Question

How do you double the period of a pendulum?

Answer 1

Time period:

1. Oscillatory motion can be defined as the to or fro motion of an object in a periodic fashion along with an equilibrium position by taking time (T) for one complete oscillation is known as a time period.

The amplitude of oscillatory motion:

1. The maximum distance traveled from the equilibrium position is known as the amplitude of that oscillatory motion.

Step 1: Given that

The motion of a simple pendulum is a periodic motion in which the bob of a pendulum is oscillating about the mean position within a particular time period.

Time period of a simple pendulum

$\Rightarrow \mathrm{T}=2\mathrm{\pi }\sqrt{\frac{L}{\mathrm{g}}}........\left(1\right)$

Here, we can observe that the time period is independent of the mass of the bob of the pendulum, and assuming $g$ constant at a particular position, the only varying quantity is the length of the pendulum.

Let us assume the initial length of the pendulum be$\left(L\right)$, so the time period will be$\left(T\right)$, now the new length is ${\mathrm{L}}^{\text{'}}$and the time period is $2T$, so

$\Rightarrow 2\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{{\mathrm{L}}^{\text{'}}}{\mathrm{g}}}........\left(2\right)$

Where, $L=$the initial length of the pendulum, $T=$time period, and ${\mathrm{L}}^{\text{'}}=$new length.

Step 2: Further solve the above expression

By dividing the equation $\left(2\right)$ by the equation $\left(1\right)$ we have,

$\begin{array}{rcl}& \Rightarrow & \frac{2\mathrm{T}}{\mathrm{T}}=\frac{2\mathrm{\pi }\sqrt{{\displaystyle \frac{{\mathrm{L}}^{\text{'}}}{\mathrm{g}}}}}{2\mathrm{\pi }\sqrt{{\displaystyle \frac{\mathrm{L}}{\mathrm{g}}}}}\\ & \Rightarrow & \sqrt{\frac{{\mathrm{L}}^{\text{'}}}{\mathrm{L}}}=2\\ & & \end{array}$

Step 3: Calculate the situation when the period of the pendulum is doubled

Squaring both sides we have,

$\begin{array}{rcl}& \Rightarrow & \frac{\mathrm{L}\text{'}}{\mathrm{L}}=4\\ & \Rightarrow & \mathrm{L}\text{'}=4\mathrm{L}\end{array}$

Hence, by increasing the length of a simple pendulum by four times the time period will be doubled.