Animated Solution for Physics - Oscillations: The length of a simple pendulum executing simple harmonic motion is increased by 21%. The percentage increase in the time period of the pendulum of increased length is

Imagine you are standing in a grand clock tower, watching a massive pendulum swing back and forth. The rhythm of this swing, known as the time period ($T$), is governed by a beautifully simple yet profound equation:

Here, $l$ represents the length of the pendulum string, and $g$ is the acceleration due to gravity. Notice what is missing from this equation? The mass of the bob! Whether you hang a feather or a bowling ball, as long as the length is the same, the time period remains identical.

Since $2\pi$ and $g$ are constants in a given location, we can establish a direct proportionality:

This tells us that the time period scales with the square root of the length. If you want to double the time period, you must make the pendulum four times as long!

In our specific problem, the pendulum's length is increased by $21\%$. Let's translate this physical change into mathematics. If the original length is $l$, the new length $l^{\prime }$ becomes:

Now, we need to find out how this new length affects our time period. We simply substitute $l^{\prime }$ back into our master equation to find the new time period, $T^{\prime }$:

This is where the magic of numbers comes into play. We can separate the numerical factor from the variables inside the square root:

Do you recognize the number $1.21$? It is the square of $1.1$ (just like $11^2=121$). Therefore, $\sqrt{1.21}=1.1$. And the remaining part of the expression, $2\pi \sqrt{\frac{l}{g}}$, is exactly our original time period $T$. Substituting this back, we get a remarkably clean relationship:

This means the new time period is $1.1$ times the original time period. To find the percentage increase, we calculate the fractional change and multiply by $100$:

So, a $21\%$ increase in length results in exactly a $10\%$ increase in the time period.

You might wonder, what if the increase was very small, say $2\%$? For small percentage changes (typically less than $5\%$), we can use the binomial approximation, which states that the percentage change in $T$ is half the percentage change in $l$:

So a $2\%$ increase in length would cause approximately a $1\%$ increase in time period. However, never use this approximation for large changes like $21\%$. Always stick to the exact square root method we used above to avoid falling into a trap!