Question
A particle executing harmonic motion is having velocities $${v_1}$$ and $${v_2}$$ at distances is $${x_1}$$ and $${x_2}$$ from the equilibrium position. The amplitude of the motion is
A.
$$\sqrt {\frac{{v_1^2{x_2} - v_2^2{x_1}}}{{v_1^2 + v_2^2}}} $$
B.
$$\sqrt {\frac{{v_1^2x_1^2 - v_2^2x_2^2}}{{v_1^2 + v_2^2}}} $$
C.
$$\sqrt {\frac{{v_1^2x_2^2 - v_2^2x_1^2}}{{v_1^2 - v_2^2}}} $$
D.
$$\sqrt {\frac{{v_1^2x_2^2 + v_2^2x_1^2}}{{v_1^2 + v_2^2}}} $$
Answer :
$$\sqrt {\frac{{v_1^2x_2^2 - v_2^2x_1^2}}{{v_1^2 - v_2^2}}} $$
Solution :
$$\eqalign{
& {v_1} = \omega \sqrt {{a^2} - x_1^2} ,{v_2} = \omega \sqrt {{a^2} - x_2^2} \cr
& {\text{We}}\,{\text{get,}}\,\,a = \sqrt {\frac{{v_1^2x_2^2 - v_2^2x_1^2}}{{v_1^2 - v_2^2}}} \cr} $$