GRE Physics Companion: Uniform Circular Motion

This companion is designed for rapid review after M01-08. GRE style circular motion questions are usually solved fastest by deciding which angular or linear quantities are given, converting revolutions or periods immediately, and then using the shortest equivalent form of the centripetal acceleration relation.

1 Fast triage

The key relations are

v = ωR,

(1)

v2- 2 ac = R = ω R,

(2)

and

2π ω = ---= 2πf. T

(3)

If a rotation rate is given in revolutions per minute, convert to revolutions per second first:

rpm- f = 60 .

(4)

Figure 1. GRE speed equation triage for uniform circular motion. Choose the form that uses the quantities already supplied.

2 Common traps

Constant speed does not imply zero acceleration. The velocity vector changes direction continuously.

The velocity is tangent to the circle; the centripetal acceleration points toward the center.

At fixed radius,

2 ac ∝ v .

(5)

At fixed angular speed,

v ∝ R, ac ∝ R.

(6)

Do not add a separate “centripetal force” to a Free-body diagram. The real forces must combine to produce the required inward net force.

Figure 2. Common circular motion traps: constant speed with nonzero acceleration, tangent velocity, inward acceleration, and the distinction between fixed radius and fixed angular speed scaling.

3 Worked GRE example 1: scaling with speed

A particle moves in a circle of fixed radius R. Its speed is doubled. By what factor does its centripetal acceleration change?

Because

2 v-- ac = R ,

(7)

and R is fixed,

ac,2 (2v)2 ac,1 = v2 = 4.

(8)

The centripetal acceleration increases by a factor of 4.

4 Worked GRE example 2: two points on one rotating disk

Points A and B lie on the same rigid disk at radii R and 2R. The disk rotates with constant angular speed ω. Compare their speeds and centripetal accelerations.

Since

v = ωR,

(9)

we have

vB = 2vA.

(10)

Since

ac = ω2R,

(11)

we also have

a = 2a . B A

(12)

The acceleration ratio is 2, not 4, because the two points have the same angular speed rather than the same linear speed.

5 GRE speed questions

A particle moves at constant speed around a circle. Its acceleration points (A) tangent to the circle (B) outward (C) inward (D) along the velocity.
If the speed of a particle in a circle of fixed radius triples, its centripetal acceleration changes by a factor of (A) 3 (B) 6 (C) 9 (D) 27.
A wheel rotates at 180 rpm. Its frequency is (A) 0.5 Hz (B) 3 Hz (C) 30 Hz (D) 180 Hz.
Two points on the same rigid disk are at radii R and 4R. Their centripetal accelerations are in the ratio (A) 1 : 1 (B) 1 : 2 (C) 1 : 4 (D) 1 : 16.
A particle moves uniformly in a circle with period T. If the period is halved while the radius is unchanged, the centripetal acceleration changes by a factor of (A) 1∕4 (B) 1∕2 (C) 2 (D) 4.
Which statement is true for uniform circular motion? (A) v ∥ a (B) v⋅a = 0 (C) a = 0 (D) the velocity vector is constant.

6 Answers and rationales

C. The centripetal acceleration always points toward the center.
C. At fixed R, a_c ∝ v².
B. 180 revolutions per minute is 3 revolutions per second.
C. At common ω, a_c = ω²R, so acceleration is proportional to radius.
D. Since a_c = 4π²R∕T², halving T multiplies a_c by 4.
B. The tangent velocity and inward acceleration are perpendicular.

References

[1] PhysicsLibrary, M01-08, Uniform Circular Motion.

[2] S. J. Ling, J. Sanny, and W. Moebs, University Physics, Volume 1, OpenStax, 2016.

"GRE Physics Companion: Uniform Circular Motion" is owned by bloftin.
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