Aim for FivePhysics 1

Mathematical routines (MR)

Clay ball sticks to a block on a spring

  • Units 4 and 7
  • 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 block of mass M = 0.48 kg rests on a horizontal, frictionless surface. It is attached to one end of a horizontal spring with spring constant k = 200 N/m; the other end of the spring is fixed to a wall. The block is at rest at the spring's equilibrium position. A ball of clay of mass m = 0.020 kg moving horizontally at v₀ = 15 m/s directly toward the wall strikes the block and sticks to it. The block–clay combination then oscillates.

Suggested time: 22 minutes

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Part (a)

2 points

Calculate the speed of the block–clay combination immediately after the collision.

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Part (b)

2 points

Derive an expression for the amplitude A of the oscillation in terms of m, M, v₀, k, and physical constants, as appropriate.

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Part (c)

2 points

Calculate the amplitude and the period of the oscillation.

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Part (d)

2 points

A student calculates the amplitude by writing (1/2)m v₀² = (1/2)kA² and gets A = 0.15 m. (i) Describe energy bar charts for the clay–block–spring system at three moments: just before the collision, just after the collision, and at maximum compression of the spring. Use the categories kinetic energy K, spring potential energy U_s, and internal (thermal) energy E_int, and give the value of each nonzero bar. (ii) Use your bar charts to identify the error in the student's reasoning.

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Part (e)

2 points

Suppose instead that the clay ball were replaced by a rubber ball of the same mass and initial velocity that bounced straight back after hitting the block. Would the amplitude of the block's oscillation be greater than, less than, or equal to the amplitude in part (c)? Justify your answer.

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