Question

A hockey stick of mass m_s and length L is at rest on the ice (which is assumed to be frictionless). A puck with mass m_p hits the stick a distance D from the middle of the stick. Before the collision, the puck was moving with speed v₀  in a direction perpendicular to the stick, as indicated in the figure. The collision is completely inelastic, and the puck remains attached to the stick after the collision.

Find the speed v_f of the center of mass of the stick+puckcombination after the collision. Express v_f in terms of the following quantities: v₀, m_p, m_s, and L.v_f =
After the collision, the stick and puck will rotate about their combined center of mass. How far is this center of mass from the point at which the puck struck? In the figure, this distance is (D-b) .  (D-b)  =
What is the angular momentum L_cm of the system before the collision, with respect to the center of mass of the final system? Express L_cm in terms of the given variables.L_cm =
What is the angular velocity w of the stick+puckcombinationafter the collision? Assume that the stick is uniform and has a moment of inertia i₀ about its center. Your answer for w should not contain the variable b.

w =

Which of the following statements are TRUE?

1) Kinetic energy is conserved.
2) Linear momentum is conserved.
3) Angular momentum of the stick+puck is conserved about the center of mass of the combined system.
4) Angular momentum of the stick+puck is conserved about the(stationary) point where the collision occurs.

Answer 1

Concepts and reason

The concepts used to solve this problem are the law of conservation of momentum, the center of mass, angular momentum, a moment of inertia, and conservation of angular momentum.

Initially from the shown figure apply the law of conservation of momentum by substituting mass and velocity before and after the collision, and then calculate the speed of the center of mass of the stick and puck combination after the collision.

Next calculate the distance from the center of mass of the point at which the puck struck, before the collision the stick is concentrated at the distance D. The distance of the center of mass from the point in which the puck hits the stick is D-b.

Next, calculate the angular momentum of the puck with respect to the center of mass of the system. By substituting mass and velocity and distance of the center of mass from the point in which the puck hits the stick is D-b.

Next, calculate the moment of inertia of the stick and puck about the center of mass, then calculate the initial angular and final angular momentum, finally use the conservation of angular momentum to find the angular velocity of the stick and puck combination after the collision.

Finally, find the true or false for the given statement and give correct information or hint about the wrong statement.

Fundamentals

According to the law of conservation of angular momentum of the stick and puck combination is,

m_pv₀ = (m_p+m_s)v_f

Here, m_p is the mass of the puck, m_s is the mass of the stick, v₀  is the initial velocity of the puck, and v_f is the final speed of the stick and puck combination.

The center of mass of the stick is,

Here, b is the center of mass and D is the distance from the middle of the stick.

The expression of the moment of inertia of the stick about the center of mass is,

Here, I_s  is the moment of inertia of the stick and I₀ is the initial moment of inertia.

The expression of the moment of inertia of puck about of the center of mass is,

Here, I_p is the moment of inertia of the puck and D-b is the distance of the shift in the center of mass with respect to the puck.

The initial angular momentum of the puck with respect to the center of mass of the system is,

Here, Lcm is the initial angular momentum of the puck with respect to the center of mass.

The total moment of inertia of the stick and puck is,

Here, I is the total moment of inertia.

The final angular momentum is,

Here, L_f is the final angular momentum of the system and w is the angular velocity of the tick and puck combination after collision.

The initial angular momentum is,

Here, L_i is the initial angular momentum of the system

According to the conservation of angular momentum is,

L_f=L_i

(1)

According to the law of conservation of angular momentum of the stick and puck combination is,

Rewrite the expression in terms of v_f.

(2)

The center of mass of the stick and puck combination is,

The distance of the center of mass from the point in which the puck hits the stick is D-b.

Substitute for b.

(3)

The initial angular momentum of the puck with respect to the center of mass of the system is,

Substitute   Dm_s/m_s+m_p for D-b to find L_cm.

(4)

The expression of the moment of inertia of the stick about the center of mass is,

Substitute for b to find I_s.

The expression of the moment of inertia of puck about of the center of mass is,

Substitute for D-b to find I_p.

The total moment of inertia of the stick and puck is,

Substitute for and for to find I.

                                                  …… (1)

The initial angular momentum is,

Substitute for D-b to find L_i.

                                                                                     …… (2)

The final angular momentum is,

                                                                                                         …… (3)

According to the conservation of angular momentum is,

Use equations (2) and (3) in the above expression.

Substitute equation (1) in the above expression to find w.

(5)

Kinetic energy is conserved.

The above statement is false because here the kinetic energy is transferred from one place to another.

Linear momentum is conserved.

Angular momentum of the stick and puck is conserved about the center of mass of the combined system.

Angular momentum of the stick and puck is conserved about the stationary point where the collision occurs.

The above statements are true because here there are no external torque acts on the system. Since the angular momentum is conserved at all points. However, we consider the center of mass of the whole system as the linear motion of the center of mass of the new system, and the rotation about the center of mass of the new system.