Free response · Unit 6 Beta

Jumping onto a merry-go-round

Mathematical Routines · 10 points

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The question

Name: ______________________   Date: ____________   Jumping onto a merry-go-round (10 points)

Diagram

Jumping onto a merry-go-round

A playground merry-go-round is a uniform disk of mass M and radius R (I = ½MR², given) that turns on a vertical axle. It starts at rest. A child of mass m runs along a line tangent to its rim at speed v and jumps on at the rim. Model the child as a point object.

RvchildTop view; the axle is at the center.Merry-go-round: uniform disk,mass M, radius R,I = ½MR² (given), at rest.Child: mass m, running atspeed v tangent to the rim.

  1. (a) Derive an expression for the angular velocity ω of the merry-go-round and child just after the child jumps on. Express your answer in terms of m, M, R, v and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. [3 points]

  2. (b) The child has mass 30 kg and runs at 4.0 m/s. The merry-go-round has mass 120 kg and radius 1.5 m. Calculate ω. [1 point]

  3. (c) After the child is on, friction at the axle exerts a constant torque of 20 N·m that slows the merry-go-round until it stops. On the axes provided, sketch the angular momentum of the child-merry-go-round system about the axle as a function of time, from just before the child jumps on (t = 0) until the merry-go-round stops. The jump takes a very short time. [2 points]

  4. (d) Calculate how long the merry-go-round takes to stop after the child jumps on. [2 points]

  5. (e) A student claims that the kinetic energy of the child-merry-go-round system is the same just before and just after the child jumps on. Do you agree or disagree? Justify your answer. [2 points]