Question

A particle moves according to the equation $x=a\sin \omega t$ and $y=a(1-\cos \omega t)$ . The path of the particle is

• A. circle
• B. parabola
• C. hyperbola
• D. cycloid
• E. ellipse

Answer 1

Answer: (D)

cycloid

$x=a\sin wt$
$y=a(1-\cos wt)or(\frac{y}{a}-1)=-\cos wt$
${\left(\frac{x}{a}\right)}^2+{(\frac{y}{a}-1)}^2=1$

$x^2+{(y-a)}^2=a^2\Leftarrow$ $Equationofcircle$

Thus the point moves on the rim of a circle which is above the origin by $a$, therefore the path traced by it is going to be cycloid.