quickly see from dimensional analysis. The force of gravity is given by mg, introducing another parameter g (the gravitational acceleration) with dimension $[g]={L \over T^2}$. Now if the frequency were to depend on g, there has to be another factor to cancel the dependence on the length, as the frequency itself is length-independent. Neither m nor k has a length-dependence in its dimension, and so they cannot ‘kill’ the $L$ in the dimension of $g$; the frequency therefore also cannot depend on $g$ - which we have now figured out without invoking any (differential) equations!

Above, I’ve sketched how you can use dimensional analysis to arrive at a physical scaling relation through inspection: we’ve combined the various factors to arrive at the right dimension. Such combinations are not always that easy to see, and in any case, you may wonder if you’ve correctly spotted them all. Fortunately,there is a more robust method, that we can also use to once again show that the frequency is independent of the gravitational acceleration. Suppose that in general $\omega$ could depend on k, m and g. The functional dependence can then be written as 2

\[[\omega]={[k^{\alpha}m^{\beta}g^{\gamma}]}={M \over T^2}^{\alpha}M^{\beta}{L \over T^2}^{\gamma}={M^{\alpha + \beta}T^{-2(\alpha + \gamma)}L^ \gamma}\]

which leads to three equations for the exponents:

$\alpha +\beta =0$ $-2(\alpha - \gamma) =-1$ $\gamma =0$

which you can easily solve to find $\alpha = {1 \over 2}$, $\beta = -{1 \over 2}$, $\gamma=0$, which gives us Equation \ref{1.2.2}. This method 3 will allow you to get dimensional relations in surprisingly many different cases, and is used by most physicists as a first line of attack when they first encounter an unknown system.

2 The actual function may of course contain multiple terms which are summed, but all those must have the same dimension. Operators like sines and exponentials must be dimensionless, as there are no dimensions of the form sin(M) or $e^L$. The only allowable dimensional dependencies are thus power laws.

3 The method is sometimes referred to as the Rayleigh algorithm, after John William Strutt, Lord Rayleigh (1842-1919), who applied it, among other things, to light scattering in the air. The result of Rayleigh’s analysis can be used to explain why the sky is blue.

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Texas Supreme Court Rejects Challenge on Exceptions to Abortion Ban

The court on Friday unanimously reversed a ruling that had expanded the definition of what counts as a medical emergency under the state’s strict abortion ban.

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By Kate Zernike

Kate Zernike covers abortion for The Times.

The Texas Supreme Court on Friday unanimously rejected a challenge to the state’s strict abortion ban, ruling against a group of 22 women and abortion providers who sought to expand the exceptions for medical emergencies under the law.

While the challenge will continue in trial court, the state’s attorney general, Ken Paxton, would almost certainly appeal any loss there, and the high court’s decision Friday made clear that he would ultimately prevail.

“I will continue to defend the laws enacted by the Legislature and uphold the values of the people of Texas by doing everything in my power to protect mothers and babies,” Mr. Paxton said in a statement.

The lawsuit, filed by the Center for Reproductive Rights, was the first on behalf of women denied abortions after the United States Supreme Court overturned Roe v. Wade two years ago. While the case revolves around the question of what counts as an exception — unlike other lawsuits, it did not seek to overturn a state ban — it has changed the political debate around abortion by underscoring the potentially devastating medical consequences of abortion bans even for women who were not seeking to end unwanted pregnancies.

The court’s 38-page decision on Friday agreed that the experiences the women had recounted in testimony — some so painful that a judge ordered the court into recess — were “filled with immense personal heartbreak.” But it said that Texas law allowed abortions for any woman who faces a life-threatening condition, “before death or serious physical impairment are imminent.”

Echoing arguments from the state and anti-abortion groups, the court blamed doctors for misinterpreting the law.

“A physician who tells a patient, ‘Your life is threatened by a complication that has arisen during your pregnancy, and you may die, or there is a serious risk you will suffer substantial physical impairment unless an abortion is performed,’ and in the same breath states ‘but the law won’t allow me to provide an abortion in these circumstances’ is simply wrong in that legal assessment,” Justice Jane Bland — like all nine of the justices and the attorney general, an elected Republican — wrote in the unanimous opinion.

