Question

A car moves with speed $v$ on a horizontal circular track of radius $R$. A head-on view of the car is shown in figure. The height of the car's centre of mass above the ground is $h$, and the separation between its inner and outer wheels is $d$. The road is dry, and the car does not skid. The maximum speed the car can have without overturning is.

• A. $\sqrt{\frac{gRd}{2h}}$
• B. $\sqrt{\frac{gRd}{h}}$
• C. $\sqrt{\frac{2gRd}{h}}$
• D. $\sqrt{\frac{2gRd}{3h}}$

Answer 1

Answer: (A)

When the car is on the point of rolling over, the normal force on its inside wheels is zero.
$\sum F_y=m{a}_y:N-mg=0$
$\sum F_x=m{a}_x:f=\frac{m{v}^2}{R}$
Take torque about the centre of
mass : $fh-N\frac{d}{2}=0$
Then by substitution,
$\frac{m{v}_{\max }^2}{R}h-\frac{mgd}{2}=0$
$\Rightarrow v_{\max }=\sqrt{\frac{gdR}{2h}}$
A wider wheelbase (larger $d$ ) and a lower centre of mass (smaller $h$ ) will reduce the risk of rollover.