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The curve obtained for a simple harmonic pendulum between the displacement of velocity will be-
1. straight line
2. parabola
3. ellipse
4. circle

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Correct Answer - Option 3 : ellipse

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 a 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) -- (1)

A is amplitude, ω is the angular frequency, ϕ is the phase difference. 

The speed in SHM is given as 

  • The Potential energy of the body in SHM  is

P = \(\frac{1}{2}mω ^{2}x^{2}\)

  • Kinetic Energy of the body in SHM is 

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

So, Speed is given as 

v 2 = ω 2 (A2 - x 2 )

CALCULATION:

Velocity is given as 

\(v = \omega \sqrt{A^2 - x^2}\)

Squaring

2 = ω 2 (A2 - x 2 )

Dividing by ω 2 A2 both side

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

\(\implies \frac{v^2}{\omega^2 A^2} + \frac{x^2}{A^2}= 1\)

This is the equation of an ellipse. 

So, the correct option is an ellipse.

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