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(a) The motion of the particle in simple harmonic motion is given by \( x=a \sin \omega t \). If its speed is \( u \), when the displacement is \( x_{1} \) and speed is \( v \), when the displacement is \( x_{2} \), show that the amplitude of the motion is
\( \mathrm{P} \)
\[
A=\left[\frac{v^{2} x_{1}^{2}-u^{2} x_{2}^{2}}{v^{2}-u^{2}}\right]^{1 / 2}
\]
W
(b) A particle is moving with simple harmonic motion in a straight line. When the distance of the particle from the equilibrium position has the values \( x_{1} \) and \( x_{2} \), the corresponding values of velocity are \( u_{1} \) and \( u_{2} \), show that the period is
\[
T=2 \pi\left[\frac{x_{2}^{2}-x_{1}^{2}}{u_{1}^{2}-u_{2}^{2}}\right]^{1 / 2}
\]
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