Question

The magnitude of displacement of a particle moving in a circle of radius $a$ with constant angular speed $\omega$ varies with time $t$ as

• A. $2a\sin \omega t$
• B. $2a\sin \frac{\omega t}{2}$
• C. $2a\cos \omega t$
• D. $2a\cos \frac{\omega t}{2}$

Answer 1

The correct option is B $2a\sin \frac{\omega t}{2}$

$R=acos\theta \left(i\right)+asin\theta \left(j\right).........1$

Ris the position vector at any time performing circular motion

$\theta$ is the angular motion

$\omega =\frac{\theta }{t}$

$\theta =\omega t........2$

from (1) and (2)

$Rt=acos\left(\omega t\right)\left(i\right)+asin\left(\omega t\right)\left(j\right)........3$

at t=0

$Rt=a\left(i\right).................4$

Displacement = $Rt-Ro=a\left(cos\right(\omega t\left)\right)\left(i\right)+asin\left(\omega t\right)\left(j\right)$

Magnitude = $\left[\right(a^2\left(cos\right(\omega t{)}^2+a^2\left(sin\omega t{)}^2\right)){]}^{\frac{1}{2}}$

$a[2a^2-2a^{2}cos(\omega t){]}^{\frac{1}{2}}$

this implies that

$2acos\left(\omega \frac{t}{2}\right)$