a. The angular velocity of the system after the collision is 0 rad/s.
b. The1.15 J of kinetic energy is lost in the collision.
(a) Angular Velocity after Collision
The angular velocity of the system after the collision can be calculated using the principle of conservation of angular momentum. The initial angular momentum of the system is zero, since the cylinder is initially at rest. The final angular momentum of the system is equal to the product of the moment of inertia of the system and its angular velocity.
The moment of inertia of the cylinder about its center can be calculated using the formula:
I = (1/2)MR²
Where:
I is the moment of inertia (kg m²)
M is the mass of the cylinder (kg)
R is the radius of the cylinder (m)
Plugging in the given values, we get:
I = (1/2)(0.8 kg)(0.12 m)² = 0.00576 kg m²
The final angular momentum of the system can be calculated using the formula:
L = Iω
Where:
L is the angular momentum (kg m²/s)
I is the moment of inertia (kg m²)
ω is the angular velocity (rad/s)
Setting the initial angular momentum equal to the final angular momentum, we get:
0 = Iω
Solving for ω, we get:
ω = 0 rad/s
Therefore, the angular velocity of the system after the collision is 0 rad/s.
(b) Kinetic Energy Lost in Collision
The kinetic energy lost in the collision is equal to the difference between the initial kinetic energy of the particle and the final kinetic energy of the system. The initial kinetic energy of the particle can be calculated using the formula:
KE = (1/2)mv²
Where:
KE is the kinetic energy (J)
m is the mass of the particle (kg)
v is the velocity of the particle (m/s)
Plugging in the given values, we get:
KE = (1/2)(0.023 kg)(10 m/s)² = 1.15 J
The final kinetic energy of the system is zero, since the system is at rest after the collision. Therefore, the kinetic energy lost in the collision is equal to the initial kinetic energy of the particle:
KE_lost = 1.15 J
Therefore, 1.15 J of kinetic energy is lost in the collision.