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 `h`, and the separation between its inner and outer wheel, is `d`. The road is dry, and the car does not skid. Show that the maximum speed the car can have without overturning is given by `v_(max)=sqrt((gRd)/(2h))`. To reduce the risk of rollover, should one increase or decrease `h`? Should one increase or decrease the width `d` of the wheel base?

Explanation

When the car is on the point of rolling over, the normal force on its inside wheels is zero.
`sumF_(y)=ma_(y):n-mg=0`
`sumF_(x)=ma_(x):f=(mv^(2))/R`
Take torque about the centre of mass
`fh-n d/2=0`
Then by substitution
`(mv_("max")^(2))/Rh-(mgd)/2=0implies v_("max")=sqrt((gdR)/(2h))`
A wider whelbase (larger `d`) and a lower centre of mass (smaller `h`) will refuce the risk of rollover.