.
Which of the two will start to roll earlier ? The co-efficient of kinetic
friction is 
.
Answer:
In case of pure rotation without translation, the velocity
of centre of mass is zero. The friction reduces the speed at the point of
contact and as such accelerates the centre of mass till the velocity of centre
of mass becomes equal to 
and
the instantaneous velocity at the contact point becomes zero. Thus the force of
friction 
mg produces an
acceleration a in the centre of mass. So the equation of motion for CM is
?(1)
The torque due to force of
friction is 
. It produces
angular retardation given by
?(2)
Rolling begins when
But 
...(3)
[From (1), 
]
Fig. 7.108
and 
or 
or 
or 
or 
for a disc: 
.
For a ring: 
.
Thus the disc begins to roll earlier
than the ring.
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