A particle executes S.H.M., the graph of velocity as a function of displacement is :

Solution

Physics sir · 4 steps

Write v in terms of x, square it, and the ellipse equation appears right in front of you.

1. Velocity in SHM. For SHM with amplitude A and angular frequency ω, the speed at displacement x is v = ω√(A² - x²).
2. Square and rearrange. Squaring gives v² = ω² (A² - x²). Divide every term by ω² A² and move the x² term to the left.

   x²/A² + v²/(ω² A²) = 1

3. Read the shape. This is x²/a² + y²/b² = 1 with semi-axes a = A along x and b = ω A along v. Since ω A is in general not equal to A, the curve is an ellipse.
4. So. The velocity versus displacement graph is an ellipse, option (c).

The idea to keep. In SHM, x and v always satisfy x²/A² + v²/(ω² A²) = 1. So v against x is an ellipse, while a against x is a straight line through the origin.

Answer from the NTA final answer key. Solution by PrepWiser.