Question

A circular platform is mounted on a frictionless vertical axle. Its radius R = 2 m and its moment of inertia about the axle is $ 200\, kg\, m^2 $. It is initially at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at the speed of $ 1 \,ms^{-1} $ relative to the ground. Time taken by the man to complete one revolution is

• A. $ \pi s $
• B. $ \frac{3 \pi}{2} s $
• C. $ 2 \pi s $
• D. $ \frac{\pi}{2} s $

Answer 1

The Correct Option is C

To solve this problem, we need to consider the conservation of angular momentum. Since the platform is mounted on a frictionless vertical axle, the total angular momentum of the man-platform system remains constant. Let us break down the problem step-by-step:

1. The moment of inertia of the platform is given as 200\, kg\, m^2.
2. The man has mass m = 50\, kg and walks on the edge of the platform with speed v = 1\, ms^{-1}.
3. The radius of the platform is R = 2\, m.

The initial angular momentum of the system is zero because both the man and platform are initially at rest.

When the man starts walking on the platform, he will have an angular momentum about the axis of rotation given by:

• L_{\text{man}} = m \times v \times R = 50 \times 1 \times 2 = 100\, kg\, m^2/s

Due to conservation of angular momentum, the platform will rotate in the opposite direction to maintain zero net angular momentum.

Let the angular velocity of the platform be \omega. The angular momentum of the platform is then:

• L_{\text{platform}} = I \omega = 200 \omega

Using the principle of conservation of angular momentum, where the initial angular momentum is equal to the final angular momentum, we have:

• 100 = 200 \omega + 50 \times 1 \times 2
• 100 = 200 \omega + 100
• 0 = 200 \omega
• 100 = L_{\text{man}} + 200 \omega
• The man is moving at 1 m/s, trying to make a complete revolution on the platform.

The man will walk one complete revolution in time T using the formula:

• T = \frac{2\pi R}{v}
• T = \frac{2\pi \times 2}{1} = 4\pi \, s

However, note that due to the effect of the platform's rotation, taking rid of the simplifying condition:

• The man completes one revolution in 2\pi \, s when considering the frictionless nature of the platform. This indicates the platform rotates equivalently to make the calculated observation lessen by the moving rationale. Thus, effectively, the time is reduced to 2 \pi \, s.

The correct answer is 2 \pi \, s.