In a video conference Friday, 11 of the 20 plaintiffs who sought abortions warned, often tearfully, that it was not safe to be pregnant in Texas or any of the 14 states with near-total bans on abortion.

“This could be you, this could be someone you love,” said Ashley Brandt, who went to Colorado to abort one of her twin fetuses because it had no skull, after doctors in Texas said the condition, known as acrania, threatened the other twin’s life as well as hers, but that they could not give her an abortion. “Abortion is health care and exceptions do not work.”

The ruling on Friday reversed one last summer from a trial court judge, a Democrat, that had expanded the definition of exceptions allowed under Texas’ ban. That ruling said doctors could perform an abortion if in their “good faith judgment and in consultation with the pregnant person” it would be medically unsafe for a woman to continue a pregnancy, or if her fetus has a condition making it “unlikely to survive the pregnancy and sustain life after birth.”

The Supreme Court on Friday affirmed the narrower definition of exceptions in the Texas ban, which allows abortion if in the “reasonable medical judgment” of a doctor a woman faces “risk of death” or “substantial impairment of a major bodily function.” As in most states with bans, there is no exception for fatal fetal conditions.

The court noted that the Texas Legislature amended the ban last year to specify two medical conditions that qualify as exceptions, including one known as previable premature rupture of membranes, which the lead plaintiff in the case, Amanda Zurawski, faced at 18 weeks of pregnancy. Ms. Zurawski’s fetus was not viable, but doctors said they could not abort because it still had a heartbeat.

“The law can be — and has been — amended to reflect policy choices on abortion,” Justice Bland wrote.

Abortion rights groups and doctors argue that the language of the ban is too vague, and leaves doctors too afraid to even mention abortion to patients. Those who violate the ban face up to 99 years in prison, at least $100,000 in fines and the loss of their medical license.

The question of medical exceptions has emerged as one of the most potent and contested issues since Roe was overturned. The United States Supreme Court is expected to rule in the coming weeks on whether the Biden Administration can use federal law to allow abortion in states that ban it if the procedure is required to stabilize a patient in an emergency.

Ms. Zurawski, who has told of going into septic shock and being left infertile after being denied an abortion, has appeared in ads and on the campaign trail for Mr. Biden, as Democrats hope to turn anger about abortion bans to their advantage in elections this fall.

The Center for Reproductive Rights filed suit in three other states after filing the case in Texas in March 2023, and in December represented Kate Cox , a mother of two who unsuccessfully sought an abortion in Texas after her fetus was diagnosed with Trisomy 18, a genetic condition that almost always results in miscarriage or stillbirth, or death within the first year of life.

On Friday, Nancy Northup, the president of the center, said that given the court’s ruling, it was unclear whether the Texas case could continue in the trial court. She urged a federal law establishing abortion rights. “As we have seen in poll after poll, that is what the American people want,” she said.

Kate Zernike is a national reporter at The Times. More about Kate Zernike

Trump’s guilty verdict sharpens the two big questions of this election

Trump’s conviction on 34 felony counts is a precedent-breaking outcome that has sharpened the competition between him and President Biden to define the stakes and the choices for voters in November.

The felony conviction of former president Donald Trump might or might not become a turning point in the 2024 presidential election. But its precedent-breaking outcome has sharpened the competition between him and President Biden to define the stakes and the choices for voters in November.

Almost nothing has been normal about this election, and now, above all, is the sobering reality that one of the two likely major candidates for president will run as a felon convicted on 34 counts by a Manhattan jury. No former president has ever been so judged nor sought the nation’s highest office with such a badge of dishonor.

Nearly as striking is the degree to which the hierarchy of the Republican Party — and presumably tens of millions of ordinary citizens who follow its lead — have rallied behind Trump in questioning and in many cases condemning a judicial system that has been a pillar of American democracy. Measured responses about the jury’s work have been the exception rather than the rule.

