Friction: Problem Solving

To prevent slipping, analyze the equilibrium conditions by drawing a free-body diagram showing gravitational, frictional, and normal forces. Resolve the bus's weight into components parallel and perpendicular to the road. Since the bus travels at constant velocity, the resultant forces in both directions equal zero. Solve these equilibrium equations using the coefficient of static friction to find the maximum angle before slipping occurs.

Three primary forces act on the bus: gravitational force (weight), normal force perpendicular to the road surface, and frictional forces at the tire-road contact points. The gravitational force is resolved into components parallel and perpendicular to the incline. These forces must satisfy equilibrium conditions since the bus maintains constant velocity, meaning the net force in all directions is zero.

As the road angle increases, the bus first reaches a slipping condition when frictional forces cannot balance the gravitational component. At a steeper angle, tipping occurs when the bus loses contact with the uphill tires. At the tipping point, no normal or frictional force acts at the upper contact. The resultant moment about the downhill tires must equal zero to prevent tipping, establishing a distinct maximum angle for stability.

The coefficient of static friction directly determines the maximum angle before slipping occurs. A higher coefficient allows greater frictional force to resist the gravitational component along the incline, permitting steeper road angles. In this problem, a coefficient of 0.5 limits the maximum angle for no slipping. This value represents the relationship between normal force and maximum available friction at the tire-road interface.

When tipping begins, the bus pivots about the downhill tires, which become the only contact point with the road. For the bus to remain stable without toppling, the resultant moment about this pivot point must equal zero. This moment balance condition accounts for the gravitational force acting at the center of mass and determines the maximum angle at which the bus maintains contact with all tires without rotating over the downhill edge.

The center of mass location determines where the gravitational force acts on the bus. As the road angle increases, the gravitational force's line of action shifts relative to the downhill contact point. When this line passes beyond the downhill tires, the moment about that point causes tipping. The specific position of the center of mass directly influences the critical angle at which the bus loses stability and begins to tip over.

A bus traveling at constant velocity satisfies equilibrium conditions because acceleration is zero. This means the net force and net moment acting on the bus must be zero in all directions. By applying these equilibrium equations to the free-body diagram, you can solve for unknown forces and determine the maximum road angle. This approach simplifies the analysis by eliminating dynamic effects and focusing on static force balance.