Question

The P.E. of a particle executing SHM at a distance x from its equilibrium position is
A) $\dfrac{1}{2}m{\omega ^2}{x^2}$
B) $\dfrac{1}{2}m{\omega ^2}{a^2}$
C) $\dfrac{1}{2}m{\omega ^2}({a^2} - {x^2})$
D) Zero

Answer 1

Hint:
Here in this question, we have to find the potential energy of the particle executing the simple harming motion from the distance x from its equilibrium position. For which we only have to use the formula of potential energy after which only we have to put the values changes from question. As a result, we get the solution to this question.


Formula used :
We are aware that the potential energy formula is,
${U_p} = \dfrac{1}{2}m{\omega ^2}{y^2}$


Complete step by step solution:
As we know that, the required formula of Potential energy executing simple harming motion is as below,
$P.E. = \dfrac{1}{2}k{x^2}$
And we also know that,
${\omega ^2} = \dfrac{k}{m}$
As from the above equation we need the value of k from the above equation, after getting the value we use it in Potential energy’s formula to get the result,
$k = {\omega ^2}m$
Putting the value we get the result as,
$P.E. = \dfrac{1}{2}m{\omega ^2}{x^2}$
As a result, we get the Potential energy of the particle executing simple harmonic motion from the distance x from the equilibrium position is $\dfrac{1}{2}m{\omega ^2}{x^2}$ .
Therefore, the correct answer is $\dfrac{1}{2}m{\omega ^2}{x^2}$ .
Hence, the correct option is (A).



Hence the correct answer is Option(A).




Note:
The major importance of Simple harmonic motion are as follows: Despite being a simplification, basic harmonic motion is nonetheless a reasonably accurate approximation. In order to describe oscillations, such as those in wind turbines and vibrations in automobile suspensions, simple harmonic motion is crucial. A guitar string, a bouncing ball, or a clock's pendulum are all examples of simple harmonic motion in daily life. These examples operate as they ought to if the amplitude, period, and frequency are understood.