Free-Body Diagrams, Equilibrium and Friction

Friction on inclined plane

Friction on an inclined plane is a self-adjusting force. A body at rest on a slope has static friction whose magnitude can take any value up to a limit, F_f \le \mu_s N. Resolve the weight into components parallel and perpendicular to the plane. The normal force is N = mg\cos\theta when no other force has a perpendicular component, and the component along the plane is mg\sin\theta. For an external force F acting parallel to the plane, equilibrium is possible if and only if |mg\sin\theta - F| \le \mu_s mg\cos\theta. This defines a band of admissible forces, and any force outside the band cannot hold the body at rest.

For the smallest force that prevents sliding down the plane, friction acts up the plane at its limiting value. The minimum force is F_{min} = mg(\sin\theta - \mu_s\cos\theta), and the upper limit of the band is mg(\sin\theta + \mu_s\cos\theta).

When two bodies touch along an inclined interface, the friction at that interface acts along the interface, and its limiting value is \mu N, where N is the normal force on that interface. The normal force is not necessarily the weight of the upper body alone, so both bodies must be resolved along and perpendicular to the interface.

When a body is pulled up a ramp at constant speed, kinetic friction acts down the ramp. The required pull is T = mg(\sin\theta + \mu\cos\theta), and a rope driven by a pulley of radius r requires torque \tau = Tr.

• impending motion — the limiting state in which friction has reached \mu_s N and the body is about to slide.
• kinetic friction — friction during sliding, F = \mu_k N. When the static and kinetic coefficients are taken equal, their limiting values coincide.
• Example: A 5 kg body on a 30° slope with \mu_s = 0.2 and g = 10\ \text{m/s}^2 has mg\sin\theta = 25 N and \mu_s mg\cos\theta \approx 8.7 N. It stays at rest for a parallel force between about 16.3 N and 33.7 N.

Free-body diagram of supported structure

A free-body diagram (FBD) is obtained by isolating the body and replacing each support or connection with the reaction it can exert. The type of support determines the number and direction of the unknowns. A smooth surface gives only a normal reaction. A pin gives two force components, a roller or smooth slot gives one perpendicular reaction, and a fixed support gives two force components plus a couple. A string or cable carries a tension along its own line, a linear spring carries a force k\delta along its axis, and a spiral spring gives a restoring couple.

The unknowns are then found from \sum F_x = 0, \sum F_y = 0 and \sum M = 0. Drawing every reaction with its correct line of action is the step that matters most in choosing the correct diagram.

A rigid bar that rotates about a hinge has displacements proportional to distance from the hinge, so a point twice as far from the hinge moves twice as much. Taking moments about the hinge converts the applied load into a spring force, and the stiffness follows from k = F/\delta at the spring point.

A frictionless support contributes only a normal reaction, and a straight pin-ended link that carries no load between its ends transmits force only along its axis. Once these reactions are included in the FBD, the vertical balance of the loaded body fixes the spring force, and the elongation follows from s = F/K.

Two-force member

A two-force member is a body loaded only at two points, with no other forces and no couple acting on it. Summing forces and moments shows that the two end forces must be equal in magnitude, opposite in direction and collinear, so they act along the line joining the two points. The shape of the member does not matter, and a bent or curved link can still be a two-force member. Recognising this early reduces the unknowns to a single axial force.

In a lever-type tool that rotates about a pivot, moment equilibrium about the pivot relates the handle force to the jaw force. The moment of each force about the pivot must be equal and opposite, so the jaw force times its arm equals the handle force times its arm. The mechanical advantage is therefore the ratio of the two arms.