Question

A particle moves in $xy$ plane according to the law $x=a\sin \omega t$ and $y=a(1-\cos \omega t)$ where $a$ and $\omega$ are constants.The particle traces

• A. a parabola
• B. a straight line equally inclined to x and y axis
• C. a circle
• D. a distance proportional to time

Answer 1

Answer: (C), (D)

The correct options are
C a circle
D a distance proportional to time

A particle moves in $xy$ plane according to the law
$x=a\sin \left(\omega t\right)$ and $y=a(1-cos(\omega t\left)\right)$ where a and $\omega$ are constants.
$\Rightarrow {\sin }^2\left(\omega t\right)=\frac{x^2}{a^2}$
$\Rightarrow {\cos }^2\left(\omega t\right)={(1-\frac{y}{a})}^2$
$\Rightarrow \frac{x^2}{a^2}+{(1-\frac{y}{a})}^2=1$
$\Rightarrow x^2+(y-a{)}^2=a^2$
The given equation is the equation of circle with centre $(0,a)$, the radius $a$
also the angular velocity is $\omega$ so distance covered$=a\omega t$