The left and right wheels of an automobile are separated by a transverse distance of 1.40 m. The center of mass of this automobile is .6 m above the ground. If the automobile is driven around a flat (no banking) curve a radius of 26.0 m with an excessive speed, it will topple over sideways. What is the speed at which it will begin to topple? Assume that the wheels do not skid.
Fcentripetal = m Ac = m v^2/r = m v^2/26
toppling moment = height Fc = (.6/26) m v^2
Righting moment = m g (1.4/2)
start to tip when toppling moment = righting moment
(.6/26)v^2 = 9.8 (1.4/2)
The car will topple over if the centripetal force, applied at the center of mass, reaches a value such that the net torque about the wheels on the OUTSIDE of the curve is zero. Under these conditions, the centripetal-force torque about a line through the outer wheel's contact points with the ground will be equal and opposite to the gravity-force torque. At that speed, the inner wheels will have no ground force on them, and will start to lift. The car will topple.
The equation to solve(for V) is
(M V^2/R) * y = M g L
where y = 0.6 m , L = 1.40 m and R = 26.0 m. Note that M cancels out. Long wheelbase (high L) and low CG (low value of y) tend to increase the toppling velocity.
The max velocity is
Sqrt(Rgl/2h)
R = 26 m
g = 9.8 m/s^2
l = 1.4 m
h = 0.6 m
To determine the speed at which the automobile will begin to topple, we need to consider the maximum centripetal force that can be exerted on the vehicle before it starts to tip over.
The first step is to calculate the maximum force that can be exerted on the vehicle without causing it to topple. This can be found by considering the torque acting on the vehicle about its outer wheel.
The torque caused by the force of gravity acting on the vehicle can be calculated as follows:
Torque = mass * acceleration due to gravity * height of the center of mass
Since the torque must be equal to zero to prevent tipping, we can set up the following equation:
Torque = max centripetal force * transverse distance between the wheels
Setting these two equal to each other will give us the equation:
mass * acceleration due to gravity * height of center of mass = max centripetal force * transverse distance
Rearranging the equation to solve for the maximum centripetal force:
max centripetal force = (mass * acceleration due to gravity * height of center of mass) / transverse distance
Now that we have the maximum centripetal force, we can use it to find the speed at which the vehicle will topple. The centripetal force is given by the equation:
max centripetal force = mass * (velocity^2 / radius of the curve)
Rearranging the equation to solve for the velocity:
velocity = sqrt((max centripetal force * radius of the curve) / mass)
Plugging in the given values, we can calculate the velocity at which the automobile will begin to topple.