Animated Solution for Physics - Physics and Measurement: If the length of the pendulum in pendulum clock increases by $0.1\%$, then the error in time per day is

Have you ever stood in front of a grand grandfather clock, mesmerized by the rhythmic, hypnotic swing of its pendulum?

It feels like magic, but it is pure, unadulterated physics. The pendulum clock is a masterpiece of classical mechanics, relying entirely on the constant time period of a swinging mass to keep accurate time. But what happens when the physical dimensions of this perfect system are slightly altered?

In this problem, we are going to explore the fascinating world of error analysis and see how a microscopic change in length can lead to a significant loss of time over a single day.

The Pendulum's Rhythm

The time period $T$ of a simple pendulum is governed by a beautifully elegant equation:

Here, $L$ is the length of the pendulum string, and $g$ is the acceleration due to gravity. Notice the square root? That is the secret sauce of this problem. The time period is directly proportional to the square root of the length.

If the length $L$ increases—perhaps due to thermal expansion on a hot summer day—the time period $T$ will also increase. This means each swing takes slightly longer, and the clock will inevitably run slow, losing precious seconds.

Error Analysis

When the length $L$ increases by a tiny fraction, we don't need to recalculate everything from scratch. We can use the power of calculus—specifically, relative error analysis.

By taking the natural logarithm of both sides of our time period equation and differentiating, the power of $\frac{1}{2}$ gracefully steps down to become a multiplier.

This gives us our master equation for small errors:

This equation tells us that the fractional error in the time period is exactly half of the fractional error in the length. It is a brilliant shortcut that saves us from tedious algebraic expansions.

Plugging in the Numbers

We are given that the length of the pendulum increases by $0.1\%$. This means:

Substituting this into our master equation, we find that the percentage error in the time period is exactly half of that:

So, the time period increases by $0.05\%$. But the question doesn't ask for a percentage. It asks for the absolute error in a single day.

The Seconds Lost

To find the absolute error, we first need to know the total time $T$ in a single day. How many seconds are in a day?

Finally, we multiply our fractional error by the total seconds in a day to find $\Delta T$:

Because the length increased, the pendulum takes longer to complete one swing. The clock is running slow, and it will lose $43.2$ seconds every single day.

Beyond Length

This is a classic and highly favored concept for JEE. But don't just stop here. Think about the physical implications.

What if the temperature increased? The metal rod would expand, increasing $L$, and the clock would lose time. What if we took the clock to the moon or the top of Mount Everest? The acceleration due to gravity $g$ would decrease. Since $g$ is in the denominator, a decrease in $g$ would also increase the time period $T$, causing the clock to run slow once again.

Mastering this interplay between physical variables and error analysis will give you a massive edge in competitive physics!