A 33.0 kg child named Lindsey runs as fast as she can and jumps onto the outer edge of a merry-go-round. The merry-go-round is initially at rest and has a mass of 78.0 kg and a radius of 2.20 m. Lindsey's linear velocity was 9.00 m/s at the moment she jumped onto the merry-go-round.
1. What is the initial angular momentum of the system (in kg m^2/s)? [Hint: Although she started with linear velocity, consider the moment just before she landed on the merry-go-round.]
2. What is the angular velocity of the merry-go-round after she jumps on it (in rad/sec)?
1. To find the initial angular momentum of the system, we first need to find the initial angular momentum of Lindsey and the initial angular momentum of the merry-go-round.
The initial angular momentum of Lindsey can be calculated using the formula:
L_initial = m * v * r
Where:
m = mass of Lindsey = 33.0 kg
v = linear velocity of Lindsey = 9.00 m/s
r = radius of the merry-go-round = 2.20 m
Plugging in the values, we get:
L_initial = 33.0 kg * 9.00 m/s * 2.20 m
L_initial = 653.4 kg m^2/s
The initial angular momentum of the system is the sum of Lindsey's initial angular momentum and the initial angular momentum of the merry-go-round, which is initially at rest. Since the merry-go-round is initially at rest, its initial angular momentum is 0.
Therefore, the initial angular momentum of the system is 653.4 kg m^2/s.
2. The angular momentum of the system is conserved, so we can set the initial angular momentum equal to the final angular momentum:
L_initial = L_final
The final angular momentum is the sum of Lindsey's final angular momentum and the merry-go-round's final angular momentum. Since they are moving together after Lindsey jumps on the merry-go-round, their angular momenta will be combined.
The final angular momentum of the system can be calculated using the formula:
L_final = (m_Lindsey + m_merry_go_round) * v_final * r
Where:
m_Lindsey = mass of Lindsey = 33.0 kg
m_merry_go_round = mass of the merry-go-round = 78.0 kg
v_final = final angular velocity of the system
r = radius of the merry-go-round = 2.20 m
Plugging in the values and setting the initial and final angular momenta equal, we get:
653.4 kg m^2/s = (33.0 kg + 78.0 kg) * v_final * 2.20 m
Solving for v_final, we get:
v_final = 653.4 kg m^2/s / (33.0 kg + 78.0 kg) / 2.20 m
v_final = 2.02 rad/s
Therefore, the angular velocity of the merry-go-round after Lindsey jumps on it is 2.02 rad/s.