Question

A simple pendulum has a length 1m and energy equal to 0.2 Joule when its amplitude is 4cm. Its energy when length is doubled is
a) 0.04J
b) 0.02J
c) 0.1J
d) 0.8J

Answer 1

Hint: At its highest point the pendulum is momentarily motionless. All of the energy in the pendulum is gravitational potential energy and there is no kinetic energy. At the lowest point the pendulum has its greatest speed. All of the energy in the pendulum is kinetic energy and there is no gravitational potential energy. However, the total energy is constant as a function of time.

Complete Step By Step Solution:
We all know that Kinetic Energy (KE) = $\dfrac{1}{2}m{{v}^{2}}$
And Potential Energy (PE) = mgh
And the total energy of pendulum is Kinetic energy + potential energy

But we are given with length , energy and amplitude , we need to write the energy in term of these quantities.

First write all the given quantities
L= 1m
Energy= 0.2J (Joule)
Amplitude (A)= 4cm

And we have a formula of energy in terms of these quantities
$E=\dfrac{1}{2}m{{A}^{2}}{{\omega }^{2}}$
Where $\omega =\dfrac{2\pi }{T}$

We know that $T=2\pi \sqrt{\dfrac{L}{g}}$

Put this value of T in $\omega $, we get
$\omega =\sqrt{\dfrac{g}{L}}$
Put this value of $\omega $ in Energy equation
$E=\dfrac{1}{2}m{{A}^{2}}\dfrac{g}{L}$

Now, the question is asking about energy when its length get doubled
Now from the above energy equation we know,
$E\propto \dfrac{1}{L}$

Therefore, let assume that ${{E}_{1}}$ is the energy when length was ${{L}_{1}}$ and ${{E}_{2}}$ be the energy when length was ${{L}_{2}}$

Since Energy is inversely proportional to length so we can write
$\dfrac{{{E}_{1}}}{{{E}_{2}}}=\dfrac{{{L}_{2}}}{{{L}_{1}}}$

Now according to question ${{L}_{2}}=2{{L}_{1}}$
So, $\dfrac{{{E}_{1}}}{{{E}_{2}}}=\dfrac{2{{L}_{1}}}{{{L}_{1}}}$
We have ${{E}_{1}}=0.2J$

Put this in Above equation,
$\dfrac{0.2J}{{{E}_{2}}}=\dfrac{2{{L}_{1}}}{{{L}_{1}}}$
On solving this we get ${{E}_{2}}=0.1J$

Hence, we can conclude that option (C) is the correct answer, that is
${{E}_{2}}=0.1J$.

Note:
In this question we have been asked for the energy when length is doubled, In these types of questions where two things are given or asked in ratio, you do not need to solve the full question. Just eliminate the common quantities from the equation and the equate both the equation, you will get the required answer.