Physics angular speed problem help?

Question

A wad of sticky clay with mass m and velocity Vi is fired at a solid cylinder of mass M and radius R. The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle at a distance d < R from the center.

find the angular speed of the system after the clay hits the cylinder

is mechanical energy conserved?

Answer 1

The wad of clay will strike the cylinder at a point that subtends an angle from the surface to the center of th w/r/t the horizontal. The sin(th)=d/R

Since d sin(th)*Vi, or Vi*d/R
At the instant of contact, this can be modeled as angular momentum of the clay as (notet that w=v/R)
C*R*Vi*d/R^2 (note that this is general for 0<=d<=R)
Where C is the mass of the clay
simplify
C*Vi*d/R

after the collision, the angular momentum is conserved
the moment of inertia of the combined system is
.5*M*R^2+C*R
so
R*(M*R/2+C)*w=C*Vi*d/R
w=C*Vi*d/(R^2*(M*R/2+C))

Mechanical energy is not conserved since it is an inelastic collision.

j