A uniform rigid cylinder of mass and radius is released from rest at the top of a…

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

Problem

A uniform rigid cylinder of mass

M

and radius

R

is released from rest at the top of a straight ramp of length

L

that makes an angle

\theta

with the horizontal. Throughout the motion the cylinder rolls without slipping. The cylinder’s moment of inertia may be written as

I = k M R^{2}

, where

k

is a dimensionless constant that depends on the mass distribution (for a solid cylinder,

k = \tfrac12

). Neglect air resistance. (a) Derive an expression for the magnitude of the cylinder’s linear acceleration down the ramp in terms of

g

\theta

, and

k

. (b) Using the result from part (a), obtain an expression for the time required for the cylinder to travel the full distance

L

down the ramp. (c) For a solid cylinder (

k = \tfrac12

) released on a ramp of length

6.0\,\text{m}

inclined at

25^{\circ}

, calculate the travel time you derived in part (b).