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Question

A uniform solid sphere is placed on a smooth horizontal surface. An impulse I is given horizontally to the sphere at a height h=4R5 above the centre line. m and R are the mass and radius of the sphere respectively.


Find angular velocity (ω) of the sphere and linear velocity (v) of the centre of mass of the sphere after the impulse.

A
v=Im
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B
ω=2ImR
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C
v=I2m
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D
ω=ImR
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Solution

The correct options are
A v=Im
B ω=2ImR
Assume that just after the impulse, linear velocity of the sphere is v and angular velocity is ω.


As we know,
Impulse = change in linear momentum
I=ΔP=PfPi
I=mv0
(Pi=0, because initially sphere is at rest.)
v=Im is the linear velocity of the COM.

and angular impulse = change in angular momentum
Linear impulse ×h=ΔL
I×4R5=I0ω0 {ΔL=LfLi}
Li=0, because initially sphere is at rest.
and, I0=25mR2

I×4R5=25mR2.ω
ω=2ImR is the angular velocity of the sphere after impulse.

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