Solid Sphere Impulse: Find Velocity, Angular Velocity, & Kinetic Energy

[ZxcvbnM2000]

Homework Statement

A uniform solid sphere has radius R and mass M. It is initially at rest but is
free to move, floating in space with nothing touching it. It suddenly receives
an impulse J at a tangent to its surface.

As a function of R, M and J, find formulae for:

(a) the linear velocity of the sphere,
(b) its angular velocity around its centre of mass
(c) its total kinetic energy after the impulse.

Homework Equations

L=Iω,J=MV,V=ωR

The Attempt at a Solution

a) All particles on the sphere have the same angular velocity and different linear velocities depending on their distance from the centre.

J=ΔP=M(V-u)=MV so V=J/M Whose velocity is this ? ( V=J/M) .Is it the C.o.M ?

b) L=Iω <=> ω=L/I = 5L/(2MR²) But how can i move from here ?

c) well if i knew how to solve b) then c) is an easy one !

 

[Doc Al]

a) All particles on the sphere have the same angular velocity and different linear velocities depending on their distance from the centre.

J=ΔP=M(V-u)=MV so V=J/M Whose velocity is this ? ( V=J/M) .Is it the C.o.M ?

Good. That's the velocity of the COM.

b) L=Iω <=> ω=L/I = 5L/(2MR²) But how can i move from here ?

What's the angular impulse?

 

[ZxcvbnM2000]

ΔL=IΔω but since it was stationary at first then ΔL=Iω

It seems to be simple but i still can't understand how to relate these two :S

 

[ZxcvbnM2000]

Actually no :S

 

[Doc Al]

ΔL=IΔω but since it was stationary at first then ΔL=Iω

Nothing wrong with that, but what is ΔL in terms of J?

 

[ZxcvbnM2000]

Hmm dL=Iω=J*R=MVR but i realized that V=ωR is not valid , why is that ?

 

[Doc Al]

Hmm dL=Iω=J*R=MVR but i realized that V=ωR is not valid , why is that ?

The angular impulse is J*R. Now you can solve for ω.

As to whether V = ωR is valid, that depends on what you mean by V. (In any case, you don't need it here.)