GRE Physics Companion: Rotational Kinematics of Rigid Bodies

GRE Physics Companion: Rotational Kinematics of Rigid Bodies

For fixed-axis rotation,

(1)

For a point at perpendicular radius r_⊥,

(2)

1 High-value GRE facts

1. One revolution is 2π radians.
2. ω = d𝜃∕dt and α = dω∕dt.
3. Under constant α, rotational kinematics mirrors one-dimensional constant-acceleration motion.
4. All points on a rigid body share the same ω and α in fixed-axis rotation.
5. Linear speed increases with perpendicular radius: v = r_⊥|ω|.
6. Tangential acceleration is a_t = r_⊥|α|.
7. Normal acceleration is a_n = r_⊥ω².
8. A point on the fixed axis has zero linear speed.
9. Constant angular speed still produces nonzero normal acceleration away from the axis.
10. Positive α does not always mean increasing angular speed.

Part I: Original GRE-style problems

Problem 1: revolutions to radians

A wheel rotates through 3 complete revolutions. Its angular displacement is

1. 3 rad
2. 3π rad
3. 6π rad
4. 9π rad
5. 12π rad

Problem 2: frequency and angular speed

A rotor spins at 10 Hz. Its angular speed is

1. 5π rad∕s
2. 10π rad∕s
3. 20π rad∕s
4. 40π rad∕s
5. 100π rad∕s

Problem 3: constant angular acceleration

A wheel starts from rest with constant angular acceleration 4 rad∕s². After 3 s its angular speed is

1. 4 rad∕s
2. 7 rad∕s
3. 8 rad∕s
4. 12 rad∕s
5. 36 rad∕s

Problem 4: angular displacement

For the wheel in Problem 3, the angular displacement after 3 s is

1. 6 rad
2. 12 rad
3. 18 rad
4. 24 rad
5. 36 rad

Problem 5: two radii

Points A and B lie on the same rotating disk at radii r and 2r. The ratio of their speeds is

1. v_B∕v_A = 1∕2
2. v_B∕v_A = 1
3. v_B∕v_A = 2
4. v_B∕v_A = 4
5. v_B∕v_A = 8

Problem 6: normal acceleration

A point is 0.50 m from the axis of a wheel rotating at 6 rad∕s. Its normal acceleration is

1. 3 m∕s²
2. 6 m∕s²
3. 12 m∕s²
4. 18 m∕s²
5. 36 m∕s²

Problem 7: tangential acceleration

A point at radius 0.40 m has angular acceleration magnitude 5 rad∕s². Its tangential acceleration magnitude is

1. 1 m∕s²
2. 2 m∕s²
3. 4 m∕s²
4. 5 m∕s²
5. 12.5 m∕s²

Problem 8: constant angular speed

A point away from the axis of a rigid body rotates with constant nonzero angular speed. Its acceleration is

1. zero
2. purely tangential
3. purely inward normal
4. parallel to its velocity
5. always outward

Problem 9: sign of angular acceleration

A wheel has ω = −8 rad∕s and α = −2 rad∕s². Its angular speed is

1. increasing
2. decreasing
3. constant
4. zero
5. impossible to determine

Problem 10: point on axis

A material point lies exactly on a fixed rotation axis. Its tangential speed is

1. |ω|
2. ω²
3. r_⊥|ω|
4. zero
5. infinite

Problem 11: radial scaling of acceleration

Two points on the same disk have radii r and 3r. At one instant the disk has the same ω and α for both points. The ratio of their total acceleration magnitudes is

1. 1∕3
2. 1
3. 3
4. 9
5. depends on the signs of ω and α

Problem 12: angular graph

The area under an angular-velocity-versus-time graph equals

1. angular acceleration
2. angular displacement
3. tangential speed
4. normal acceleration
5. torque

Part II: Complete worked solutions

Solution 1

Each revolution is 2π radians:

(3)

Answer: (C).

Solution 2

(4)

Answer: (C).

Solution 3

(5)

Answer: (D).

Solution 4

Δ𝜃	= ω0t + αt2	(6)
	= (4)(9)	(7)
	= 18 rad.	(8)

Answer: (C).

Solution 5

All points share the same angular speed:

(9)

Thus

(10)

Answer: (C).

Solution 6

(11)

Answer: (D).

Solution 7

(12)

Answer: (B).

Solution 8

Constant angular speed means

(13)

so tangential acceleration is zero. Normal acceleration remains

(14)

Answer: (C).

Solution 9

The signed angular velocity is negative and becomes more negative because α < 0. Therefore |ω| increases. Answer: (A).

Solution 10

On the axis,

(15)

Hence

(16)

Answer: (D).

Solution 11

At fixed ω and α,

(17)

Thus total acceleration magnitude is proportional to radius:

(18)

Answer: (C).

Solution 12

Since

(19)

integration gives

(20)

Answer: (B).

2 GRE checklist

1. Convert revolutions and degrees to radians before using rotational-kinematics equations.
2. Solve angular motion first when all points share the same fixed-axis rotation.
3. Convert to point motion using r_⊥.
4. Keep tangential and normal acceleration distinct.
5. Constant ω does not mean zero acceleration away from the axis.
6. Check signs of both ω and α before deciding whether angular speed increases.
7. For constant α, use the same algebraic structure as constant-acceleration translation.
8. Distinguish common angular velocity from position-dependent linear velocity.

References

References