What is the final angular velocity of the system after the collision?

• hidemi
In summary, the conversation discusses the concept of conservation of energy in a scenario where two bodies collide and stick together. It is mentioned that the glue holding them together absorbs the energy that would have been associated with a rebound. It is also noted that if the question were rephrased to include a perfectly elastic bounce, then conservation of energy could be established. The conversation concludes with a thank you for the clarification.

hidemi

Homework Statement
A particle of mass m = 0.10 kg and speed v0 = 5.0 m/s collides and sticks to the end of a uniform solid cylinder of mass M = 1.0 kg and radius R = 20 cm. If the cylinder is initially at rest and is pivoted about a frictionless axle through its center, what is the final angular velocity (in rad/s) of the system after the collision?
(A) 8.1
(B) 2.0
(C) 6.1
(D) 4.2
(E) 10
Relevant Equations
Li = Lf = Iω
I calculated as attached and got it right. However, I just wonder why we can't use conservation of energy as the question has already specified 'frictionless', meaning no energy loss and energy distributed to the rotation only.

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hidemi said:
why we can't use conservation of energy
You should never assume conservation of mechanical energy without good cause.
The scenario in this question is a coalescence: the two bodies stick together after colliding. Imagine what would happen if they did not do so, e.g. if it were a rubber ball hitting a protrusion from a concrete drum. Clearly the ball would bounce off. The glue that holds them together in the actual question has therefore absorbed the energy that would have been associated with that rebound.

haruspex said:
You should never assume conservation of mechanical energy without good cause.
The scenario in this question is a coalescence: the two bodies stick together after colliding. Imagine what would happen if they did not do so, e.g. if it were a rubber ball hitting a protrusion from a concrete drum. Clearly the ball would bounce off. The glue that holds them together in the actual question has therefore absorbed the energy that would have been associated with that rebound.
Oh I see.
If the question rephrases a bit, the ball hits the cylinder and bounces off as well as the frictionless remains, then the conservation of energy can be established. Let me know if I'm right.

hidemi said:
If the question rephrases a bit, the ball hits the cylinder and bounces off as well as the frictionless remains, then the conservation of energy can be established.
If it is a perfectly elastic bounce, yes.

hidemi
haruspex said:
If it is a perfectly elastic bounce, yes.
Thank you so much.

1. What factors determine the final angular velocity of a system after a collision?

The final angular velocity of a system after a collision is determined by the initial angular velocities of the objects involved, the masses and moments of inertia of the objects, and the type of collision (elastic or inelastic).

2. How is the final angular velocity calculated after a collision?

The final angular velocity can be calculated using the conservation of angular momentum equation: Iω = I₁ω₁ + I₂ω₂, where I is the moment of inertia and ω is the angular velocity. This equation takes into account the initial angular velocities and moments of inertia of the objects before the collision.

3. What is the difference between an elastic and an inelastic collision in terms of final angular velocity?

In an elastic collision, both the linear and angular momenta are conserved, resulting in a final angular velocity that is equal to the initial angular velocity. In an inelastic collision, some of the kinetic energy is lost and the final angular velocity will be less than the initial angular velocity.

4. Can the final angular velocity of a system after a collision be greater than the initial angular velocity?

No, the final angular velocity of a system after a collision cannot be greater than the initial angular velocity. This is due to the conservation of angular momentum, which states that the total angular momentum of a system remains constant in the absence of external torques.

5. How does friction affect the final angular velocity of a system after a collision?

Friction can cause a decrease in the final angular velocity of a system after a collision. This is because friction produces a torque that opposes the motion of the objects, resulting in a loss of kinetic energy and a decrease in angular velocity.