GRE Physics Companion: Units and Dimensional Analysis

This companion entry is attached to M00-03: Units and Dimensional Analysis in Mechanics. The main entry develops the physics and mathematics in a normal instructional sequence. This companion focuses only on the kinds of rapid recognition, elimination, scaling, and dimensional checks that are useful in timed GRE-style mechanics problems.

1 What to recognize quickly

A dimensional-analysis problem can often be reduced to one of four questions:

1. Do the two sides of an equation have the same dimensions?
2. Which answer choice has the required dimensions?
3. How does a quantity scale when one variable changes?
4. Which combination of variables is dimensionless?

In mechanics, the most useful base dimensions are

(1)

Frequently used derived dimensions are

(2)

(3)

(4)

(5)

(6)

2 GRE shortcut 1: eliminate by dimensions before calculating

Suppose a speed is known to depend only on gravitational acceleration g and a length L. Before doing any derivation, compare dimensions:

(7)

Taking a square root gives

(8)

which is the dimension of speed. Any answer choice with a different dimensional form can be eliminated immediately.

3 GRE shortcut 2: scaling is often faster than substitution

If

(9)

and x changes by a factor r while y changes by a factor s, then

(10)

For geometrically similar objects,

(11)

(12)

(13)

At constant density, mass also scales as s³.

4 Worked GRE example 1: choose the only possible period

A hypothetical oscillation period T depends only on a mass m and a spring constant k, where

(14)

Which expression can have the dimensions of time?

1. mk
2. m∕k
3. 
4. 

Since

(15)

we obtain

(16)

Therefore the correct choice is C.

5 Worked GRE example 2: scaling without recomputing

A drag-force scale obeys

(17)

The density is unchanged, the area increases by a factor of 3, and the speed doubles. Then

(18)

So the force scale increases by a factor of 12.

6 GRE-speed questions

MECH-GRE-FOUND-001

Which expression has the dimensions of speed?

1. gL
2. g∕L
3. 
4. 

MECH-GRE-FOUND-002

A family of geometrically similar objects has constant density. If all linear dimensions double, the mass changes by a factor of

1. 2
2. 4
3. 8
4. 16

MECH-GRE-FOUND-003

For small oscillations, a pendulum period scales as T ∝. If L increases by a factor of 9, T changes by a factor of

1. 3
2. 9
3. 1/3
4. 81

MECH-GRE-FOUND-004

From Newton’s gravitational law, the dimensions of G are

1. ML⁻²T⁻²
2. L³M⁻¹T⁻²
3. M⁻¹LT⁻¹
4. ML²T⁻¹

MECH-GRE-FOUND-005

Which quantity is dimensionless?

1. v∕g
2. v²∕(gL)
3. gL
4. vL

MECH-GRE-FOUND-006

Suppose a drag-force scale obeys F ∝ ρAv². If density is unchanged, area triples, and speed doubles, the force scale changes by a factor of

1. 6
2. 8
3. 12
4. 24

MECH-GRE-FOUND-007

The dimensions of impulse are the same as the dimensions of

1. energy
2. momentum
3. power
4. acceleration

MECH-GRE-FOUND-008

A quantity has dimensions L²T⁻². Which is a possible interpretation?

1. acceleration
2. speed
3. specific energy, energy per unit mass
4. force

MECH-GRE-FOUND-009

For geometrically similar bodies of fixed density, the ratio A∕V scales with linear size L as

1. L
2. L²
3. 1∕L
4. 1∕L²

MECH-GRE-FOUND-010

If a characteristic time obeys T ∝ L^1∕2g^−1∕2 and g is unchanged, increasing L by a factor of 16 changes T by a factor of

1. 2
2. 4
3. 8
4. 16

7 Answer key and brief rationale

1. C. has dimensions LT⁻¹.
2. C. Mass scales with volume, hence as the cube of length.
3. A. = 3.
4. B. From F = Gm₁m₂∕r².
5. B. v² and gL have the same dimensions.
6. C. The factor is 3 × 2² = 12.
7. B. Impulse has dimensions force times time, equal to momentum.
8. C. Energy per mass has dimensions L²T⁻².
9. C. Area scales as L², volume as L³.
10. B. = 4.

8 Use with the main article

Use this entry after M00-03, not instead of it. The main article supplies the conceptual and mathematical development; this companion supplies timed recognition and elimination practice.

References