Question

A particle is travelling in a circular path of radius $4m$. At certain instant the particle is moving at $20m/s$ and its acceleration is at an angle of ${37}^o$ from the direction to the centre of the circle as seen from the particle.
(i) At what rate is the speed of the particle increasing?
(ii) What is the magnitude of the acceleration?

Answer 1

Given that

A particle is travelling in a circular path of radius $4$ m. At certain instant the particle is moving at $20$ m/s and its acceleration is at an angle of ${37}^o$ from the direction to the centre of the circle

Radial acceleration, $a_c=\frac{v^2}{r}=\frac{20\ast 20}{4}=100m/s^2$

Also $tan{37}^o=\frac{a_t}{a_c}$

where $a_t$ is the tangential acceleration

$⟹\frac{3}{4}=\frac{a_t}{100}$

$⟹a_t=75m/s^2$

So the rate at which speed of the particle is increasing, $a_t=75m/s^2$

Magnitude of acceleration ,$a=\sqrt{a_c^2+a_t^2}=\sqrt{{100}^2+{75}^2}=125m/s^2$