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The total vibrational energy of a particle in S.H.M. is $E$. Its kinetic energy at half the amplitude from mean position will be:
A) $ \dfrac{E}{2} $
B) $ \dfrac{E}{3} $
C) $ \dfrac{E}{4} $
D) $ \dfrac{{3E}}{4} $

Last updated date: 18th Jun 2024
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Hint: Kinetic energy is defined as the energy of mass in motion. We have to find out Kinetic energy of half amplitude from the mean position. We can calculate kinetic energy of half amplitude by using the formula of kinetic energy. For finding the kinetic energy the first thing is to know the amplitude of the particle, mean position of particle, mass of the particle by knowing all this thing we can easily calculate.

Formula used:
The formula of Kinetic energy is $KE= \dfrac{1}{2}m \omega ^2 \sqrt{A^2 - x^2}$
Kinetic energy- Kinetic energy can be defined as it is simply the energy of mass in motion. It also tells us about energy density. Or in other words Kinetic energy can be defined as the faster an object is moving the more Kinetic energy is consumed. This is called Kinetic energy.
Where, m is mass, and $\omega $ is magnitude of velocity.

Complete step by step solution:
We have to find out that amplitude is half of the mean position.
Amplitude- Amplitude is defined as the maximum displacement of a point on a wave. Which you can think of as the degree or intensity of change.
The formula of finding amplitude is $x = \dfrac{{Amplitude}}{2} $
Now, by using the formula of Kinetic energy.
Now, substituting all the value,
$ KE = \dfrac{1}{2}m{\omega ^2}\left( {{A^2} - \dfrac{{{A^2}}}{4}} \right) $
Now, we will take L.C.M-
Now, take $\dfrac{3}{4}$ outside
Now we know that total energy $ \left( E \right) = \dfrac{1}{2}m{\omega ^2}{A^2} $
Now, put the value in the following.

The correct is $ \dfrac{{3E}}{4} $.

Note: The main thing is to remember is to know the amplitude of the particle, mean position of the particle, mass of particle. Knowing all these things we can find the required answer. As we know kinetic energy is a scalar quantity. Kinetic energy does not have any direction. For example, moving cars, bullets from guns. The standard unit of the kinetic energy is Joule. Where $1Joule$ is equivalent to $1Kgm^2/s^2$. Kinetic energy is directly proportional to the square of angular velocity. These points must be remembered while solving the question.