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

Correct answer: D. [x_0^2+s_0^2+ 2 x_0 s_0]^ 1 / 2 Two SHMs are separated at angle 45^ . So, resultant amplitude is aligned & A_R= (s_0+x_0 45^ )^2+(x_0 45^ )^2 & = (s_0^2+x_0^2 ^2 45^ +2 x_0 s_0 45^ +x_0^2 ^2 45^ ) & = s_0^2+x_0^2+ 2 x_0 s_0 aligned

JIPMER 2015

Exam: JIPMER
Year: 2015
Subject: Physics
Chapter: Oscillations

JIPMER 2015 · Physics · Oscillations

A particle is subjected to two simple harmonic motions along X...

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

Options

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

Correct answer

D. $\left[x_0^2+s_0^2+\sqrt{2} x_0 s_0\right]^{1 / 2}$

Step-by-step 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}

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