Sarah, with mass m and velocity v, runs toward a playground merry-go-round and jumps on at its edge. Sarah and the merry-go-round (mass M, radius R, and I = cMR²) then spin together with a constant angular velocity w_f. If Sarah's initial velocity is tangent to the circular merry-go-round, what is w_f?

Apply angular momentum conservation. Sarah's angular momentum before the collision equals the angular momentum of the system after the collision.

Sarah's linear momentum p can be transformed to an angular momentum L much like a force is turned into a torque.

The magnitude of the angular momentum is L = r p sin(q), where q is the angle between r and p.

In this example Sarah's initial angular momentum is L_i = Rmv.

Applying angular momentum conservation:

Total angular momentum before = total angular momentum after

Rmv + 0 = I_total w_f

I_total = The moment of inertia of the merry-go-round plus Sarah's moment of inertia.

I_total = cMR² + mR²

Therefore:

wf

=

R m v Itotal

=

m v cMR + mR

Numerical example

Let's say m = 25 kg; v=4 m/s; M = 50 kg; R = 2 m; c = ½

wf

=

100 50 + 50

=

1 rad/s