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 :
.

Options

A. `sqrt((gRd)/(2 h))`
B. `sqrt((gRd)/(h))`
C. `sqrt((2 gRd)/(h))`
D. `sqrt((2 gRd)/(3 h))`

Answer

A

Explanation

(a) 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 = (mv^2)/( R)`
Take torque about the centre of mass :
`f h - N (d)/(2) = 0`
Then by substitution,
`(mv^2_(max))/( R) h -(mgd)/(2) = 0 rArr v_(max) = sqrt((gdR)/(2 h))`
A wider wheelbase (larger `d`) and a lower centre of mass (smaller `h`) will reduce the risk of rollover.
.