Aim for FivePhysics 1

Mathematical routines (MR)

Hanging block unwinding a heavy pulley

  • Unit 5
  • 10 points
  • About 22 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem where you use math to analyze a physical situation. You draw or sketch a representation such as a free-body diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and then make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator allowed. The CED suggests 20–25 minutes.

The question

A pulley is a uniform solid disk of mass M = 2.0 kg and radius R = 0.10 m, so its rotational inertia about its center is (1/2)MR². It is mounted on a fixed horizontal axle through its center, and friction at the axle is negligible. A light string is wrapped around the rim of the pulley, and a block of mass m = 0.50 kg hangs from the free end. The block is released from rest, and the string unwinds without slipping. Use g = 9.8 m/s² (answers that use g = 10 m/s² are also accepted).

Suggested time: 22 minutes

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Part (a)

2 points

Describe (i) the free-body diagram of the block and (ii) a force diagram of the pulley that shows where each force is exerted on the pulley. Give the direction of each force, and compare the lengths of the arrows on the block.

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Part (b)

3 points

Derive an expression for the magnitude of the acceleration a of the block in terms of m, M, and g, as appropriate.

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Part (c)

3 points

Calculate (i) the acceleration of the block, (ii) the speed of the block after it has fallen 0.80 m, and (iii) the angular speed of the pulley at that moment.

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Part (d)

2 points

The solid-disk pulley is replaced with a thin ring (hoop) of the same mass and radius, whose rotational inertia is MR². A student claims the block's acceleration will be smaller with the ring. Do you agree or disagree? Justify your answer.

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