example of Rotational Work, Power, and Energy Transfer

GRE Physics Companion: Rotational Work, Power, and Energy Transfer

The essential rotational energy relations are

(1)

(2)

and

(3)

1 High-value GRE facts

1. Constant torque gives W = τΔ𝜃.
2. Variable torque requires W = ∫ τ d𝜃.
3. The area under a τ versus 𝜃 graph is work.
4. rotational power is P = τω for a fixed axis.
5. Torque and energy may share N m dimensions but are different quantities.
6. Flywheel energy scales as ω².
7. Constant power implies τ = P∕ω.
8. A torsional spring stores κ𝜃².
9. Viscous rotational damping gives P = −bω².
10. An ideal transmission conserves mechanical power.

Part I: Original GRE-style problems

Problem 1: constant torque work

A constant torque of 6 N m acts through 10 rad. The work is

1. 0.6 J
2. 6 J
3. 16 J
4. 60 J
5. 600 J

Problem 2: rotational power

A shaft rotates at 80 rad∕s while transmitting torque 25 N m. The power is

1. 0.32 kW
2. 2.0 kW
3. 3.2 kW
4. 20 kW
5. 200 kW

Problem 3: torque-angle graph

Torque increases linearly from zero at 𝜃 = 0 to τ₀ at 𝜃 = 𝜃₀. The work is

1. 0
2. τ₀∕𝜃₀
3. τ₀𝜃₀
4. τ₀𝜃₀
5. 2τ₀𝜃₀

Problem 4: flywheel speed scaling

A flywheel’s angular speed doubles while its moment of inertia remains constant. Its rotational kinetic energy is multiplied by

1. 1∕2
2. 2
3. 4
4. 8
5. 16

Problem 5: torsional spring energy

A torsional spring has constant κ and is twisted through angle 𝜃. Its stored energy is

1. κ𝜃
2. κ𝜃²
3. κ𝜃
4. κ𝜃²
5. 2κ𝜃²

Problem 6: constant power motor

A motor delivers constant power P at angular speed ω. Its torque is

1. Pω
2. P∕ω
3. ω∕P
4. P∕ω²
5. independent of ω

Problem 7: damping power

A rotational damper obeys τ_d = −bω. Its power is

1. +bω
2. −bω
3. +bω²
4. −bω²
5. −bω³

Problem 8: ideal transmission

An ideal transmission reduces angular speed by a factor of 5. If input torque is τ_in, output torque is

1. τ_in∕25
2. τ_in∕5
3. τ_in
4. 5τ_in
5. 25τ_in

Problem 9: braking angle

A rotor with moment of inertia I and initial angular speed ω₀ is stopped by constant braking torque magnitude τ_b. The stopping angle is

1. Iω₀∕τ_b
2. Iω₀²∕(2τ _b)
3. 2Iω₀²∕τ _b
4. τ_b∕(Iω₀²)
5. I∕τ_b

Problem 10: belt power

A tangential belt force F acts at radius R of a Pulley rotating at ω. The transmitted power is

1. Fω
2. FR
3. FRω
4. Fω∕R
5. FR∕ω

Problem 11: constant torque from rest

A rotor with I = 5 kg m² starts from rest. Net torque 20 N m acts through 10 rad. The final angular speed is closest to

1. 4.0 rad∕s
2. 6.3 rad∕s
3. 8.9 rad∕s
4. 20 rad∕s
5. 40 rad∕s

Problem 12: torque versus energy

Which statement is correct?

1. Torque and energy are identical because both use N m.
2. Torque is a rotational moment, while work is a scalar energy transfer.
3. Torque is measured in joules.
4. Work has a direction in space.
5. Torque can be added directly to kinetic energy.

Part II: Complete worked solutions

Solution 1

(4)

Answer: (D).

Solution 2

(5)

Answer: (B).

Solution 3

The graph is a triangle:

(6)

Answer: (C).

Solution 4

Since K = Iω², doubling ω multiplies K by 4. Answer: (C).

Solution 5

(7)

Answer: (D).

Solution 6

From P = τω,

(8)

Answer: (B).

Solution 7

(9)

Answer: (D).

Solution 8

Ideal power conservation gives

(10)

If ω_out = ω_in∕5, then

(11)

Answer: (D).

Solution 9

(12)

so

(13)

Answer: (B).

Solution 10

Since τ = FR,

(14)

Answer: (C).

Solution 11

(15)

Then

(16)

so

(17)

Answer: (C).

Solution 12

Torque is a moment of force. Work is the scalar integral of torque through angular displacement. Answer: (B).

2 GRE checklist

1. Use radians in torque-angle work.
2. Use area under a τ versus 𝜃 graph for variable torque.
3. Use P = τω for fixed-axis power.
4. Use W_net = Δ(Iω²) for rotational state changes.
5. For flywheels, remember the ω² dependence.
6. For constant power, use τ = P∕ω.
7. For damping, check that power is negative.
8. For ideal transmissions, conserve power rather than torque.

References

References