A wad of sticky clay with mass ma and velocity vi is fired at a solid cylinder o

Question

A wad of sticky clay with mass ma 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 and at a distance d < R from the center. (a) Find the angular speed of the system just after the clay strikes and sticks to the surface of the cylinder. (b)Is the mechanical energy of the clay-cylinder system constant in this process? (c) Is the momentum of the clay-cylinder system constant in this process?

Explanation / Answer

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<R, it has a tangential component of velocity of
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

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