Q. A car is travelling with linear velocity $v$ on a circular road of radius $r.$ If the speed is increasing at the rate of $ams^{- 2}$ , then the resultant acceleration will be:-
NTA AbhyasNTA Abhyas 2022
Solution:
We have given that the velocity of the car is increasing. Therefore, the motion of the car is a non-uniform circular motion. In the non-uniform circular motion, the total acceleration of the car is the sum of its tangential acceleration and radial acceleration (centripetal acceleration). That is, $\overset{ \rightarrow }{a}_{n e t}=\overset{ \rightarrow }{a}_{c p}+\overset{ \rightarrow }{a}_{t}$
$\Rightarrow a_{n e t}=\sqrt{a_{c p}^{2} + a_{t}^{2}}$ $=\sqrt{\left(\frac{v^{2}}{r}\right)^{2} + a^{2}}$ $=\sqrt{\frac{v^{4}}{r^{2}} + a^{2}}$ .
$\Rightarrow a_{n e t}=\sqrt{a_{c p}^{2} + a_{t}^{2}}$ $=\sqrt{\left(\frac{v^{2}}{r}\right)^{2} + a^{2}}$ $=\sqrt{\frac{v^{4}}{r^{2}} + a^{2}}$ .