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
so
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

To find the maximum speed at which the car can move without overturning, we need to consider the forces acting on the car and analyze the conditions for equilibrium.

1. First, let's consider the gravitational force acting on the car. The weight of the car acts downward at the center of mass. The magnitude of this force is given by the equation: F_gravity = m * g, where m is the mass of the car and g is the acceleration due to gravity.

2. Next, let's consider the normal force. The car is moving in a circular path, so there must be a force acting inward to keep it on the track. This force is provided by the normal force, which acts perpendicular to the road surface. The magnitude of this force is given by: F_normal = m * a_c, where m is the mass of the car and a_c is the centripetal acceleration.

3. Now let's consider the maximum force of friction. Since the road is dry and the car does not skid, the frictional force acts in the opposite direction of the motion and has a magnitude of: F_friction_max = μ * F_normal, where μ is the coefficient of friction between the tires and the road.

4. Finally, let's analyze the conditions for equilibrium. The car will not overturn if the net torque about its center of mass is zero. The torque due to the gravitational force acts in the clockwise direction and is given by: τ_gravity = F_gravity * h. Similarly, the torque due to the maximum frictional force acts in the counterclockwise direction and is given by: τ_friction_max = F_friction_max * d.

Setting the net torque equal to zero: τ_gravity = τ_friction_max, we have the equation:

F_gravity * h = F_friction_max * d

Substituting the expressions for F_gravity, F_friction_max, and a_c into the equation, and rearranging for the maximum speed v:

(m * g * h) = (μ * m * a_c * d)

Now we can cancel out the mass m from both sides of the equation:

g * h = μ * a_c * d

Since a_c = v^2 / R (centripetal acceleration formula), we can substitute this into the equation:

g * h = μ * (v^2 / R) * d

Now we solve for the maximum speed v:

v^2 = (g * h * R) / (μ * d)

Finally, taking the square root of both sides of the equation, we get:

v = sqrt((g * h * R) / (μ * d))

So, the maximum speed the car can have without overturning is given by this equation.