Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): If the amplitude of a simple harmonic oscillator is doubled, the area enclosed by its velocity-displacement graph becomes two times its original value. Reason (R): The velocity-displacement graph of a simple harmonic oscillator is an ellipse whose semi-major and semi-minor axes are both directly pr

Step-by-step solution

The velocity v of a simple harmonic oscillator at displacement x is given by v^2 = ^2(A^2 - x^2) , which can be rewritten as v^2 ( A)^2 + x^2 A^2 = 1 . This is the equation of an ellipse in the x - v plane. The semi-axes of this ellipse are A (along the x -axis) and A (along the v -axis). Both axes are directly proportional to the amplitude A . Thus, Reason (R) is true. The area enclosed by the ellipse is given by ( semi-major axis ) ( semi-minor axis ) = (A)( A) = A^2 . Since the area is proportional to A^2 , doub