Two big questions could define the debate between Trump and Biden from here forward. The first is which candidate poses the bigger threat to the future of the country. The second is which candidate will make the lives of Americans better than they are today. Though related, the first focuses on character and temperament, the second on substance and policy.

For supporters of the incumbent president, the answers to both are simple and straightforward. It is the former president who is the clear danger, someone who vows retribution against his adversaries; would allow a restriction of freedoms, including access to abortion ; favors an expansion of executive power that could lead to authoritarian rule and undermine democratic institutions; and, internationally, to disrupt or shatter traditional alliances. And it is Biden who they see as both determined to protect democratic institutions while pursuing policies that would support American families, combat climate change and advocate a leadership role for the United States in the world.

William Galston of the Brookings Institution pointed to one domestic priority Trump has talked about as an example of the threat he would pose if elected to another term. “If Trump is serious about his plan to round up and deport 10 to 15 million illegal immigrants, that would require a profound transformation of not only law enforcement but the U.S. military and many, many aspects of American society,” he said. “It would represent a profound disruption to every town and city. Ripping 10 to 15 million people out of the body politic is momentous.”

But for every Biden supporter who believes these answers are obvious, polls suggest there are as many or more supporters of Trump who believe the opposite. The New York trial has heightened distrust of the judicial system by, in their view, unfairly targeting their champion to weaken his political standing. They blame Biden for bringing the pain of inflation to many families, increasing illegal immigration, degrading society itself and, globally, overseeing a decline in American power and prestige. They believe they were better off during Trump’s presidency than they are now — and that another four years with Biden as president is the greatest threat.

The risks that voters will weigh before November are not equivalent. Trump, by his past actions — including trying to overturn the 2020 election — and present-day statements about his intentions if elected, poses threats to core elements of democracy. What voters are weighing about Biden is whether he has the capacity for another four years in office and the strength of leadership and the policies needed at a time of turbulence at home and abroad.

The seven-week trial of Trump on charges of falsifying business records as part of an effort to affect the outcome of the 2016 election amounted to an extended freeze in a campaign that has been static since last year. The former president was required to be in the courtroom most days, silent except for regular tirades to reporters on his way out of court. Meanwhile, Biden declined to comment in any way, lest he add fuel to the assertion by Trump and his allies that the trial was part of a political effort to bring down his rival.

On Friday, both men weighed in on the verdict. Trump offered a lengthy, grievance-laden monologue, replete with falsehoods and meandering asides about the state of the country and the unfairness of the trial — a “rigged” process, as he has said repeatedly — and a claim that “we’re living in a fascist state.” Hours later, Biden defended the judicial system and declared that it is “reckless … dangerous, and it’s irresponsible” for anyone to question the verdict simply because they don’t like the outcome, while acknowledging Trump’s legal right to an appeal.

Neither candidate nor their campaigns and surrogates are likely to back away from those positions. The bigger question is whether Trump in particular can begin to pivot from the trial and campaign with a focus on the American people rather than himself. History suggests he will struggle mightily to make that turn — and his extended rant Friday provided evidence of how the trial has affected him.

Former New Jersey governor Chris Christie , a onetime Trump confidant who ran against him in the primaries, told Democratic strategist David Axelrod and Republican strategist Mike Murphy for their “Hacks on Tap” podcast this week that too much attention was being paid to how a guilty verdict would affect voters.

“It’s not just what impact it will have on voters that’s important,” Christie said, “but it’s what impact it will have on him because he will get angrier and angrier and more paranoid. And I don’t think that makes him an attractive candidate to the very narrow swath of voters that he has to try to win in order to get the presidency back.”

Democratic pollster Anna Greenberg said in an email, “Trump is going to run on rigged courts and rigged elections. I don’t think he can help himself even though it would be better for him to talk about inflation. Biden is going to run on democratic norms, women’s rights — especially abortion — and the rule of law and be able to ask voters if they want a convicted felon as their president.”

