Q110 / 180 · JIPMER 2015 · Physics

A particle is subjected to two simple harmonic motions along $X$-axis while other is along a line making angle $45^{\circ}$ with the $X$-axis. The two motions are given by $x=x_0 \sin \omega t$ and $s=s_0 \sin \omega t$. The amplitude of resultant motion is

  • A
    $x_0+s_0+2 x_0 s_0$
  • B
    $\sqrt{x_0^2+s_0^2}$
  • C
    $\sqrt{x_0^2-s_0^2+2 x_0 s_0}$
  • Correct answer
    D
    $\left[x_0^2+s_0^2+\sqrt{2} x_0 s_0\right]^{1 / 2}$

Solution

Two SHMs are separated at angle $45^{\circ}$. So, resultant amplitude is $\begin{aligned} & A_R=\sqrt{\left(s_0+x_0 \cos 45^{\circ}\right)^2+\left(x_0 \sin 45^{\circ}\right)^2} \\ & \quad=\sqrt{\left(s_0^2+x_0^2 \cos ^2 45^{\circ}+2 x_0 s_0 \cos 45^{\circ}+x_0^2 \sin ^2 45^{\circ}\right)} \\ & \quad=\sqrt{s_0^2+x_0^2+\sqrt{2} x_0 s_0}\end{aligned}$

Hence, the correct answer is option: 'D'.

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