Flat Belts: Problem Solving

The belt-to-surface contact angle, measured in radians, is a critical input for determining belt tensions using the friction relationship. In this problem, the contact angle is 140 degrees, calculated from the system's geometry where the belt wraps around pulleys with different radii. This angle, combined with the coefficient of static friction of 0.4, is substituted into the tension expression to find T1 from the maximum allowable T2 of 1000 N.

When pulley B rotates clockwise, it creates a tension difference between the two ends of the belt—T2 minus T1. This tension difference acts at different radii on pulley A, generating a moment. The moment is calculated by multiplying the tension difference by the radius of pulley A, which is 30 cm. This moment represents the power transmission capability of the belt system.

A free-body diagram of pulley A is drawn showing both tension forces acting at the pulley's radius. Moment equilibrium requires that the net moment equals zero at static conditions. By substituting the calculated tension values and the pulley radius into the moment equation, the maximum moment is determined. In this case, the maximum moment on pulley A is 186.921 N·m.

The coefficient of static friction of 0.4 determines how much tension can be transmitted between the belt and pulley surfaces without slipping. This value is substituted into the belt tension relationship along with the contact angle to calculate T1 from the maximum allowable T2. A higher friction coefficient would allow greater tension transmission, while a lower coefficient would reduce it.

Pulley radius directly determines the moment arm for the tension forces. Pulley A has a radius of 30 cm, while pulley B has 10 cm. The larger radius of pulley A means that the same tension difference produces a greater moment. The maximum moment is calculated by multiplying the tension difference by pulley A's radius of 0.3 m.

The belt-to-surface contact angle is calculated from the system's geometry, including the radii of both pulleys and the angle between the belt and horizontal at the pulleys. In this problem, with pulley radii of 30 cm and 10 cm and a 20-degree angle to the horizontal, the resulting contact angle is 140 degrees. This geometric relationship is essential for accurate tension calculations.

When the maximum allowable tension T2 is 1000 N, the calculated value of T1 is 376.93 N. This T1 value is determined by substituting the belt-to-surface contact angle in radians and the coefficient of static friction of 0.4 into the friction-based tension relationship. The tension difference of 623.07 N between T2 and T1 then generates the maximum moment on pulley A.