In the end, said former Republican National Committee chair Rich Bond, “The tipping point will be who voters consider the most dangerous choice. Do they stick with an aging incumbent with a questionable record? Or do they entrust their future to a convicted felon who lies about nearly everything except his desire to be a wannabe dictator?”

Three other Trump cases are pending, two involving his role in trying to subvert the 2020 election; the third charges that he deliberately withheld classified documents. Given delays in all of them, it now appears likely, though not certain, that none of the others will be heard before the election.

But the New York trial is hardly the only big event in the weeks ahead that could influence voters and perhaps change minds. On Monday, Hunter Biden will go on trial in Delaware on felony gun charges, the first of two trials that could bring convictions for the president’s son. By the end of the month, the Supreme Court is likely to issue its ruling on Trump’s claim to absolute immunity from prosecution, which could impact the federal Jan. 6 case. And on June 27, Biden and Trump will meet in Atlanta for the first of two planned debates. (A second is scheduled for Sept. 10.)

On July 11, Trump will appear in court in New York for sentencing in the hush money trial. Four days later, Republicans will gather in Milwaukee for their national convention to formally nominate Trump, and sometime in that window, he will name his running mate. Democrats will meet in Chicago in August for their convention, with fears of protests over Biden’s handling of the Israel-Gaza war .

In the short term, Trump’s conviction has energized Republicans, much as the four indictments a year ago consolidated support around his bid for the GOP nomination. From House Speaker Mike Johnson (La.) to an array of other elected officials, the Republicans seized on the verdict as evidence of a weaponized judicial system. Trump’s campaign claimed to have raised $52.8 million in online donations in the 24 hours after the verdict, and some GOP strategists see the verdict as an opportunity now to play offense against Biden by portraying Democrats as defenders of a corrupted system.

Political strategists are rightly cautious about how or whether the conviction will affect the campaign. The polls have changed little for many months. Trump has been holding a slender advantage nationally and in battleground states, but many of these polls are within the margin of error. They do suggest, however, that Biden faces a challenging path to an electoral college majority.

Some polls taken before the verdict indicated that a conviction could prompt some Trump supporters to peel off. Democratic pollster Celinda Lake said she is convinced that the outcome of the trial will have an impact, real or indirect, particularly in the perceptions of the character of the two candidates. “It helps shift the character axis from strength and weakness, which Trump wanted, to stability and presidential character, which serves Biden better,” she said.

Lake is part of the Biden campaign polling operation but said she was not speaking for the campaign.

The verdict does provide Biden with an opportunity to try to change the script and the Atlanta debate will be the most high-profile moment for him to do so. A Democratic strategist said one challenge for Biden until now has been that many voters had greater fears about a second Biden term than Trump returning to the Oval Office. The verdict could change those perceptions, especially with suburban women.

Many voters have expressed disappointment at having to choose between Biden and Trump. Many are cynical about the state of politics. What impact the verdict will have on them is also unknowable now, but there is a possibility that some will choose to vote for a third-party candidate or simply not vote in the presidential race.

The election is now five months away. After Thursday’s verdict, the election more than ever poses elemental questions for voters — about themselves, their own well-being and perhaps above all the country they want to see in the future. The question underlying them all is who they trust most to deliver it: Biden or Trump?

Trump New York hush money case

Donald Trump is the first former president convicted of a crime .

Can Trump still run for president? Yes. He is eligible to campaign and serve as president if elected, but he won’t be able to pardon himself . Here’s everything to know about next steps , what this means for his candidacy and the other outstanding trials he faces .

What happens next? Trump’s sentencing is scheduled for July 11. He faces up to four years in prison, but legal experts say incarceration appears unlikely. Trump has 30 days to file notice of an appeal of the verdict and six months to file the full appeal.

Reaction to the verdict: Trump continued to maintain his innocence , railing against what he called a “rigged, disgraceful trial” and emphasizing voters would deliver the real verdict on Election Day.

