Question

Tangential acceleration of a particle moving in a circle of radius $1\text{m}$ varies with time $t$ as shown in the figure (initial velocity of particle is zero). Time after which total acceleration of particle makes an angle of ${30}^{\circ }$ with radial acceleration is

• A. $4\text{s}$
• B. $4/3\text{s}$
• C. $2^{2/3}\text{s}$
• D. $\sqrt{2}\text{s}$

Answer 1

The correct option is C $2^{2/3}\text{s}$

From the given equation we can write the line equation as
$a_t=\sqrt{3}t$
So,
$\frac{dv}{dt}=\sqrt{3}t$
${\int }_0^{v}dv={\int }_0^t\sqrt{3}tdt$
$v=\frac{\sqrt{3}{t}^2}{2}$
and
$a_r=\frac{v^2}{r}$
$a_r=\frac{3t^4}{4}$

$\tan {30}^{\circ }=\frac{a_t}{a_r}$
$\frac{1}{\sqrt{3}}=\frac{\sqrt{3}{t}^4}{3t^4}$
On solving we get $t=2^{2/3}$