A uniform solid cylinder of mass and radius rolls without slipping across a…

PHYSICS · Multi-part · CBSE · Physics · System of Particles and Rotational Motion · System of Particles and Rotational Motion

Problem

A uniform solid cylinder of mass

1.5\,{\rm kg}

and radius

0.10\,{\rm m}

rolls without slipping across a horizontal warehouse floor at

3.0\,{\rm m\,s^{-1}}

. It then ascends a rough ramp that is oriented at

25^{\circ}

above the horizontal. The ramp is sufficiently rough to prevent any slipping, and energy losses due to friction and air resistance are negligible. The moment of inertia of a solid cylinder about its central axis is

I = \tfrac{1}{2}MR^{2}

. (a) Derive an algebraic expression for the maximum vertical height

h_{\text{max}}

that the cylinder climbs before momentarily coming to rest on the ramp. Express your answer in terms of the given speed

v = 3.0\,{\rm m\,s^{-1}}

, the gravitational acceleration

g

, and any necessary constants. (b) Calculate the numerical value of

h_{\text{max}}

for the cylinder. (c) For comparison, suppose the same cylinder were sliding up the ramp without any rotation at the same initial speed of

3.0\,{\rm m\,s^{-1}}

. Determine the ratio

h_{\text{roll}}/h_{\text{slide}}

of the heights reached in the rolling and sliding cases. (d) If the cylinder were replaced by a thin hoop of the same mass and radius, would the height reached while rolling (with no slipping) be greater than, less than, or equal to

h_{\text{max}}

found in part (b)? Justify your answer quantitatively.

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