Question

A car moves on a horizontal track of radius $r$, the speed is increasing at a constant rate $a$, coefficient of friction between road and tyre is $\mu ,$ the speed at which car will skid is

• A. $\left[\right({\mu }^2{g}^2+a^2)r^2{]}^{1/4}$
• B. $\sqrt{\mu gr}$
• C. $\left[\right({\mu }^2{g}^2-a^2)r^2{]}^{1/4}$
• D. $\sqrt{ar}$

Answer 1

The correct option is C $\left[\right({\mu }^2{g}^2-a^2)r^2{]}^{1/4}$

Given, $a_t=a$
$a_C=\frac{V^2}{r}$
(Here $V$ keeps increasing due to $a_t$ )
$F_{net}=\sqrt{(ma{)}^2+{\left(\frac{m{V}^2}{r}\right)}^2}$
For the car to just start skidding,
$F_{net}=\mu mg$

$\Rightarrow \mu mg=\sqrt{(ma{)}^2+{\left(\frac{m{V}^2}{r}\right)}^2}$
$\Rightarrow {\mu }^2{g}^2=a^2+\left(\frac{V^4}{r^2}\right)$

Solving for $V$,
$V={\left[\right({\mu }^2{g}^2-a^2\left)r^2\right]}^{\frac{1}{4}}$