Question 49: Oscillations

Question

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . The length of the string required for the pendulum to perform 40 oscillations in the same time duration is
$\_\_\_\_$ cm . [Assume that the mass of the pendulum remains same.]

Accepted Answer

The time period (T) of a simple pendulum is the time taken to complete one full oscillation.
For a simple pendulum of the length of the string $(\mathrm{L})$ and the acceleration due to gravity $(\mathrm{g})$ the formula for the time period is,
$$ \mathrm{T}=2 \pi \sqrt{\frac{\mathrm{~L}}{\mathrm{~g}}} $$
So, the time period is directly proportional to the square root of the length $(T \propto \sqrt{L})$.
$$ \mathrm{T}^2 \propto \mathrm{~L} \Rightarrow \propto \mathrm{~T}^2 $$
Case - 1
The initial length of the string is $\mathrm{L}_1=30 \mathrm{~cm}$
$\mathrm{N}_1=20$ is the number of oscillations completed in total time $\mathrm{t}_1=10 \mathrm{~s}$
So, the initial time period of oscillation is
$$ \mathrm{T}_1=\frac{	ext { Total Time }}{	ext { Number of Oscillations }}=\frac{10}{20}=0.5 \mathrm{~s} $$
Case - 2
$\mathrm{N}_2=40$ is the number of oscillations completed in the same time $\mathrm{t}_2=\mathrm{t}_1=10 \mathrm{~s}$
So, the final time period of oscillation is,
$$ \mathrm{T}_2=\frac{	ext { Total Time }}{	ext { Number of Oscillations }}=\frac{10}{40}=0.25 \mathrm{~s} $$
We need to find the required length $\left(\mathrm{L}_2ight)$ corresponding to the final length of the string, Using the proportionality $\mathrm{L} \propto \mathrm{T}^2$
$$ \frac{\mathrm{L}_2}{\mathrm{~L}_1}=\left(\frac{\mathrm{T}_2}{\mathrm{~T}_1}ight)^2 $$
$$\Rightarrow $$ $\frac{\mathrm{L}_2}{30}=\left(\frac{0.25}{0.50}ight)^2=\frac{1}{4}$
$$\Rightarrow $$ $$ \frac{\mathrm{L}_2}{30}=\frac{1}{4} \Rightarrow \mathrm{~L}_2=\frac{30}{4} \mathrm{~cm}=7.5 \mathrm{~cm} $$
Therefore, the required length of the string is 7.5 cm .
Hence, the correct option is (B).

Answer

0.75

Answer

7.5

Answer

15

Answer

120

Oscillations

A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . The length of the string required for the pendulum to perform 40 oscillations in the same time duration is \_\_\_\_ cm . [Assume that the mass of the pendulum remains same.]

Choose the correct answer: