Question

Friction between the skaters and the ice is negligible.

Ann has a mass of M, a moment of inertia of 3I_oand has a velocity of v_a = -2voi + 7voj. She is spinning with an angular velocity 10ω_o up.

Bob has a mass of 2M, a moment of inertia of 5I_o and has a velocity of vb = 4voi + voj. He is spinning with an angular velocity 2ω_o down.

The two ice dancers join together (collide) and hold onto each other taking time Δt for this to occur. Assume that the moment of inertia of the pair joined together is equal to the sum of their individual moments of inertia.

Answer the following questions in terms of M, I_o, v_o, ω_o,and Δt.

(A) What quantities are conserved during the collision (when the skaters join together)? List all conserved quantities that are useful to calculate the velocity and angular velocity after the collision.  

(B) What is the velocity of the pair after joining together? Write in component vector form.

(C) What is the y component of the average force that Ann exerts on Bob during the collision?

(D) What is the final angular velocity of the pair after joining together? (Give magnitude of the angular velocity and whether it is up or down.)

(E) What is the total rotational kinetic energy after the collision?

Answer 1

(A)

quantities that are conserved during collision are:

angular momentum,linear momentum and the energy

(B)

let v be the velocity of the pair after joining together

so , applying conservation of linear momentum

M *Va + 2M *Vb = (M+2M)*V = 3M*V

so , Va +2Vb = 3V

-2voi + 7voj +2(4voi + voj) = 3V

6voi +9voj = 3V

so V = (6voi +9voj)/3 = 2voi +3voj

So,the velocity of the pair after joining together =2voi +3voj

(C)

force = mass(final velocity - initial velocity)/time taken

So,average force that Ann exerts on Bob during the collision = M(2voi +3voj-(-2voi + 7voj))/Δt = M(4voi -4voj)/Δt

So,the y component of the average force that Ann exerts on Bob during the collision = -4Mvo/Δt

(D)

let w be the angular velocity of the pair after joining together

so , applying conservation of angular momentum

3Io*10ωo + 5Io*-2ωo = (3Io +5Io)w

30 Ioωo - 10 Ioωo = 8 Iow

so, 20ωo = 8w    and w = 20ωo/8 =2.5ωo

So,the final angular velocity of the pair after joining together = 2.5ωo up (since w comes out to be positive)

(E)

total rotational energy = 1/2(total moment of inertia)(angular velocity of the pair after joining together)

= 1/2(3Io +5Io)2.5ωo

= 10Ioωo