Angular momentum and oscillation of disk

[jonnyboy]

[SOLVED] Angular momentum and oscillation of disk

A large solid disk of mass M and radius R is mounted on a fixed axis through its center using ideal bearings. A small projectile of mass m_1 traveling with velocity v_1 collides tangentially to the slight extension and sticks to the larger disk causing it to rotate about its center and oscillate

a. Determine the maximum angle the disk will rotate (theta_max)
b. Determine the frequency of oscillation ( assume theta_max is small enough so that the angle approximation is valid)

If someone would know what steps I need to take or where to start would be a great help.

 

[Astronuc]

When the mass m₁ with some momentum and kinetic energy impacts the disk of mass M, the disk will rotate to some angle at which the smaller mass is at some height (change in potential energy) where the rotation stops. Then the disk will rotate in the opposite direction, and oscillate.

So one has to determine by that appropriate conservation equation(s), the height at which m₁ stops, and related that to the maximum angle of rotation.

Then one must determine the appropriate equation of motion from which one will obtain an expression for frequency.

 

[ogb p]

The first step in solving this problem would be to determine the initial angular momentum of the system before the collision. This can be calculated using the formula L = Iω, where I is the moment of inertia of the disk and ω is its initial angular velocity.

Next, we can use the conservation of angular momentum to find the final angular velocity of the disk after the collision. Since the projectile sticks to the disk, the total angular momentum of the system will remain constant.

Once we have the final angular velocity, we can use the simple harmonic motion equation ω = √(k/m) to find the frequency of oscillation, where k is the spring constant of the oscillating disk and m is its mass.

To find the maximum angle of rotation, we can use the small angle approximation θ_max = Aω^2/g, where A is the amplitude of oscillation and g is the acceleration due to gravity.

Overall, the key steps in solving this problem involve using conservation of angular momentum, simple harmonic motion equations, and small angle approximations. It is also important to carefully consider the initial conditions and assumptions made in the problem.