A uniform solid sphere of mass M and radius R is placed on a ramp inclined at an

Question

A uniform solid sphere of mass M and radius R is placed on a ramp inclined at an angle
? to the horizontal. The coe?cient of static friction between sphere and ramp is µs .
Find the maximum value of ? for which the sphere will roll without slipping, starting
from rest, in terms of the other quantities. (Hint one: you will ?nd the moment of
inertia of a sphere on page 208 of Giancoli. Hint two: write the “no slip” condition as
|?acm| = Ra where ?acm is the linear acceleration of the center of mass of the sphere.)

Explanation / Answer

Velocity of center of mass, v = r

Let

= Angle of inclination

M a = m g sin - f ...(1)

where

a = linear acceleration

and

f = Frictional force

The friction decreases the translation, but accelerates rolling.

Torque of static friction wrt axis of through the center of mass,

= R f = I ...(2)

where

I = moment of inertia of solid sphere = (2/5) M R^2

= Angular acceleration

For pure rolling,

a = R

   = R ( R f / I )   [ From (2) ]

   = R^2 f / I

By substituting in (1)

M [ R^2 f / I ] = M g sin - f

f [ 1 + (MR^2 / I) ] = M g sin

But,  MR^2 / I = 5/2 [ since I = (2/5) M R^2]

So, f [ 1 + (5/2) ] = M g sin

f = (2/7) M g sin

But, f  ( less than or equal to ) s N = tan * M g cos

So, (2/7) M g sin ( less than or equal to ) tan * M g cos

                     tan ( less than or equal to ) (7/2 ) tan

For pure rolling,

The tangent of angle must not exceed (7/2) times of tangent of angle at which the block just

starts to move.

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