Question

A race car starts from rest on a circular track. The car increases its speed at a constant rate a; as it goes once around the track. Find the angle that the total acceleration of the car makes- with the radius connecting the center of the track and the car- at the moment the car completes the circle.

Answer 1

Here, we will use kinematics

for, tangential acceleration

x = 1/2a_Tt²

here x = one complete revolution = 2$\pi$r

so,

2$\pi$r = 1/2 * a_T * t²

a_T = 4$\pi$r / t²

Now, we can find final velocity after one revolution

v = a_T * t

v =  4$\pi$r / t² * t

v = 4$\pi$r / t

so,

centripetal acceleration, a_c = v² / r = 16$\pi$²r² / t²r = 16$\pi$² r / t²

now,

to find the angle

$\theta$ = arctan ( 4$\pi$r / t² / 16$\pi$² r / t² )

$\theta$ = arctan ( 1 / 4$\pi$)

$\theta$ = arctan ( 0.07957)

$\theta$ = 4.549 degree