A 20-kg child running at 1.4 m/s jumps onto a playground merry-go-round that has inertia 180 kg and radius 1.6 m. She is moving tangent to the platform when she jumps, and she lands right on the edge. Ignore any friction in the axle about which the platform rotates.
To understand what happens when the child jumps onto the merry-go-round, we need to analyze the conservation of angular momentum.
Angular momentum is a property of rotating objects and is defined as the product of the moment of inertia and angular velocity. The law of conservation of angular momentum states that the total angular momentum of a system remains constant unless acted upon by an external torque.
In this case, the child transfers some of her angular momentum to the merry-go-round. Before the jump, the child's angular momentum is zero since she is not rotating. However, the merry-go-round is rotating, so it has some angular momentum.
After the child lands on the merry-go-round, both the child and the merry-go-round will start to rotate with a final common angular velocity.
To find the final angular velocity, we can use the principle of conservation of angular momentum. The total angular momentum before the jump should be equal to the total angular momentum after the jump.
The child's initial angular momentum is given by:
L_child = I_child * ω_child
where
L_child is the angular momentum of the child,
I_child is the moment of inertia of the child, and
ω_child is the angular velocity of the child.
Since the child is not rotating initially, L_child = 0.
The merry-go-round's initial angular momentum is given by:
L_merry-go-round = I_merry-go-round * ω_merry-go-round
where
L_merry-go-round is the angular momentum of the merry-go-round,
I_merry-go-round is the moment of inertia of the merry-go-round, and
ω_merry-go-round is the initial angular velocity of the merry-go-round.
We know the values of I_merry-go-round (180 kg) and ω_merry-go-round (unknown).
After the child lands on the merry-go-round, the final angular momentum of the system is given by:
L_final = (I_child + I_merry-go-round) * ω_final
where
L_final is the final angular momentum of the system,
I_child is the moment of inertia of the child,
I_merry-go-round is the moment of inertia of the merry-go-round, and
ω_final is the final angular velocity of the system.
Since angular momentum is conserved, we can set L_final equal to L_initial:
(I_child + I_merry-go-round) * ω_final = L_merry-go-round
Now we can solve for ω_final:
ω_final = L_merry-go-round / (I_child + I_merry-go-round)
Substituting the given values, calculate ω_final and then you will find the final angular velocity of the system.