Question

A particle is subjected to two simple harmonic motions along X-axis while other is along a line making angle 45${}^o$ 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. $x_0+s_0+2x_0{s}_0$
• B. $\sqrt{x_0^2+s_0^2}$
• C. $\sqrt{x_0^2-s_0^2+2x_0{s}_0}$
• D. $[x_0^2+s_0^2+\sqrt{2}{x}_0{s}_0{]}^{1/2}$

Answer 1

Answer: (D)

$[x_0^2+s_0^2+\sqrt{2}{x}_0{s}_0{]}^{1/2}$

Along x-axis the motion is: $x=x_{0}sin\omega t+\frac{s_{0}sin\omega t}{\sqrt{2}}$

Along y - axis the motion is: $y=\frac{s_{0}sin\omega t}{\sqrt{2}}$
Resultant motion is $=\sqrt{{\left(\frac{s_{0}sin\omega t}{\sqrt{2}}\right)}^2+{(\frac{s_{0}sin\omega t}{\sqrt{2}}+x_{0}sin\omega t)}^2}$
Thus, the amplitude is $=\sqrt{{x_0}^2+{s_0}^2+\sqrt{2}{x}_0{s}_0}$