Question

of mass m and radius R is freely rotating at angular velocityw,spinning clockwise The moment of inertia of a disk is ImR2.) While it is rotating, someone when seen from above. ( ps a thin ring of mass m and also of radius R on top of it. When the ring lands, it is initially ot otating, but friction quickly causes it to rotate along with the disk, but the rotation of the disk slows down. a) What principle of physics erplains uhy the rotation slous doun once the ring is dropped on top of the disk? (10 points) b) What is the final angular velocity of the disk and ring? (20 points) Now, suppose that a bird of mass 3m flies in and lands on the disk. It lands a distance R/2 south of the center of the disk, and is flying eastward when it lands. The bird is flying at exactly the right speed to stop the platform from rotating once it lands. c) Find the initial speed of the bird ts in terms of m, R, and w. (20 points)

Answer 1

a)

This can be explained on the basis of conservation of angular momentum. The product of moment of inertia and angular velocity must remain same before and after dropping the ring. As the ring is dropped, the total moment of inertia becomes equal to the sum of moment of inertia of the disk and the ring. hence total moment of inertia increase. To keep the product of moment of inertia and angular velocity constant, the angular velocity decrease.

b)

I_disk = moment of inertia of disk = (0.5) m R²

I_ring = moment of inertia of ring = m R²

Before ring is dropped :

I_i = initial total moment of inertia = I_disk = (0.5) m R²

w_i = initial angular velocity = w

After the ring is dropped :

I_f = final total moment of inertia = I_disk + I_ring = (0.5) m R² + mR² = (1.5) m R²

w_f = final angular velocity = ?

Using conservation of angular momentum

I_i w_i = I_f w_f

(0.5) m R² w = (1.5) m R² w_f

(0.5) w = (1.5) w_f

w_f = 0.33 w

c)

Angular momentum of bird = Angular momentum of disk + ring

m v_b (R/2) = (1.5) m R² w_f

m v_b (R/2) = (1.5) m R² (0.33) w

v_b = 2 (0.33) (1.5) R w

v_b = (0.99) R w