A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about i

Q: Why is angular momentum conserved here?

Because there is no external torque acting on the platform-person system about the axis of rotation. The forces involved (like the person's footsteps) are internal to the system.

Q: Do we need to convert rpm to rad/s?

Not necessarily. Since we are equating $I_1 \omega_1 = I_2 \omega_2$, the conversion factor ($\frac{2\pi}{60}$) will cancel out on both sides. You can safely use rpm directly.

Q: Why is the person's final moment of inertia zero?

The moment of inertia of a point mass is $mr^2$. When the person reaches the center, their distance $r$ from the axis is exactly zero, making their inertia zero.

Q: Does the kinetic energy remain conserved in this process?

No! The rotational kinetic energy actually increases. The person does work against the centrifugal force to reach the center, which adds energy to the system.

Q: What if the person walks along the rim instead of towards the center?

If they walk along the rim, their distance $R$ doesn't change, so the moment of inertia remains constant. However, their movement would create an internal torque, changing the platform's speed depending on their direction of walking.

Question

A person of 80 kg mass is standing on the rim of a circular platform of mass 200 kg rotating about its axis at 5 rpm. The person now starts moving towards the centre of the platform. What will be the rotational speed (in rpm) of the platform when the person reaches its centre?

Sigma Percentile · Accepted Answer

### The Setup: A Spinning Platform

Imagine a heavy circular platform spinning smoothly at $5 	ext{ rpm}$. A person of mass $80 	ext{ kg}$ is standing right at the edge of this $200 	ext{ kg}$ platform. This is a classic physics scenario that perfectly demonstrates one of the most beautiful laws of nature.

When the person starts walking towards the center, they are changing the mass distribution of the entire system. But how does this affect the rotation?

### The Master Principle: Conservation of Angular Momentum

Since there is no external twisting force, or torque, acting on the whole system (the platform plus the person), the total angular momentum must remain perfectly conserved. This is the exact same principle that ice skaters use to spin faster when they pull their arms in!

Mathematically, we write this as:

$$I_1 \omega_1 = I_2 \omega_2$$

Here, $I$ represents the moment of inertia, and $\omega$ represents the angular velocity.

### Calculating the Inertia

First, let's calculate the initial moment of inertia ($I_1$). It's the sum of the platform's inertia (which is a solid disc) and the person's inertia (who is at a distance $R$ from the center).

$$I_1 = \frac{M R^2}{2} + m R^2$$

When the person reaches the exact center, their distance from the rotation axis becomes zero. So, their contribution to the moment of inertia completely vanishes! We only have the platform's inertia left for our final state ($I_2$).

$$I_2 = \frac{M R^2}{2} + m(0)^2 = \frac{M R^2}{2}$$

### The Final Spin

Let's plug these expressions into our conservation equation. Notice how we don't even need the actual value of the radius, because $R^2$ is present in every single term.

$$\left(\frac{M R^2}{2} + m R^2ight) \omega_1 = \left(\frac{M R^2}{2}ight) \omega_2$$

We can safely cancel out $R^2$ from both sides. Now, let's substitute the given masses: $200 	ext{ kg}$ for the platform and $80 	ext{ kg}$ for the person, along with the initial speed of $5 	ext{ rpm}$.

$$\left(\frac{200}{2} + 80ight) 	imes 5 = \left(\frac{200}{2}ight) 	imes \omega_2$$

Simplifying the numbers, $200$ divided by $2$ is $100$. Adding $80$ gives us $180$. On the right side, we just have $100$.

$$180 	imes 5 = 100 	imes \omega_2$$

$$900 = 100 	imes \omega_2$$

Dividing by $100$ gives us exactly $9$. So, the platform speeds up to $9 	ext{ rpm}$!

### The Energy Paradox

The platform spins faster, which means the rotational kinetic energy has actually increased. You might wonder, where did this extra energy come from if no external forces were involved?

The answer lies in the person's movement. As the platform spins, the person experiences an outward centrifugal force. To walk towards the center, they must exert a force and do mechanical work against this centrifugal force. This internal work done by the person is exactly what gets converted into the extra rotational kinetic energy of the system. Fascinating, isn't it?

Enter Numerical Value:

Visualized Solution

Visualizing the Setup

Conservation of Angular Momentum

Initial Moment of Inertia

Final Moment of Inertia

Substituting the Values

Canceling the Radius

Simplifying the Equation

Final Calculation

The Energy Paradox

The Sigma Insight: Conservation of Angular Momentum

The Setup

The Master Principle

Calculating the Inertia

The Final Spin

The Energy Paradox

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