A 1.53 kg mass hangs on a rope wrapped around a frictionless disk pulley of mass 7.07 kg and radius 66.0 cm. What is the angular acceleration of the pulley?

Question

BlackTom AI · Accepted Answer

Apply Newton's second law to the hanging mass and rotational dynamics to relate linear acceleration a and angular acceleration α (α = a/R).

1) Relevant concepts
- For the hanging mass: ΣF = m a (downward positive): mg − T = m a.
- For the frictionless solid disk pulley: torque τ = I α, with τ = T R and I (solid disk) = (1/2) M R^2.
- No slipping: a = α R, so α = a / R.

2) Step-by-step reasoning
Write the two equations:
(1) mg − T = m a.
(2) T R = I α = (1/2) M R^2 (a / R) → T R = (1/2) M R a → T = (1/2) M a.

Substitute T from (2) into (1):
mg − (1/2) M a = m a.

Collect terms in a:
mg = m a + (1/2) M a = a (m + M/2).

Solve for linear acceleration a:
a = mg / (m + M/2).

Then angular acceleration α = a / R.

3) Numerical calculation (all intermediate steps)
m = 1.53 kg, M = 7.07 kg, R = 0.66 m, g = 9.80 m/s^2.
Denominator: m + M/2 = 1.53 + 7.07/2 = 1.53 + 3.535 = 5.065 kg.
mg = 1.53 × 9.80 = 14.994 N.
a = 14.994 / 5.065 = 2.959 m/s^2.
α = a / R = 2.959 / 0.66 = 4.484 rad/s^2 ≈ 4.49 rad/s^2.

4) Why this is correct
This result follows directly from balancing the net force on the hanging mass and the torque on the disk, properly including the disk’s rotational inertia (I = 1/2 MR^2). The disk’s inertia reduces the acceleration compared with the case of a massless pulley, producing the computed α.

5) Why the other options are incorrect
- 7.98 and 9.87 rad/s^2 are too large; they correspond to neglecting the pulley’s rotational inertia or using incorrect relations between torque, I, and α. Including the disk’s I reduces α to the value above.
- zero is wrong because there is an unbalanced weight mg causing motion; the system does accelerate.

6) Conclusion
The angular acceleration of the pulley is 4.49 rad/s^2, which matches the provided correct answer.

A 1.53 kg mass hangs on a rope wrapped around a frictionless disk pulley of mass 7.07 kg and radius 66.0 cm. What is the angular acceleration of the pulley?单项选择题

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