A wad of sticky clay of massmand velocity vi is fired at a solid cylinder of massM and radius
R. The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through the
center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance d,
less than R, from the center.
(a) What is the angular momentum of the system before the clay strikes the cylinder (give both
magnitude and direction)?
(b) Find the angular speed of the system immediately after the clay strikes and sticks to the surface
of the cylinder.
(c) What is the difference in the mechanical energy before and after the collision?
To answer these questions, we need to consider the principles of conservation of angular momentum and conservation of mechanical energy.
(a) The angular momentum of an object can be calculated by multiplying its moment of inertia by its angular velocity.
The angular momentum of the clay initially is given by:
L1 = m*v1*d, where m is the mass of the clay and v1 is its velocity.
Since the cylinder is initially at rest, its initial angular momentum is zero.
Therefore, the total initial angular momentum of the system is:
L_initial = L1 + L_cylinder = m*v1*d, where L_cylinder is the initial angular momentum of the cylinder.
The direction of the angular momentum vector depends on the given orientation of the system. If we assume a counterclockwise rotation to be positive, the direction of L_initial will be perpendicular to the plane of motion.
(b) When the clay sticks to the cylinder, the angular momentum of the system remains conserved. Therefore, the total angular momentum after the collision is equal to the initial angular momentum.
Let the angular speed of the system after the collision be ω. The moment of inertia of the cylinder is given by I = 0.5*M*R^2.
The final angular momentum of the system is given by:
L_final = L_clay + L_cylinder = (m + M) * ω * R
Setting L_initial = L_final, we can solve for ω:
m*v1*d = (m + M) * ω * R
ω = (m*v1*d) / ((m + M) * R)
(c) To find the difference in mechanical energy before and after the collision, we need to consider the initial mechanical energy and the final mechanical energy.
The initial mechanical energy is the sum of the kinetic energy of the clay and the potential energy due to its position:
E_initial = 0.5*m*v1^2 + m*g*d, where g is the acceleration due to gravity.
The final mechanical energy is the sum of the kinetic energy of the system after the collision and the potential energy due to its new position:
E_final = 0.5*(m + M)*R^2*ω^2 + (m + M)*g*R
The difference in mechanical energy is given by:
ΔE = E_final - E_initial
Substituting the value of ω from part (b), we can evaluate the difference in mechanical energy.