A man stands on a rotating platform, with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform is 30 revolutions per minute. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90cm to 20cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg m². (a) What is his new angular speed? (Neglect friction.) (b) Is kinetic energy conserved in the process? If not, from where does the change come about?
Given, the holding weight is 5 kg, the initial angular speed of the platform is 30 rpm and the change in the location of the weight from the axis is 90 cm to 20 cm. The moment of inertia of the man together with the platform is 7.6 kg-m 2 .
(a)
Write the expression for the initial moment of inertia due to the mass at each hand.
I = m r 0 2 + m r 0 2 = 2 m r 0 2
Here, m is the holding mass and r 0 is the initial radial distance of the mass from the axis of rotation.
Substituting the given values in the above equation, we get:
I = 2 × ( 5 kg ) × ( 90 cm ) 2 = 2 × ( 5 kg ) × ( 90 cm × 1 m 100 cm ) 2 = 8.1 kg-m 2
Total initial moment of inertia of the combined system is,
I 0 = 8.1 + 7.6 = 15.7 kg-m 2
The expression for initial angular momentum of the combined system is:
L 0 = I 0 ω 0
Here, ω 0 is the initial angular speed of the platform.
The expression for final moment of inertia due to the mass at each hand is:
I ′ = m r f 2 + m r f 2 = 2 m r f 2
Here, r f is the final radial distance of the holding mass from the axis of rotation.
Substituting the given values in the above equation, we get:
I ′ = 2 × ( 5 kg ) × ( 20 cm ) 2 = 2 × ( 5 kg ) × ( 20 cm × 1 m 100 cm ) 2 = 0.4 kg-m 2
Total final moment of inertia of the combined system is,
I f = 7.6 + 0.4 = 8.0 kg-m 2
The expression for final angular momentum of the combined system is:
L f = I f ω f
Here, ω f is the final angular speed of the platform.
Since, there are no non-conservative forces acting on the system, the angular momentum remains conserved.
The expression for the law of conservation of angular momentum is:
L 0 = L f
Substituting the given values of L 0 and L f in the above equation, we get:
I 0 ω 0 = I f ω f .
Solving for ω f the equation is rearranged as:
ω f = ( I 0 I f ) ω 0
Substituting the calculated values in the above equation, we get:
ω f = ( 15.7 8 ) ( 30 ) = 58.875 rev / min ≈ 59 rev / min
Thus, the new angular speed of the platform is 59 rev/min.
(b)
Since some amount of work is required to change the position of the holding mass, the kinetic energy is not conserved in this process. The net change in the kinetic energy comes from the work done by the man to change the position of the holding mass.
Given, the holding weight is 5 kg, the initial angular speed of the platform is 30 rpm and the change in the location of the weight from the axis is 90 cm to 20 cm. The moment of inertia of the man together with the platform is
(a)
Write the expression for the initial moment of inertia due to the mass at each hand.
Here,
Substituting the given values in the above equation, we get:
Total initial moment of inertia of the combined system is,
The expression for initial angular momentum of the combined system is:
Here,
The expression for final moment of inertia due to the mass at each hand is:
Here,
Substituting the given values in the above equation, we get:
Total final moment of inertia of the combined system is,
The expression for final angular momentum of the combined system is:
Here,
Since, there are no non-conservative forces acting on the system, the angular momentum remains conserved.
The expression for the law of conservation of angular momentum is:
Substituting the given values of
Solving for
Substituting the calculated values in the above equation, we get:
Thus, the new angular speed of the platform is 59 rev/min.
(b)
Since some amount of work is required to change the position of the holding mass, the kinetic energy is not conserved in this process. The net change in the kinetic energy comes from the work done by the man to change the position of the holding mass.
