T0 is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to 1

Q: Why doesn't the mass of the bob affect the time period?

The restoring force for a pendulum is proportional to its mass ($mg \sin\theta$), and its inertia is also proportional to its mass ($ma$). When you equate them, the mass cancels out entirely!

Q: What if the pendulum was taken to the moon?

The time period depends on the acceleration due to gravity ($g$). Since gravity on the moon is weaker (about $1/6$th of Earth's), the time period would increase, making the pendulum swing slower.

Q: Why do we take the square root of the length ratio?

The formula for the time period is $T = 2\pi\sqrt{\frac{l}{g}}$. Because the length $l$ is inside the square root, any change to the length will affect the time period by the square root of that change.

Q: Does the amplitude of the swing matter?

For small angles (typically less than 15 degrees), the time period is independent of the amplitude. This property is called isochronism. However, for very large swings, the formula becomes more complex and amplitude does matter.

Q: What happens if the length is increased by 16 times instead?

If the length is increased by a factor of 16, the new time period would be $\sqrt{16} = 4$ times the original time period. It would swing much slower!

Question

$T_0$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, then the modified time period is

Options:

(A) $T_0$

(B) $8\pi T_0$

(C) $4T_0$

(D) $\frac{1}{4}T_0$

Sigma Percentile · Accepted Answer

The simple pendulum is one of the most elegant and fundamental systems in physics. It’s a classic example of simple harmonic motion, and its time period formula is a powerful tool that unlocks many interesting problems. In this question, we are going to explore exactly how the length of the pendulum dictates its rhythm.

### The Simple Pendulum's Rhythm

Imagine a simple pendulum swinging back and forth. The time it takes to complete one full oscillation is called its time period, denoted by $T_0$. The beauty of the simple pendulum lies in its formula:

$$T_0 = 2\pi\sqrt{\frac{l}{g}}$$

Here, $l$ is the length of the string, and $g$ is the acceleration due to gravity. Notice something fascinating: the mass of the bob doesn't even appear in this equation! The time period is entirely governed by the length of the pendulum and the local gravity.

### Shrinking the Pendulum

The problem presents a scenario where the length of this pendulum is drastically reduced. Specifically, the new length $l'$ is made $\frac{1}{16}$ times its initial value.

$$l' = \frac{l}{16}$$

When we shorten the pendulum, intuition tells us it should swing faster, meaning its time period should decrease. But by exactly how much? To find out, we simply plug our new length into the time period formula.

### The Mathematical Magic

Let's write the equation for the new time period, $T'$:

$$T' = 2\pi\sqrt{\frac{l'}{g}}$$

Substituting $l' = \frac{l}{16}$ into the equation, we get:

$$T' = 2\pi\sqrt{\frac{\frac{l}{16}}{g}}$$

Now, we can separate the constant factor from the variables. The square root of $\frac{1}{16}$ is exactly $\frac{1}{4}$. Let's pull this fraction out of the square root:

$$T' = \frac{1}{4} \left( 2\pi\sqrt{\frac{l}{g}} \right)$$

Look closely at the term inside the parentheses. It is exactly our original time period, $T_0$!

$$T' = \frac{1}{4} T_0$$

And there we have it. By reducing the length to $\frac{1}{16}$ of its original value, the time period is reduced to $\frac{1}{4}$ of its original value. This perfectly illustrates the square-root proportionality between time period and length.

Animated Solution for Physics - Oscillations: $T_{0}$ is the time period of a simple pendulum at a place. If the length of the pendulum is reduced to $\frac{1}{16}$ times of its initial value, then the modified time period is

Select Answer:

Visualized Solution

Initial State

Final State

New Time Period

Substitution

Simplification

Final Answer

The Way Forward

The Sigma Insight: Simple Harmonic Motion (SHM)

The Simple Pendulum's Rhythm

Shrinking the Pendulum

The Mathematical Magic

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