posted by Mischa on .
A car moves with speed v on a horizontal circular track of radius R. The height of the car's center of mass is h, and the separation between the inner and outer wheels is d. The road is dry, and the car does not skid. Find the maximum speed the car can have without overturning.
acceleration = v^2/R
side force = m v^2/R
overturning moment = m v^2 h/R
righting moment = m g d/2
m g d/2 = m v^2 h/R
v^2 = g d R /(2h)
v = sqrt [ g d R / (2 h) ]
Consider the moment about the wheels farthest from the center of the track. If the car is about to top over, there will be no weight or friction force on the inside wheels.
Imageine that you al=re lookng at the car head-on and considere the moments acting on it.
The moment due to the car's weight
M g d/2 will be equal to the oppositely directed moment due to the centripetal force acting through the center of mass,
M V^2 h/R
Therefore V^2 = g d R/(2h)
That will tell you the maximum stable velocity, V