Question

A car of mass $m$ moves in a horizontal circular path of radius $r$ meter. At an instant its speed is $v\,m/s$ and is increasing at a rate a $m/s^{2}$ , then the acceleration of the car is:

• A. $\sqrt{a\left( \frac{{{v}^{2}}}{r} \right)}$
• B. $\sqrt{{{a}^{2}}+{{\left( \frac{{{v}^{2}}}{r} \right)}^{2}}}$
• C. $\frac{{{v}^{2}}}{r}$
• D. $a$

Answer 1

Answer: (B)

$\sqrt{{{a}^{2}}+{{\left( \frac{{{v}^{2}}}{r} \right)}^{2}}}$

Answer 2

In the non-uniform circular motion, car will have radial and tangential acceleration.
In the non-uniform circular motion of the car, the car will have radial $\left(a_{R}\right)$ and tangential acceleration $\left(a_{T}\right)$ and both these accelerations will be perpendicular to each other.
Radial or centripetal acceleration, $a=\frac{v^{2}}{r}$

and tangential acceleration, $a_{T}=a$.
$\therefore$ Magnitude of resultant acceleration is
$a'=\sqrt{a_{R}^{2}+a_{T}^{2}}$
$a'=\sqrt{\left(\frac{v^{2}}{r}\right)^{2}+a^{2}}$