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:
