Suppose a 69 kg person stands at the edge of a 7.8 m diameter merry-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1600 kg*m^2. The turntable is at rest initially, but when the person begins running at a speed of 3.7 m/s (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable. Please help with step-by-step explanation.
The total angular momentum remains zero because of the frictionless bearings. The person and the turntable rotate about the axis of rotation in opposite directions.
Let w be the angular velocity of the turntable after "the person" begins running. His (or her) angular momentum about the axis is
M V R = M (3.7 - R w) *R
Note that we have to use the speed V with respect to land, not the turntable. That is why R w has to be subtracted from the velocity with respect to the turntable.
Solve this equation for the angular velocity w:
I w = M (3.7 - R w) *R
I w (1 + MR^2)= 3.7 M R
good
To solve this problem, we can use the principle of conservation of angular momentum.
The initial angular momentum of the system is given by L_i = (moment of inertia of turntable) * (angular velocity of turntable)
The final angular momentum of the system is given by L_f = (moment of inertia of turntable + moment of inertia of person) * (angular velocity of turntable)
Since there is no external torque acting on the system, the angular momentum is conserved. Therefore, we can equate the initial and final angular momenta:
L_i = L_f
Using the given values:
L_i = (1600 kg*m^2) * (angular velocity of turntable)
L_f = (1600 kg*m^2 + 69 kg * (7.8 m/2)^2) * (angular velocity of turntable)
Substituting these values and rearranging the equation:
(1600 kg*m^2) * (angular velocity of turntable) = (1600 kg*m^2 + 69 kg * (7.8 m/2)^2) * (angular velocity of turntable)
Simplifying the equation:
1600 kg*m^2 * (angular velocity of turntable) = (1600 kg*m^2 + 69 kg * (3.9 m)^2) * (angular velocity of turntable)
1600 kg*m^2 * (angular velocity of turntable) - (1600 kg*m^2 + 69 kg * (3.9 m)^2) * (angular velocity of turntable) = 0
Rearranging and factoring out the angular velocity of turntable:
(1600 kg*m^2 - (1600 kg*m^2 + 69 kg * (3.9 m)^2)) * (angular velocity of turntable) = 0
(-69 kg * (3.9 m)^2) * (angular velocity of turntable) = 0
Dividing both sides by (-69 kg * (3.9 m)^2):
(angular velocity of turntable) = 0
Therefore, the angular velocity of the turntable is 0.
To calculate the angular velocity of the turntable, we can use the principle of conservation of angular momentum. The initial angular momentum of the system is zero since the turntable is at rest.
1. Calculate the initial moment of inertia of the system:
The moment of inertia is given as 1600 kg*m^2.
2. Calculate the initial angular momentum of the person:
The initial angular momentum of the person can be calculated as the product of the person's moment of inertia and their initial angular velocity, which is zero since they are at rest:
Angular momentum(person)_initial = moment of inertia(person) * angular velocity(person)_initial
However, since the person is initially at rest, the initial angular velocity of the person is zero, so the initial angular momentum of the person is also zero.
3. Calculate the final moment of inertia of the system:
The final moment of inertia of the system is the sum of the moment of inertia of the person and the turntable:
moment of inertia(system)_final = moment of inertia(person) + moment of inertia(turntable)
moment of inertia(system)_final = 0 kg*m^2 + 1600 kg*m^2 = 1600 kg*m^2
4. Calculate the final angular momentum of the person:
The final angular momentum of the person can be calculated as the product of their moment of inertia and their final angular velocity:
Angular momentum(person)_final = moment of inertia(person) * angular velocity(person)_final
Since the person is running around the edge of the turntable, their final angular velocity is the same as the angular velocity of the turntable, which we want to find.
5. Apply the principle of conservation of angular momentum:
According to the principle of conservation of angular momentum, the total angular momentum of the system remains constant before and after the person starts running:
Angular momentum(system)_initial = Angular momentum(system)_final
Since the initial angular momentum of the system is zero, this equation becomes:
Angular momentum(person)_initial = Angular momentum(person)_final
Since the initial angular momentum of the person is zero, we can simplify the equation to:
0 = moment of inertia(person) * angular velocity(person)_final
Rearranging the equation gives:
angular velocity(person)_final = 0
6. Solve for the final angular velocity of the turntable:
Since the person's final angular velocity is zero, the turntable's final angular velocity will be the same as the person's final angular velocity:
Angular velocity(turntable)_final = angular velocity(person)_final = 0
Therefore, the angular velocity of the turntable is 0 rad/s.