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
v 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.
