Question

A particle follows two SHM's. One along the y axis and the other along the x axis.What is the trajectory of the particle of combined SHM's. If $A_1\ne A_2$ and $\varphi =\frac{\pi }{2}$ the nature of path followed by the particle is

• A. straight line with positive slope
• B. parabola symmetric about y-axis
• C. Circle in x-y plane
• D. An ellipse in x-y plane

Answer 1

The correct option is D An ellipse in x-y plane

Let equation of the particle moving along y axis $y=A_{1}sin\left(wt\right)$

$⟹sinwt=\frac{y}{A_1}$ ........(1)

Now equation of the particle moving along x axis $x=A_{2}sin(wt+\varphi )$

Given : $\varphi =\frac{\pi }{2}$

Thus $x=A_{2}sin(wt+\frac{\pi }{2})=A_{2}cos\left(wt\right)$

$⟹coswt=\frac{x}{A_2}$ ............(2)

Squaring and adding (1) and (2), $si{n}^{2}wt+co{s}^{2}wt=\frac{y^2}{A_1^2}+\frac{x^2}{A_2^2}$

$\therefore$ $\frac{y^2}{A_1^2}+\frac{x^2}{A_2^2}=1$ which is an equation of an ellipse.