
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
162.6k+ views
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.
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.
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