A 3.6 m diameter merry-go-round is rotating freely with an angular velocity of 0.78 rad/s. Its total moment of inertia is 1700 kg m^2. Four people standing on the ground, each of mass 62 kg, suddenly step onto the edge of the merry-go-round.
What is the angular velocity of the merry-go-round now?
Express your answer using two significant figures.
What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?
When they step on, the I increases by 4*62*1.8^2 (mr^2)
so, wnew=wold* Iold/Inew
In the second case, Inew would be decreased, ..
To determine the angular velocity of the merry-go-round after four people step onto the edge, we need to consider the conservation of angular momentum.
The initial angular momentum of the system is given by:
H_initial = I_initial * ω_initial
where:
H_initial = initial angular momentum
I_initial = moment of inertia of the merry-go-round
ω_initial = initial angular velocity of the merry-go-round
Given that the diameter of the merry-go-round is 3.6 m, the radius is half of that, which is 1.8 m. Hence, the moment of inertia is:
I_merry_go_round = (1/2) * m * r^2
Substituting the given values:
I_merry_go_round = (1/2) * (1700 kg m^2) * (1.8 m)^2
The mass of each person is 62 kg, and we have 4 people. The moment of inertia contributed by the people is:
I_people = m * r^2
Substituting the given values:
I_people = (4 * 62 kg) * (1.8 m)^2
The total moment of inertia of the system is the sum of the moment of inertia of the merry-go-round and the people:
I_total = I_merry_go_round + I_people
Now, we can calculate the initial angular momentum by multiplying I_total with ω_initial:
H_initial = I_total * ω_initial
To find the angular velocity of the merry-go-round after the people stepped onto the edge, we need to consider the conservation of angular momentum. Since no external torque is acting on the system, the total angular momentum remains constant.
When the people step onto the merry-go-round, their moment of inertia increases, which means the angular velocity of the merry-go-round must decrease to keep the angular momentum constant.
The final angular momentum of the system can be calculated as:
H_final = I_total * ω_final
Since the angular momentum is conserved, H_initial = H_final. Therefore, we can solve for ω_final.
ω_final = H_initial / I_total
Substituting the values we calculated earlier, you can find the final angular velocity of the merry-go-round by dividing the initial angular momentum by the total moment of inertia. Round the result to two significant figures.
If the people were initially on the merry-go-round and then jumped off, the moment of inertia contributed by the people would become zero. As a result, the total moment of inertia of the system would decrease, causing the final angular velocity to increase to conserve the angular momentum.