Aim for FivePhysics C: Mechanics

Mathematical routines (MR)

Block unwinding a string from a solid cylinder

  • Units 2, 5 and 6
  • 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, including calculus, 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. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 20–25 minutes. This question type started in May 2025, so Physics C: Mechanics free-response questions from 2024 and earlier are built differently.

The question

A uniform solid cylinder of mass M = 4.0 kg and radius R = 0.20 m is mounted on a fixed, frictionless horizontal axle through its center. Its rotational inertia about the axle is 12MR2\tfrac{1}{2}MR^2. A light string is wrapped many times around the cylinder, and a block of mass m = 1.0 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 a free-body diagram for the block and a force diagram for the cylinder after the block is released. For each force, name the object that exerts it and give its direction. For the cylinder, also state where each force is exerted. Compare the lengths of the arrows on the block.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

0 / 2,500 characters

Part (b)

2 points

Starting with Newton's second law in translational and rotational form, derive an expression for the magnitude of the acceleration a of the block in terms of m, M, R, and g, as appropriate.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (c)

2 points

Calculate the magnitude of the acceleration of the block and the tension in the string.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (d)

2 points

Using conservation of energy, calculate the speed of the block after it has fallen 1.5 m from rest.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (e)

2 points

The solid cylinder is replaced with a thin-walled hollow cylinder (a hoop) of the same mass M and radius R, with rotational inertia MR2MR^2. The experiment is repeated. Will the acceleration of the block be greater than, less than, or equal to the acceleration found in part (c)? Justify your answer.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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