Friction and Circular Motion: Laws, Formulas and PYQs

Friction and circular motion are the two parts of mechanics a student meets every day without naming them: the grip of a shoe on the floor, the squeal of a brake, the lean of a cyclist on a bend, the raised outer rail on a curved railway track. They belong together in one chapter because the force that turns a car on a level road is friction itself. Examiners ask this chapter in two ways, as a definition and as a one-step formula, so the fastest marks here come from knowing what each force is called, which way it points, and the four or five relations that connect them.

Friction: what it is and why it exists

Friction is the force that opposes relative motion, or the tendency of relative motion, between two surfaces in contact. It always acts along the surfaces, that is tangentially, and always against the direction in which the body moves or tries to move. No surface is truly smooth: seen closely, every surface is a landscape of ridges and hollows, and the real area of contact is only a tiny part of the apparent area. Friction arises from the interlocking of those irregularities and from the attraction between the molecules that touch at those few points, which is why a highly polished surface can show more friction rather than less once the molecules come very close.

Types of friction and the laws

• Static friction acts when a body at rest is pushed but does not move. It is self-adjusting: it grows exactly as much as the applied force, up to a maximum.
• Limiting friction is that maximum value, the friction just before the body begins to slide, and it equals the coefficient of static friction multiplied by the normal reaction.
• Kinetic or dynamic friction acts once the body is sliding and is less than limiting friction, which is why a box is harder to start than to keep moving.
• Rolling friction acts when a body rolls over a surface and is much smaller than sliding friction, which is the whole reason for the wheel.

The experimental laws are short. Limiting friction is proportional to the normal reaction between the surfaces. It does not depend on the apparent area of contact, so a brick dragged on its broad face and on its narrow face needs the same pull. It depends on the nature and roughness of the two surfaces. Kinetic friction is nearly independent of speed at ordinary speeds, and the coefficient of kinetic friction is smaller than the coefficient of static friction. The coefficient of friction is a ratio of two forces and therefore has no unit and no dimensions.

Angle of friction and angle of repose

If the normal reaction and the limiting friction are added as vectors, the resultant makes an angle with the normal that is called the angle of friction, and the tangent of that angle equals the coefficient of static friction. The angle of repose is the angle to which an inclined plane must be raised before a body placed on it just begins to slide down, and its tangent also equals the coefficient of static friction. The two angles are therefore numerically equal, a result that is asked directly in the paper.

Friction in use

Friction is both a servant and a tax. Without it we could not walk, since walking depends on the backward push of the foot gripping the ground; brakes could not stop a wheel, a nail would not hold in wood, a belt would not drive a pulley, and a pencil would not mark paper. Against that, friction wears out machine parts, turns useful work into heat and wastes a large part of the energy supplied to an engine. It is reduced by lubricants, which replace solid contact with a fluid film, by ball bearings, which turn sliding friction into rolling friction, by polishing, and by streamlining a body so that the fluid parts smoothly around it. It is increased where grip is wanted, by cutting treads into tyres, by spreading sand on a slippery track and by roughening a handle.

Circular motion

A body moving on a circular path at constant speed is still accelerating, because velocity is a vector and its direction changes at every instant. That acceleration points towards the centre of the circle and is called centripetal acceleration; its magnitude is the square of the speed divided by the radius, which is also the square of the angular velocity multiplied by the radius. The force that produces it, the centripetal force, is the mass multiplied by that acceleration and also points to the centre. The important question in any numerical problem is what supplies that force: gravity for a satellite, the tension of a string for a stone whirled in a sling, the electrostatic attraction of the nucleus for an electron in the Bohr model, and friction between the tyres and the road for a car taking a turn.

Centrifugal force, banking and leaning

A passenger in a turning bus feels thrown outwards. That outward force, the centrifugal force, is not a force acting on the passenger in the ground frame at all; it is a pseudo force that has to be introduced only when the motion is described from inside the rotating frame, and it is equal in size and opposite in direction to the centripetal force. The real explanation in the ground frame is inertia: the passenger tends to continue in a straight line while the bus turns under them. Machines such as the centrifuge, the cream separator and the spin dryer are explained using centrifugal effects. Because friction alone is often not enough to turn a fast vehicle, a road is banked, that is tilted inward on a curve, so that a component of the normal reaction supplies the centripetal force; for the same reason the outer rail of a curved railway track is raised above the inner rail, and a cyclist leans inward on a bend so that the reaction of the ground has a horizontal component pointing to the centre of the turn.