A girl (mass M) standing on the edge of a frictionless merry-go-round (radius R, rotational inertia I) that is not moving. She throws a rock (mass m) in a horizontal direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is v. Calculate (a) the angular speed of the merry-go- round and (b) the linear speed of the girl after the rock is thrown.

a)

b)

Explanation:

a)

Since there are no external torques acting on the system, the total angular momentum must remain constant.

At the beginning, the merry-go-round and the girl are at rest, so the initial angular momentum is zero:

Later, after the girl throws the rock, the angular momentum will be:

where:

is the moment of inertia of the merry-go-round

is the moment of inertia of the girl, where

M is the mass of the girl

R is the distance of the girl from the axis of rotation

is the angular speed of the merry-go-round and the girl

is the angular momentum of the rock, where

m is the mass of the rock

v is its velocity

Since the total angular momentum is conserved,

So we find:

And the negative sign indicates that the disk rotates in the direction opposite to the motion of the rock.

b)

The linear speed of a body in rotational motion is given by

where

is the angular speed

r is the distance of the body from the axis of rotation

In this problem, for the girl, we have:

is the angular speed

is the distance of the girl from the axis of rotation

Therefore, her linear speed is:

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