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Step-by-Step Solution
Step 1: Understand the Physical Setup
A circular platform of radius $R = 2\,\text{m}$ is mounted on a frictionless vertical axle. Its moment of inertia (about the same vertical axis) is $I = 200\,\text{kg}\,\text{m}^2$. A man of mass $m = 50\,\text{kg}$ stands at the edge of the platform. He begins walking with a speed of $1\,\text{m/s}$ relative to the ground.
Step 2: Apply Conservation of Angular Momentum
Initially, both the platform and the man are at rest; hence, the total initial angular momentum is
$$ L_i = 0. $$
Once the man starts walking, if the platform rotates with angular velocity $\omega$ in the opposite direction, the man has an angular momentum $m v R$ (trying to rotate the system in one direction), and the platform has an angular momentum $I \omega$ (in the opposite direction). By conservation of angular momentum,
$$ L_f = m v R - I\,\omega = 0 \quad \text{(taking directions into account)}, $$
so
$$ m v R = I \omega. $$
Step 3: Solve for the Platform's Angular Velocity
From
$$ m v R = I \omega, $$
substitute the given values: $m = 50\,\text{kg}$, $v = 1\,\text{m/s}$, $R = 2\,\text{m}$, and $I = 200\,\text{kg}\,\text{m}^2$:
$$ 50 \times 1 \times 2 = 200 \,\omega, $$
which simplifies to
$$ 100 = 200\,\omega \quad \Longrightarrow \quad \omega = 0.5\,\text{rad/s}. $$
Step 4: Determine the Man’s Actual Speed Around the Center
The tangential speed of the platform’s edge due to its rotation is
$$ \omega R = 0.5 \times 2 = 1\,\text{m/s}. $$
Since the man walks with $1\,\text{m/s}$ (in one direction) while the platform edge moves at $1\,\text{m/s}$ in the opposite direction relative to the man’s direction of walking, the man’s total speed relative to the ground around the center becomes
$$ v_{\text{net}} = v + \omega R = 1 + 1 = 2\,\text{m/s}. $$
Step 5: Calculate the Time for One Revolution
The circumference of the circular path the man must follow is
$$ \text{Circumference} = 2\pi R = 2\pi \times 2 = 4\pi\,\text{m}. $$
The time taken for one revolution is
$$ t = \frac{\text{Distance}}{\text{Speed}} = \frac{4\pi}{2} = 2\pi\,\text{s}. $$
Final Answer
The time taken by the man to complete one revolution is $2\pi$ seconds.
