A man stands on a platform that is rotating (without friction) with an angular speed of 0.9 rev/s. His arms are outstretched and he holds a weight in each hand. The rotational inertia of the system of man, weights, and platform about the central axis is 4.0 kg·m2. Assume that by moving the weights the man decreases the rotational inertia of the system to 2.0 kg·m2.(a) What is the resulting angular speed of the platform?
Question
A man stands on a platform that is rotating (without friction) with an angular speed of 0.9 rev/s. His arms are outstretched and he holds a weight in each hand. The rotational inertia of the system of man, weights, and platform about the central axis is 4.0 kg·m2. Assume that by moving the weights the man decreases the rotational inertia of the system to 2.0 kg·m2.(a) What is the resulting angular speed of the platform?
Solution
This problem can be solved using the law of conservation of angular momentum. The angular momentum before the man moves the weights is equal to the angular momentum after he moves the weights, because no external torques are acting on the system.
The formula for angular momentum is L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular speed.
Before the man moves the weights, the moment of inertia is 4.0 kg·m2 and the angular speed is 0.9 rev/s. We need to convert this speed to rad/s by multiplying by 2π, because the formula for angular momentum uses rad/s. So, ω = 0.9 rev/s * 2π rad/rev = 5.65 rad/s.
So, the initial angular momentum L_initial = I_initial * ω_initial = 4.0 kg·m2 * 5.65 rad/s = 22.6 kg·m2/s.
After the man moves the weights, the moment of inertia is 2.0 kg·m2. We can find the final angular speed ω_final by rearranging the formula for angular momentum to ω = L/I.
So, ω_final = L_final / I_final = 22.6 kg·m2/s / 2.0 kg·m2 = 11.3 rad/s.
To convert this back to rev/s, we divide by 2π rad/rev, so the final angular speed is 11.3 rad/s / 2π rad/rev = 1.8 rev/s.
So, the resulting angular speed of the platform is 1.8 rev/s.
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