At what distance from the equilibrium position is the kinetic energy equal to the potential energy?

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Concept:

• Simple Harmonic Motion (SHM): The Simple Harmonic Motion is studied to discuss the periodic Motion Mathematically. 
  • In Simple Harmonic motion, the motion is between two extreme points, and the restoring force responsible for the motion tends to bring the object to mean position. 
  • The motion of a Simple pendulum and a block attached to spring are common examples of SHM.

Mathematically, SHM is Defined as:

x =  A Sin (ωt + ɸ), 

x is the displacement of the body from mean Position, at time t. ɸ is phase Difference. 

A is Amplitude of Motion, that is the Maximum distance the body in SHM can move from mean Position.

ω is Angular Speed = \(\omega = \frac{2\pi }{T}\)

T is the time period of Motion, 

• Potential Energy of the body in SHM  is

P = \(\frac{1}{2}m\omega ^{2}x^{2}\)

• Kinetic Energy of the body in SHM is 

K = \(\frac{1}{2}m\omega ^{2}(A^{2}-x^{2})\)

• Total Energy of the Body in SHM (E) 

E= \(\frac{1}{2}m\omega ^{2}A^{2}\)

Calculation:

Given,

Potential Energy = Kinetic Energy at some displacement x from mean position.

\(\frac{1}{2}m\omega ^{2}x^{2}\) = \(\frac{1}{2}m\omega ^{2}(A^{2}-x^{2})\)

⇒ \(\frac{1}{2}m\omega ^{2}x^{2} = \frac{1}{2}m\omega ^{2}A^{2}- \frac{1}{2}m\omega ^{2}x^{2}\)

⇒\(2\times \frac{1}{2}m\omega ^{2}x^{2} = \frac{1}{2}m\omega ^{2}A^{2}\)

⇒\(x^{2} = \frac{1}{2}A^{2}\)

⇒ \(x = \pm \frac{A}{\sqrt{2}} \)

So, Option  A / √ 2 is the correct option.

• Potential Energy is maximum at Extreme positions while Kinetic Energy is Maximum at mean Position.
• Potential Energy is zero at mean Position while Kinetic Energy is zero at Extreme Positions.

Let's discuss the concepts related to Oscillations and Energy in Simple Harmonic Motion. Explore more from Physics here. Learn now!

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