A 3.6 diameter merry-go-round is rotating freely with an angular velocity of 0.73 . Its total moment of inertia is 1600 . Four people standing on the ground, each of mass 60 , suddenly step onto the edge of the merry-go-round.What is the angular velocity of the merry-go-round now?What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?
To determine the angular velocity of the merry-go-round after the people have stepped onto the edge, we can use the principle of conservation of angular momentum.
The initial angular momentum of the merry-go-round can be calculated using the formula:
Angular momentum = Moment of inertia * Angular velocity
Initially, the angular momentum is given by:
L_initial = 1600 * 0.73
Now, when the four people step onto the edge, they will induce a rotational effect due to their change in moment of inertia. We need to consider the moment of inertia of the people as they step onto the merry-go-round.
Since the people are standing on the edge, their moment of inertia can be approximated as:
Moment of inertia of people = Mass of each person * Radius of merry-go-round^2
= 60 * (1.8)^2
= 194.4
Now, let's calculate the total moment of inertia of the system after the people have stepped on:
Total moment of inertia = Moment of inertia of merry-go-round + Moment of inertia of people
= 1600 + 194.4
= 1794.4
Since angular momentum is conserved, the final angular momentum is equal to the initial angular momentum:
L_final = L_initial
Now we can solve for the final angular velocity:
L_final = Total moment of inertia * Final angular velocity
L_initial = L_final
1600 * 0.73 = 1794.4 * Final angular velocity
Final angular velocity = (1600 * 0.73) / 1794.4
Final angular velocity ≈ 0.653 rad/s
So, the angular velocity of the merry-go-round after the people have stepped onto the edge is approximately 0.653 rad/s.
Now let's consider the scenario where the people are initially on the merry-go-round and then jump off in a radial direction relative to the merry-go-round.
When the people jump off radially, their moment of inertia is removed from the system. Thus, the total moment of inertia will only include the moment of inertia of the merry-go-round.
Using the same formula as before, the total moment of inertia is:
Total moment of inertia = 1600
Since angular momentum is conserved, the final angular momentum is equal to the initial angular momentum:
L_final = L_initial
L_initial = 1600 * 0.73
Now we can solve for the final angular velocity:
L_final = Total moment of inertia * Final angular velocity
L_initial = L_final
1600 * 0.73 = 1600 * Final angular velocity
Final angular velocity = (1600 * 0.73) / 1600
Final angular velocity ≈ 0.73 rad/s
So, the angular velocity of the merry-go-round after the people have jumped off radially is approximately 0.73 rad/s.