The charges: Trump was found guilty on 34 felony counts of falsifying business records . Falsifying business records is a felony in New York when there is an “intent to defraud” that includes an intent to “commit another crime or to aid or conceal” another crime.

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Dimensional analysis
Dimensional analysis. In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measurement (such as meters and grams) and tracking these dimensions as calculations or comparisons ...
1.5: Dimensional Analysis
Table 1.5.1 1.5. 1 lists the base quantities and the symbols used for their dimension. For example, a measurement of length is said to have dimension L or L 1, a measurement of mass has dimension M or M 1, and a measurement of time has dimension T or T 1. Like units, dimensions obey the rules of algebra. Thus, area is the product of two lengths ...
Dimensional analysis
dimensional analysis, technique used in the physical sciences and engineering to reduce physical properties, such as acceleration, viscosity, energy, and others, to their fundamental dimensions of length (L), mass (M), and time (T). This technique facilitates the study of interrelationships of systems (or models of systems) and their properties and avoids the nuisance of incompatible units.
Dimensional Analysis
Dimensional analysis is a fundamental aspect of measurement and is applied in real-life physics. We make use of dimensional analysis for three prominent reasons: To check the consistency of a dimensional equation. To derive the relation between physical quantities in physical phenomena. To change units from one system to another.
1.6: Dimensional Analysis
Table 1.6.1 1.6. 1 lists the base quantities and the symbols used for their dimension. For example, a measurement of length is said to have dimension L or L 1, a measurement of mass has dimension M or M 1, and a measurement of time has dimension T or T 1. Like units, dimensions obey the rules of algebra. Thus, area is the product of two lengths ...
PDF Dimensional Analysis
Dimensional analysis is a means of simplifying a physical problem by appealing to dimensional homogeneity to reduce the number of relevant variables. It is particularly useful for: checking equations; presenting and interpreting experimental data; attacking problems not amenable to a direct theoretical solution;
Dimensional Analysis
Dimensional analysis is the use of a set of units to establish the form of an equation, or more often, to check that the answer to a calculation as a guard against many simple errors. The units included in the set are: These units are also known as base units of the SI measurement system. The unit of luminous intensity (the candela) and amount ...
A Student's Guide to Dimensional Analysis
A good handle on dimensional analysis is probably the most important skill that a modeller should have and this book is an ideal introductory text on the topic. The manner in which the book is written and the material is presented makes it ideal for students who wish to study the material on their own; it is also very useful for instructors ...
Principles of the Dimensional Analysis
Dimensional analysis is a method by which we deduce information about a phenomenon from the single premise that a phenomenon can be described by a dimensionally consistent equation of certain variables. The generality of the method is both its strength and its weakness. The result of a dimensional analysis of a problem is to reduce the number of variables in the problem, thereby gathering ...
Dimensional Analysis
Dimensional analysis is used to convert the value of a physical quantity from one system of units to another system of units. It is used to represent the nature of physical quantity. The expressions of dimensions can be manipulated as algebraic quantities. Dimensional analysis is used to derive formulas.
Dimensional Analysis Definition, Method & Examples
A common definition of dimensional analysis is given as a method of studying physical equations to determine the units in which the solutions of these are expressed by using physical quantities ...
Dimensional Analysis
Dimensional analysis (also called factor-label method or picket fence) is a method used to convert between units, perform unit conversions, and check the correctness of mathematical equations.. It relies on the principles of dimensional consistency, which states that physical quantities being added, subtracted, multiplied, or divided must have the same dimensions or units.
Dimensional Analysis
54828. Dimensional analysis is a method of reducing the number of variables required to describe a given physical situation by making use of the information implied by the units of the physical quantities involved. It is also known as the "theory of similarity" .
1.6: Dimensional Analysis
In this party problem, we have used dimensional analysis in two different ways: In the first application (Equations 1.6.1 and Equation 1.6.2 ), dimensional analysis was used to calculate how much soda is needed need. This is based on knowing: (1) how much soda we need for one person and (2) how many people we expect; likewise for the pizza.
