Question

A car on a circular track with a radius of 0.3km accelerates from rest with a constant angular acceleration of magnitude 5×10^-3 rad/s. a)How long does it take the car to make one lap around the truck.
b) what is the total acceleration of the car when it has completed half of the lap?

Answer 1

a) One lap is $\phi=2\pi$ radians. The time required to make one complete lap is

$$t=\sqrt{\frac{2\phi}{\alpha}}=\sqrt{\frac{2\cdot2\pi}{5\cdot10^{-3}}}=50.1\text{ s}.$$

b) Find the tangential acceleration:

$$a_t=\alpha R.$$

Find the centripetal acceleration:

$$a_c=\frac{v^2}{R}=\frac{2a_tL}{R},$$

where $v$ is the velocity after the car made one half of the circle accelerating at the rate $a_t$:

$$v=\sqrt{2a_tL}=\sqrt{2a_t(\pi R)},\rightarrow\\ a_c=\frac{2a_t\pi R}{R}=2\pi\alpha R.$$

The centripetal and tangential accelerations are always perpendicular to each other when considered in circular motion.

Now draw the situation:

The total acceleration $a$ at 1/2 of the lap is

$$a=\sqrt{a_c^2+a_t^2}=\alpha R\sqrt{1+4\pi^2}=9.54\text{ m/s}^2.$$