Dimensional Consistency Examples: 47 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Dimensional Consistency.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The principle that every term added or equated in a valid equation must share the same physical dimensions or units.

You can't add meters to seconds — dimensionally inconsistent equations don't make physical sense.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Dimensional consistency requires every term in an equation to carry the same units.

Common stuck point: The procedure for dimensional consistency is the easy part; the trap is adding terms with different units. Asking "Does every term being added or equated carry the same units?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does every term being added or equated carry the same units?

Worked Examples

Example 1

easy

Is the equation

v = d + t

(velocity = distance + time) dimensionally consistent?

Answer

No, dimensionally inconsistent.

First step

Step 1:

[v] = \text{m/s}

[d] = \text{m}

[t] = \text{s}

Full solution

2
Step 2: $\text{m} + \text{s}$ — you can't add meters and seconds!
3
Step 3: Not dimensionally consistent. The equation must be wrong.

Dimensional consistency requires all terms being added or set equal to have the same units. You can only add quantities of the same dimension — this is a fundamental check for equation validity.

Example 2

medium

Check:

E = mc^2

where

E

is energy (kg·m²/s²),

m

is mass (kg),

c

is speed (m/s).

Example 3

easy

Check

A = \pi r h

for the lateral area of a cylinder, with

r, h

in meters.

Example 4

medium

A student writes

v^2 = v_0^2 + 2as

. Verify dimensional consistency, with

v, v_0

in m/s,

a

\text{m/s}^2

s

in m.

Example 5

medium

Show that

\frac{1}{2}mv^2

and

mgh

have the same units, justifying conservation of energy.

Example 6

medium

Pressure can be defined as

P = \rho g h

. Verify the units give pascals, with

\rho

\text{kg}/\text{m}^3

g

\text{m/s}^2

h

in m.

Example 7

hard

Find the units of the rate constant

k

\frac{d[A]}{dt} = -k[A]^2

, where

[A]

is in mol/L and

t

in s.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A = l + w

a valid formula for area?

Example 2

medium

In the equation

x^2 + 3x = 10

(where

x

is in meters), are all terms dimensionally consistent?

Example 3

easy

Is the equation

\text{distance} = \text{speed} \times \text{time}

dimensionally consistent? (speed in m/s, time in s)

Example 4

easy

Can you add 5 meters and 3 seconds in a valid equation?

Example 5

easy

A formula gives area as

A = \ell w

with

\ell, w

in meters. What are the units of

A

Example 6

easy

v = a t

dimensionally consistent? (

a

in m/s²,

t

in s,

v

a velocity)

Example 7

easy

What is wrong dimensionally with

\text{length} = \text{length} + \text{area}

Example 8

easy

x

is measured in seconds, what are the units of

x^2

Example 9

easy

E = mc^2

with

m

in kg and

c

in m/s, what are the units of

E

Example 10

easy

Are the two sides of

T = 2\pi\sqrt{L/g}

dimensionally consistent? (

L

in m,

g

in m/s²,

T

a time)

Example 11

medium

A student writes

\text{force} = \text{mass} + \text{acceleration}

. Explain the dimensional error and give the correct relation.

Example 12

medium

Check whether

s = \tfrac12 a t^2

is dimensionally consistent and state the units of

s

. (

a

in m/s²,

t

in s)

Example 13

medium

A formula reads

A = \pi r^2 + 2\pi r

. Identify the dimensional inconsistency. (

r

in meters)

Example 14

medium

Using dimensional analysis, determine the units of

k

F = kx

where

F

is force (N = kg·m/s²) and

x

is length (m).

Example 15

medium

\rho = \frac{m}{V}

consistent with

\rho

having units kg/m³? (

m

in kg,

V

in m³)

Example 16

medium

P = \frac{F}{A}

(pressure), with

F

in N and

A

in m², what are the units of

P

, and what is that unit called?

Example 17

medium

Two formulas claim to give the same quantity: (A)

E = mgh

and (B)

E = \tfrac12 m v^2

. Show both have the same units. (

m

kg,

g

m/s²,

h

v

m/s)

Example 18

medium

Determine the units of the constant

G

F = G\frac{m_1 m_2}{r^2}

. (

F

in N,

m

in kg,

r

in m)

Example 19

medium

What are the units of

\frac{1}{2}mv^2

with

m

in kg and

v

in m/s, and is the

\frac12

relevant to the unit check?

Example 20

challenge

A proposed formula is

v = \sqrt{2gh + k}

where

v

is speed,

g

acceleration,

h

height. What must the units of

k

be, and why?

Example 21

challenge

Use dimensional analysis to guess how the period

T

of a pendulum depends on length

L

and gravity

g

, assuming

T = L^a g^b

Example 22

challenge

Explain why

\sin(x)

requires

x

to be dimensionless, and identify the error in

\sin(t)

where

t

is in seconds.

Example 23

easy

A formula says

\text{volume} = \text{length} \times \text{area}

. With length in m and area in

\text{m}^2

, what are the units of volume?

Example 24

easy

d = v + t

dimensionally consistent, where

d

is distance (m),

v

is speed (m/s),

t

is time (s)?

Example 25

easy

Can you add 200 cm to 3 m in a single sum?

Example 26

easy

Identify the inconsistent term:

E = mgh + v

, with

m

in kg,

g

\text{m/s}^2

h

in m,

v

in m/s.

Example 27

easy

x

is in seconds, what are the units of

1/x

Example 28

medium

A claim says

W = Fd + P

, where

W

and

Fd

are energy. What must the units of

P

be?

Example 29

medium

In the equation

E = hf

, with

E

in joules (

\text{kg}\cdot\text{m}^2/\text{s}^2

) and

f

\text{s}^{-1}

, find the units of

h

Example 30

medium

A formula

L = \alpha T^2

relates length (m) to temperature (K). Find the units of

\alpha

Example 31

medium

An equation reads

y = x^2 + 3x

with

x

in meters. For this to be dimensionally consistent, what must be true about the constant

3

Example 32

medium

If two equations

A = B

and

A = C

both hold and are dimensionally consistent, must

B

and

C

share units?

Example 33

medium

Power has units of W = J/s. If energy is in joules and time in seconds, is

P = E - t

dimensionally consistent?

Example 34

hard

A claim says the centripetal acceleration is

a = v^2 / r^2

. Use dimensions to test the claim. (

v

in m/s,

r

in m,

a

\text{m/s}^2

Example 35

hard

Given

E = \frac{1}{2}I\omega^2

with

\omega

in rad/s (

=\text{s}^{-1}

) and

E

in joules, find the units of

I

Example 36

hard

A proposed model says heat flux

q

q = -k\, dT/dx

, with

q

\text{W/m}^2

T

in K,

x

in m. Find the units of

k

Example 37

hard

A student claims

T = \sqrt{L/g} + L

for a pendulum period. Why is this dimensionally inconsistent?

Example 38

hard

Why must the argument of

e^x

be dimensionless? Show this for

N = N_0 e^{-\lambda t}

Example 39

challenge

Using only dimensions, guess how the speed of a transverse wave on a string depends on tension

T

(units N) and linear mass density

\mu

(units kg/m).

Example 40

challenge

Why can dimensional consistency confirm

E = mc^2

is plausible but not prove a missing constant like

1/2

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Related Concepts

Equations Dimensional Reasoning

Background Knowledge

These ideas may be useful before you work through the harder examples.

equations