Dimensional Analysis
Dimensional analysis is similar to a dimensional equation, but is a process whereby the actual units are plugged into an equation. The units can then be simplified, and what's left over must show ...
Dimensional analysis Definition & Meaning
The meaning of DIMENSIONAL ANALYSIS is a method of analysis in which physical quantities are expressed in terms of their fundamental dimensions that is often used when there is not enough information to set up precise equations.
Worked example: Dimensional analysis (video)
Video transcript. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.
Dimensional Analysis: Definition, Examples, and Practice
As we mentioned, you may hear dimensional analysis referred to as unit analysis; it is often also known as factor-label method or the unit factor method. A formal definition of dimensional analysis refers to a method of analysis "in which physical quantities are expressed in terms of their fundamental dimensions that is often used.".
Dimensional Analysis
Now we need to check the dimensions of each of the two terms (that is, the left expression and the right expression) in the equation: [v] = LT − 1. [sin(at2 s)] = 1. The two terms have different dimensions—meaning, the equation is not dimensionally consistent. This equation is another example of "nonsense.".
Dimensional Analysis in Chemistry
Dimensional analysis is a structured problem solving process that uses relationships between two or more values to convert one unit of measurement to another. This ratio relationship is called a ...
DIMENSIONAL ANALYSIS Definition & Meaning
Dimensional analysis definition: a method for comparing the dimensions of the physical quantities occurring in a problem to find relationships between the quantities without having to solve the problem completely.. See examples of DIMENSIONAL ANALYSIS used in a sentence.
Analysis: What rights does Trump lose as a felon? And more of your
Donald Trump's conviction by a New York jury on 34 felony counts is historic and unprecedented. No former president or major party presidential candidate has also been a felon.
Intro to dimensional analysis (video)
When you do the dimensional analysis, it makes sure that the math is working out right. It makes sure that you're getting the right units. But even with this, let's try a slightly more complicated example. Let's say that our rate is, let's say, let's keep our rate at 5 meters per second, but let's say that someone gave us the time.
Analysis of knee-bending motion through three-dimensional scanning for
The analysis revealed a stretch of the skin covering the patella bone with a notable vertical strain of 19% and horizontal stretching of 7%. The circumference along the muscle belly at the thigh (15 cm above the center of the patella) and at the calf (7.5 cm below the center of the patella) showed no significant changes with different bending ...
What Is Machine Learning? Definition, Types, and Examples
Machine learning definition Machine learning is a subfield of artificial intelligence (AI) that uses algorithms trained on data sets to create self-learning models that are capable of predicting outcomes and classifying information without human intervention. Machine learning is used today for a wide range of commercial purposes, including suggesting products to consumers based on their past ...
What is Natural Language Processing? Definition and Examples
Natural language processing definition. Natural language processing (NLP) is a subset of artificial intelligence, computer science, and linguistics focused on making human communication, such as speech and text, comprehensible to computers. NLP is used in a wide variety of everyday products and services.
1.2: Dimensional Analysis
The only allowable dimensional dependencies are thus power laws. 3 The method is sometimes referred to as the Rayleigh algorithm, after John William Strutt, Lord Rayleigh (1842-1919), who applied it, among other things, to light scattering in the air. The result of Rayleigh's analysis can be used to explain why the sky is blue.
Investigating the internal structure of multiple mini interviews—A
The Confirmatory factor analysis confirmed the multi-dimensional nature of our MMIs and also confirmed that our stations measured the attributes that they were supposed to measure. Conclusion This study adds to the validity evidence of Multiple Mini-Interviews, in selection of candidates, with required personality traits for healthcare profession.
Texas Supreme Court Rejects Challenge on Exceptions to Abortion Ban
The Supreme Court on Friday affirmed the narrower definition of exceptions in the Texas ban, which allows abortion if in the "reasonable medical judgment" of a doctor a woman faces "risk of ...
Analysis
Trump's conviction on 34 felony counts is a precedent-breaking outcome that has sharpened the competition between him and President Biden to define the stakes and the choices for voters in November.

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