Not my strongest topic, and I'm stuck on part (b).
So I know that
L and
T are conserved, and for rotations we have
L=Iω and
T=21Iω2.
The answer says that it should be
ω=I+ml2I+ma2ω0 but shouldn't it be capital
M instead? Since we are considering the tube rather than the bead?
Also, here are my thoughts; we let
I be the moment of inertia about the axis of rotation, then the moment of inertia through a parallel axis which goes through the bead is initially
I0=I+Ma2.
Then we do the same thing for when the bead is at a distance
l from the centre of the tube and we get that
I(l)=I+Ml2.
Therefore for angular momentum we got
L0=(I+Ma2)ω0 and
L(l)=(I+Ml2)ω.
Since angular momentum is conserved we get that
L0=L(l) which implies that
ω=I+Ml2I+Ma2ω0.