a turntable rotates about a fixed axis, making one revolution in 10s . the moment of inertia of the turntable about it's axis is 1200kg/m^2 . a man of mass 80kg , initially standing at the center of the turntable, runs out along a radius. what is the angular velocity of the turntable when the man is 2m from the center?

Question

a turntable rotates about a fixed axis, making one revolution in 10s . the moment of inertia of the turntable about it's axis is 1200kg/m^2 . a man of mass 80kg , initially standing at the center of the turntable, runs out along a radius. what is the angular velocity of the turntable when the man is 2m from the center?

Answer 1

To find the angular velocity of the turntable when the man is 2m from the center, we can use the principle of conservation of angular momentum. Angular momentum is conserved when no external torques act on the system.

The initial angular momentum of the turntable and the man together is given by the equation:

L1 = (moment of inertia of the turntable) * (angular velocity of the turntable)

The final angular momentum of the turntable and the man is given by the equation:

L2 = (moment of inertia of the turntable) * (final angular velocity of the turntable) + (moment of inertia of the man) * (final angular velocity of the man)

Since angular momentum is conserved, L1 = L2.

Initially, the man is standing at the center of the turntable, so his initial angular velocity is 0. Therefore, the initial angular momentum of the man is 0.

We can write the equation for the initial angular momentum of the system as:

L1 = (moment of inertia of the turntable) * (angular velocity of the turntable) + (moment of inertia of the man) * (initial angular velocity of the man)

Since initial angular velocity of the man is 0, this simplifies to:

L1 = (moment of inertia of the turntable) * (angular velocity of the turntable)

To find the final angular velocity of the turntable, we rearrange the equation:

(angular velocity of the turntable) = L1 / (moment of inertia of the turntable)

Now, we substitute the values given in the question into the equation. The moment of inertia of the turntable is 1200 kg/m^2, and the moment of inertia of the man can be approximated by considering the man as a point mass rotating about the axis. The moment of inertia of a point mass rotating about an axis is given by the equation:

(moment of inertia of the man) = (mass of the man) * (distance of the man from the axis)^2

Plugging in the values:

(moment of inertia of the man) = (80 kg) * (2 m)^2 = 320 kg.m^2

Now we substitute the values into the equation:

(angular velocity of the turntable) = L1 / (moment of inertia of the turntable) = 0 / (1200 kg/m^2) = 0 rad/s

Therefore, the angular velocity of the turntable when the man is 2m from the center is 0 rad/s.