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Step-by-Step Solution
Step 1: Identify the principle to be used
Since there is no external torque acting on the platform-person system (assuming negligible friction at the axis), the angular momentum of the system remains constant. Hence, we use the law of conservation of angular momentum:
I_1 \omega_1 = I_2 \omega_2
Step 2: Write expressions for initial and final moments of inertia
Let:
M = 200 \text{ kg} be the mass of the platform
m = 80 \text{ kg} be the mass of the person
R be the radius of the platform
\omega_1 be the initial angular speed (in rad/s)
\omega_2 be the final angular speed (in rad/s)
Because the platform is a solid disc, its moment of inertia about its central axis is:
I_{\text{platform}} = \frac{1}{2} M R^2.
Initially, the person stands at the rim (radius R ), so the person’s moment of inertia about the platform’s axis is:
I_{\text{person, initial}} = m R^2.
Therefore, the total initial moment of inertia I_1 is:
I_1 = \frac{1}{2} M R^2 + m R^2.
In the final situation, the person stands at the center. The radius of the person’s circular path becomes 0 , making the person’s moment of inertia zero about the axis. Hence, the total final moment of inertia I_2 is:
I_2 = \frac{1}{2} M R^2.
Step 3: Apply conservation of angular momentum
According to angular momentum conservation:
I_1 \omega_1 = I_2 \omega_2.
Substituting the expressions for I_1 and I_2 :
\left( \frac{1}{2} M R^2 + m R^2 \right) \omega_1 = \left( \frac{1}{2} M R^2 \right) \omega_2.
Step 4: Solve for the final angular speed
Factor out R^2 and simplify:
\left( \frac{M}{2} + m \right) \omega_1 = \frac{M}{2} \,\omega_2.
Rearrange to get:
\omega_2 = \left( \frac{\frac{M}{2} + m}{\frac{M}{2}} \right)\,\omega_1 = \left( 1 + \frac{2m}{M} \right) \omega_1.
Substitute m = 80 \text{ kg} and M = 200 \text{ kg} :
\omega_2 = \left( 1 + \frac{2 \times 80}{200} \right) \omega_1 = \left(1 + 0.8\right)\omega_1 = 1.8\,\omega_1.
Step 5: Convert angular speed from rad/s to rpm (or directly scale the rpm)
Initially, the platform’s rotational speed is 5 \text{ rpm} . If we treat rpm as a direct measure of rotational speed for scaling purposes:
f_2 = 1.8 \times f_1 = 1.8 \times 5 = 9 \text{ rpm}.
Thus, when the person reaches the center, the platform rotates at 9 rpm.
Final Answer: 9 rpm
