The figure shows a 100 kg block being released from rest from a height of 1.0 m. It then takes 0.53 s to reach the floor. What is the mass of the block on the left? There is no friction or mass in the pulley.

Question

BlackTom AI · Accepted Answer

Use the kinematic relation for constant acceleration from rest: a = 2h / t^2. Here h = 1.0 m and t = 0.53 s, so the downward acceleration of the 100 kg block is
a = 2(1.0) / (0.53)^2 = 2 / 0.2809 = 7.119 m/s^2 (≈ 7.12 m/s^2).

1) Relevant concepts
- Kinematics for constant acceleration (s = 1/2 at^2 or a = 2s/t^2) to find the system acceleration.
- Atwood-machine dynamics with a massless, frictionless pulley: the same tension acts on both masses and the magnitudes of their accelerations are equal. For two masses M (100 kg, going down) and m (unknown, going up), Newton’s second law gives (M − m) g = (M + m) a.

2) Full reasoning step-by-step
- From kinematics we found a = 7.119 m/s^2.
- From dynamics: (M − m) g = (M + m) a.

3) Algebraic solution (showing intermediate steps)
(M − m) g = (M + m) a
Rearrange to solve for m:
M − m = (a/g)(M + m)
M − m = (a/g) M + (a/g) m
Bring m terms to one side:
M − (a/g) M = m + (a/g) m = m(1 + a/g)
Thus m = M(1 − a/g) / (1 + a/g).

Plug numbers: a/g = 7.119 / 9.80 = 0.72644.
1 − a/g = 0.27356; 1 + a/g = 1.72644.
m = 100 × (0.27356 / 1.72644) = 100 × 0.15844 = 15.844 kg ≈ 15.8 ≈ 16 kg.

4) Why this is correct
The kinematic measurement fixes the acceleration; the Atwood-machine equation links that acceleration to the two masses. Solving yields the computed m. The massless, frictionless pulley assumption ensures the simple relation (M − m) g = (M + m) a is valid.

5) Why the other choices are incorrect
Plugging any other option (11, 13, 14 kg) into the Atwood relation gives a different acceleration and therefore a different fall time inconsistent with the observed 0.53 s; they do not satisfy the measured kinematics.

6) Conclusion
The calculated mass on the left is about 15.8 kg, which matches the provided correct choice: 16 kg.

The figure shows a 100 kg block being released from rest from a height of 1.0 m. It then takes 0.53 s to reach the floor. What is the mass of the block on the left? There is no friction or mass in the pulley.单项选择题

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