A circular platform is free to rotate in a horizontal plane about a vertical axis passing through it

Q: Why is the angular momentum conserved?

Because there is no external torque acting on the system about the vertical axis of rotation. The forces involved (gravity, normal reaction) are vertical and their torques about the vertical axis are zero.

Q: Why does the moment of inertia change?

The moment of inertia depends on the mass distribution. As the tortoise moves, its distance from the axis of rotation changes, altering the total moment of inertia of the system.

Q: Why is the graph not a straight line (triangle)?

The distance squared $r(t)^2$ is a quadratic function of time. Since $\omega(t)$ is inversely proportional to $I(t)$, the relationship is $\omega(t) \propto \frac{1}{A + Bt^2}$, which is a non-linear bell-shaped curve, not a linear one.

Q: What happens when the tortoise reaches the midpoint of the chord?

At the midpoint, the tortoise is closest to the center. The moment of inertia is at its minimum, and consequently, the angular velocity reaches its maximum peak.

Q: What if the tortoise walked along the diameter instead?

If it walked along the diameter, the minimum distance would be zero (at the center). The moment of inertia would drop even further, causing a much sharper and higher peak in the angular velocity.

Question

A circular platform is free to rotate in a horizontal plane about a vertical axis passing through its centre. A tortoise is sitting at the edge of the platform. Now, the platform is given an angular velocity $\omega_0$. When the tortoise move along a chord of the platform with a constant velocity (with respect to the platform). The angular velocity of the platform $\omega(t)$ will vary with time $t$ as

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a massive circular platform spinning smoothly in a horizontal plane. A tortoise, initially resting at the edge, decides to take a stroll. Instead of walking along the rim or straight to the center, it walks along a straight chord with a constant relative velocity.

This simple act sets off a fascinating chain of physical events. As the tortoise moves, it alters the mass distribution of the entire system.

Because the platform is free to rotate and there are no external twisting forces (torques) acting on it about the vertical axis, a fundamental principle of physics comes into play: the conservation of angular momentum.

### The Master Equation

The total angular momentum $L$ of the system is the product of its total moment of inertia $I(t)$ and its angular velocity $\omega(t)$.

Since $L$ is conserved, we can write the master equation:

$$L = I(t) \omega(t) = 	ext{constant}$$

This equation tells us a beautiful story. The angular velocity $\omega(t)$ is inversely proportional to the moment of inertia $I(t)$. If the inertia goes down, the platform must spin faster to compensate. If the inertia goes up, the platform slows down.

The total moment of inertia is the sum of the platform's inertia (which is constant) and the tortoise's inertia (which changes).

$$I(t) = I_{	ext{platform}} + m r(t)^2$$

Here, $r(t)$ is the distance of the tortoise from the center of the platform at any given time $t$.

### The Mathematical Shape of the Spin

As the tortoise walks from the edge of the platform along the chord, it gets closer and closer to the center. Its distance $r(t)$ decreases until it reaches the exact midpoint of the chord.

At this midpoint, the tortoise is at its closest approach to the center. Consequently, the moment of inertia $I(t)$ hits its absolute minimum. According to our master equation, this is the exact moment the platform spins the fastest!

After passing the midpoint, the tortoise continues towards the opposite edge. Its distance $r(t)$ starts increasing again, causing the moment of inertia to rise and the platform's spin to slow down.

But what is the exact mathematical shape of this speed-up and slow-down? Let's use the Pythagorean theorem. The square of the distance $r(t)^2$ can be expressed as a quadratic function of time:

$$r(t)^2 = a^2 + (x_0 - vt)^2$$

Where $a$ is the perpendicular distance from the center to the chord, and $v$ is the tortoise's velocity.

Substituting this back into our angular velocity equation, we get:

$$\omega(t) = \frac{L}{I_{	ext{platform}} + m(a^2 + (x_0 - vt)^2)}$$

### Final Conclusion

Look closely at the denominator. It contains a quadratic term $(x_0 - vt)^2$. This means the angular velocity $\omega(t)$ does not change linearly. It does not form sharp, straight lines like a triangle.

Instead, it forms a smooth, non-linear, bell-shaped curve. It gradually rises to a smooth peak as the tortoise reaches the midpoint, and then gradually falls as the tortoise walks away.

Comparing this physical reality with the given options, the graph that perfectly captures this non-linear rise and fall is graph (c).

**The correct answer is Graph (c).**

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Visualized Solution

$System Setup$

$Conservation of Angular Momentum$

$Moment of Inertia$

$Variation of Distance r (t)$

$Variation of ω (t)$

$Mathematical Form of ω (t)$

$Analyzing the Graph$

$Final Conclusion$

The Sigma Insight: Conservation of Angular Momentum

Analyzing the Setup

The Master Equation

The Mathematical Shape of the Spin

Final Conclusion

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } System Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conservation of Angular Momentum

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Moment of Inertia

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Variation of Distance r(t)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Variation of ω(t)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Mathematical Form of ω(t)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Graph

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Conclusion