Question

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

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

Answer 1

Answer: (C)

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

Answer 2

Radial acceleration ${{a}_{r}}=\frac{{{v}^{2}}}{r}$ Tangential acceleration ${{a}_{t}}=a$ $\therefore$ Resultant acceleration $a'=\sqrt{a_{r}^{2}+a_{t}^{2}+2{{a}_{r}}{{a}_{t}}\cos \theta }$ But here $\theta =90{}^\circ$ $\therefore$ $\cos \theta =\cos 90{}^\circ =0$ and $a'=\sqrt{a_{r}^{2}+a_{t}^{2}}=\sqrt{{{\left( \frac{{{v}^{2}}}{r} \right)}^{2}}+{{a}^{2}}}$