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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 bar does not skid. The maximum speed the car can have without overturning is.

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$\sqrt{\frac{gRd}{2 h}}$

$\sqrt{\frac{gRd}{h}}$

$\sqrt{\frac{2 gRd}{h}}$

$\sqrt{\frac{2 gRd}{3 h}}$

answer is A.

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Detailed Solution

When the car is on the point of rolling over, the normal force on its inside wheels is zero.
$\sum_{} F_{y} = {ma}_{y} : N - mg = 0$

$\sum_{} F_{x} = {ma}_{x} : f = \frac{{mv}^{2}}{R}$

Take torque about the centre of mass:
$fh - N \frac{d}{2} = 0$
Then by substitution,
$\frac{{mv}_{\max}^{2}}{R} h - \frac{mgd}{2} = 0 \Rightarrow v_{\max} = \sqrt{\frac{gdR}{2 h}}$

A wider wheelbase (larger d) and a lower centre of mass (smaller h) will reduce the risk of rollover.

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