Concepts: friction and the common forces in mechanics

4.1 Contact forces

Most everyday forces are contact forces: the normal reaction of a surface, tension in a string, the force of a spring, and friction. The normal reaction is perpendicular to the surface and adjusts itself so that the body does not sink into the surface. On a horizontal floor it equals the weight when nothing else pushes vertically, but on an incline it equals only the perpendicular component of the weight.

Tension pulls along the string and has the same magnitude throughout an ideal string of negligible mass with a frictionless pulley. A spring exerts a force proportional to its extension or compression, with the constant of proportionality called the spring constant. A free body diagram should show each of these forces at the point where it acts on the body, not the forces the body exerts on other objects.

4.2 Static and kinetic friction

Friction arises from interactions between the surfaces in contact. Static friction acts while there is no relative sliding and adjusts itself to balance the applied force, up to a maximum value. Once the applied force exceeds that maximum, the body starts to slide and kinetic friction takes over. The laws of friction are empirical: friction is proportional to the normal reaction, roughly independent of the area of contact, and the kinetic value depends on the pair of surfaces.

The maximum static friction is the coefficient of static friction times the normal reaction, and the kinetic friction is the coefficient of kinetic friction times the same normal reaction. Because the kinetic coefficient is usually the smaller of the two, a body is harder to start moving than to keep moving. The friction opposes the intended relative motion in both cases, and its direction must be fixed before the equations are written.

4.3 A block on an inclined plane

On an incline of angle theta the weight has a component mg sin theta down the slope and mg cos theta perpendicular to it. The normal reaction equals mg cos theta when no other perpendicular force acts, so the maximum static friction is the coefficient times mg cos theta. The block stays at rest while the downslope component of the weight stays below that maximum, which gives the angle of repose condition tan theta equal to the coefficient.

Once the angle exceeds the angle of repose, the block slides with acceleration g(sin theta minus the kinetic coefficient times cos theta). If a force is applied up the slope, the friction can act down the slope, and if the block is pushed up, kinetic friction acts down the slope. Fixing the direction of friction from the intended relative motion before writing the equations is what keeps the signs correct. If the applied force is not enough to move the block, the friction simply equals the downslope component and the acceleration stays zero, so the static case needs its own check before the sliding formula is used. A quick way to test is to compute the required friction and compare it with the maximum before concluding that motion begins. The maximum itself never depends on the applied force, only on the normal reaction and the nature of the two surfaces, so changing the push does not change the ceiling until sliding begins.

4.4 Worked example: measuring a friction coefficient

A block slides down a rough incline of 37 degrees and the angle of the incline is raised until the block just begins to slide. If that happens at 30 degrees, the coefficient of static friction is the tangent of 30 degrees, about 0.577. The measurement needs only a protractor and no force meter.

If the same block is then released at 45 degrees and slides with an acceleration of 2 metres per second squared with g equal to 10, the equation a equals g(sin 45 minus mu cos 45) gives mu equal to 1 minus a over g sin 45, which after substitution is about 0.72 for the kinetic coefficient. The two values are different because static and kinetic friction are different, and the problem must say which one is wanted.

4.5 Reducing friction

Friction cannot be removed from everyday machines, so it is reduced by design. Ball bearings replace sliding contact with rolling contact, whose resistance is much smaller. Air cushions and lubrication put a layer of fluid between the surfaces so that they never touch directly. Polishing the surfaces and choosing suitable materials also reduce the coefficient. The same friction that wastes energy in a machine is essential for walking, driving and braking, so the aim is control rather than elimination. A surface that is too slippery is dangerous for the same reason a surface that is too rough wastes fuel, and the design of tyres, shoe soles and machine parts all balances the two.