Free response · Unit 6 Beta

A cylinder rolls down a ramp

Translation Between Representations · 12 points

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The question

Name: ______________________   Date: ____________   A cylinder rolls down a ramp (12 points)

Diagram

A cylinder rolls down a ramp

A uniform solid cylinder of mass M and radius R (I = ½MR², given) is released from rest at the top of a ramp of height H and rolls down without slipping. Air resistance and rolling friction are negligible. The system is the cylinder and Earth.

Hsolid cylinder, mass M, radius R, I = ½MR²hoop (part d)I = MR²

  1. (a) Draw energy bar charts for the cylinder-Earth system, one just after release at the top and one just before the cylinder reaches the bottom, with the gravitational potential energy Ug zero at the bottom. Use the columns Ug, Ktrans (translational kinetic energy) and Krot (rotational kinetic energy). Bar heights should show relative sizes. [3 points]

  2. (b) Derive an expression for the speed v of the cylinder’s center at the bottom of the ramp, in terms of H and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. [3 points]

  3. (c) On the axes provided, sketch Ug, Ktrans and Krot of the system as functions of the distance the cylinder has traveled along the ramp, from the top (0) to the bottom (D). Label each line. [3 points]

  4. (d) A hoop of the same mass and radius (I = MR², given) is released from the same height and also rolls without slipping. A student says it reaches the bottom with the same speed as the cylinder, because it has the same mass and falls the same height. Using your bar charts, your derivation and your graph, explain why the student is wrong and how the three representations agree with each other. [3 points]