Question

A pendulum is executing simple harmonic motion, and its maximum kinetic energy is ${K_1}$. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is ${K_2}$, Then?

Answer 1

In this question, we first find the maximum kinetic energy of the simple pendulum, and then we double the length of the pendulum, after which we find the maximum kinetic energy is ${K_2}$.

Complete step by step answer:
The initial maximum kinetic energy of a simple pendulum $= {K_1}$
The maximum kinetic energy of a simple pendulum when the length is doubled $= {K_2}$
We know the maximum kinetic energy of a simple pendulum at a point is given by the formula
$KE = \dfrac{1}{2}m{\omega ^2}{A^2} - - (i)$

Here A is the linear amplitude of a simple pendulum as shown in the diagram, which can be written as
$A = l\theta$
Now substitute the value of the linear amplitude of simple pendulum in equation (i)

$$KE = \dfrac{1}{2}m{\omega ^2}{\left( {l\theta } \right)^2} \\\ = \dfrac{1}{2}m{\omega ^2}{l^2}{\theta ^2} \\\$$

Now since we know the angular frequency is given as$\omega = \sqrt {\dfrac{g}{l}}$, hence we can further write

$$KE = \dfrac{1}{2}m\dfrac{g}{l}{l^2}{\theta ^2}\omega \\\ = \dfrac{1}{2}mgl{\theta ^2}\omega \\\$$

Hence we can write ${K_1} = \dfrac{1}{2}mgl{\theta ^2}\omega - - (ii)$
Now it is said that the length of the pendulum is doubled so $l' = 2l$
So the maximum kinetic energy of the pendulum becomes

$${K_2} = \dfrac{1}{2}mg2l{\theta ^2} \\\ = mgl{\theta ^2} \\\$$

Hence we can write
${K_2} = mgl{\theta ^2} - - (iii)$
Now to find the change in Kinetic energy, divide equation (ii) by equation (iii)
$\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{\dfrac{1}{2}mgl{\theta ^2}\omega }}{{mgl{\theta ^2}}}$
By solving
$\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{1}{2}$
Hence the kinetic energy ${K_2} = 2{K_1}$

Note:
Students must note that if the length of the simple pendulum is doubled, then its maximum kinetic energy also gets doubled. Maximum kinetic energy at a point is given by the formula $KE = \dfrac{1}{2}m{\omega ^2}{A^2}$, where $A$ is the linear amplitude, $\omega$ is the angular frequency.