Practice Dimensional Consistency in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

medium

A claim says

W = Fd + P

, where

W

and

Fd

are energy. What must the units of

P

be?

Example 3

medium

A student writes

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

. Explain the dimensional error and give the correct relation.

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

hard

A student claims

T = \sqrt{L/g} + L

for a pendulum period. Why is this dimensionally inconsistent?

Example 6

medium

If two equations

A = B

and

A = C

both hold and are dimensionally consistent, must

B

and

C

share units?

Example 7

easy

A formula gives area as

A = \ell w

with

\ell, w

in meters. What are the units of

A

Example 8

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 9

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.

Example 10

challenge

Explain why

\sin(x)

requires

x

to be dimensionless, and identify the error in

\sin(t)

where

t

is in seconds.

Example 11

easy

x

is in seconds, what are the units of

1/x

Example 12

medium

A formula reads

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

. Identify the dimensional inconsistency. (

r

in meters)

Example 13

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 14

easy

v = a t

dimensionally consistent? (

a

in m/s²,

t

in s,

v

a velocity)

Example 15

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 16

hard

Why must the argument of

e^x

be dimensionless? Show this for

N = N_0 e^{-\lambda t}

Example 17

medium

\omega

is angular speed in rad/s, the units of

\omega t

for

t

in s are ____.

Example 18

medium

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

P = E - t

dimensionally consistent?

Example 19

easy

x

is measured in seconds, what are the units of

x^2

Example 20

medium

Using dimensional analysis, determine the units of

k

F = kx

where

F

is force (N = kg·m/s²) and

x

is length (m).

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Related Concepts

Equations Dimensional Reasoning