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The static friction angle, denoted as φ, is calculated using the coefficient of static friction between the screw and gear. Specifically, the static friction angle is the angle whose tangent equals the coefficient of static friction. In this problem, with a coefficient of 0.3, you determine φ by taking the inverse tangent of 0.3. This angle is fundamental to analyzing frictional forces on screws and determining whether the shaft will self-lock.

The lead angle is determined by the relationship between the screw's lead and its mean radius. It equals the ratio of the lead to the circumference of the shaft. For a screw with a 10-millimeter lead and 20-millimeter mean radius, you calculate the circumference and then find the lead angle. This angle is critical for comparing against the static friction angle to determine if upward impending motion will occur.

The axial force developed in the shaft is determined using a formula that incorporates the applied torsional moment, the static friction angle, the lead angle, and the mean radius of the screw. For upward impending motion, you substitute these values into the relationship between torque and axial force. The resulting axial force represents the force acting along the shaft's axis that causes the plate gear to rotate when the torsional moment is applied.

The resisting torque on the plate gear equals the product of the shaft's axial force and the mean radius of the gear. Once you calculate the axial force from the applied torsional moment, multiply it by the gear's mean radius of 35 millimeters to obtain the resisting torque. This resisting torque represents the maximum torque that can be overpowered by the applied torsional moment before the system reaches equilibrium.

A screw shaft is self-locking when the static friction angle is greater than the lead angle. In this condition, the shaft remains locked even if the applied torsional moment is removed. Self-locking occurs because friction prevents the screw from unwinding or rotating backward. This property is crucial in mechanical applications where you need to ensure the system maintains its position without continuous force application.

To determine if an applied torsional moment can overcome the resisting torque, compare the magnitude of the applied moment to the calculated resisting torque. If the applied torsional moment exceeds the resisting torque, the plate gear will rotate. In this problem, an 8 newton-meter moment is applied, and you calculate whether it surpasses the resisting torque determined from the axial force and gear radius.

The coefficient of static friction between the screw and gear determines the static friction angle, which directly influences whether the shaft self-locks and how much resisting torque develops. A higher coefficient increases the friction angle, making self-locking more likely. In this system with a coefficient of 0.3, friction opposes relative motion between the screw and gear, affecting the force transmission and the overall mechanical advantage of the system.