Friction

Before calculating friction, you must find the normal reaction force R — the force the surface exerts perpendicular to itself on the object. On a horizontal surface with no vertical applied force, R equals the weight: R = mg. On an inclined plane at angle θ, the component of weight perpendicular to the plane gives R = mg cos θ. If an additional force has a perpendicular component (e.g. pushing down at an angle), that must be included in R as well.

When an object is at rest, static friction F_s adjusts to exactly balance the applied force trying to move it, up to a maximum value. The condition is F_s ≤ μ_s R, where μ_s is the coefficient of static friction. The object remains stationary as long as the applied force does not exceed μ_s R. At the point of limiting equilibrium (on the verge of moving), F_s = μ_s R. This maximum value is always greater than kinetic friction for the same surfaces.

Once an object is sliding, kinetic friction F_k = μ_k R opposes the direction of motion and remains approximately constant regardless of speed (within the standard model). Here μ_k is the coefficient of kinetic friction, and typically μ_k < μ_s for the same pair of surfaces. The direction of kinetic friction is always opposite to the velocity of the object relative to the surface.

Once you know the friction force (static or kinetic), apply Newton's second law F_net = ma in the direction of motion. For a horizontal surface: F_applied − F_friction = ma. For an inclined plane: mg sin θ − μ_k mg cos θ = ma, which simplifies to a = g(sin θ − μ_k cos θ). Always draw a free-body diagram first, resolving all forces along and perpendicular to the surface before applying the equations.

A crucial skill is determining whether an object actually moves. Compare the applied force (or the driving force component along the surface) to the maximum static friction μ_s R. If the applied force is less than or equal to μ_s R, the object stays still and F_friction equals the applied force. If it exceeds μ_s R, the object accelerates and you use kinetic friction μ_k R in Newton's second law.

• F — Friction force (Newtons (N)): The force opposing relative sliding motion between two surfaces, or the maximum static friction at the point of limiting equilibrium.
• \mu — Coefficient of friction (Dimensionless): A dimensionless constant depending on the materials of the two surfaces in contact. Use μ_s for static friction and μ_k for kinetic friction; both are typically between 0 and 1, though values above 1 are possible for very rough surfaces.
• R — Normal reaction force (Newtons (N)): The force exerted by the surface perpendicular to itself on the object. On a horizontal surface R = mg; on a slope at angle θ, R = mg cos θ.

Problem. A block of mass 5 kg is pulled along a rough horizontal floor by a horizontal force of 20 N. The coefficient of kinetic friction between the block and the floor is μ_k = 0.3. Find (a) the friction force acting on the block, and (b) the acceleration of the block. (Take g = 9.8 m/s².)

Answer. Friction force = 14.7 N; Acceleration = 1.06 m/s²

• Newton's Second Law — Friction appears as a force in F_net = ma. You must combine the friction force with all other forces to find the net force and hence the acceleration of an object on a rough surface.
• Equilibrium of Forces — When an object on a rough surface is stationary (static friction), the forces balance and the system is in equilibrium. Understanding equilibrium is essential for limiting-friction problems.
• Motion on Inclined Planes — Friction on a slope requires resolving the weight into components parallel and perpendicular to the plane, and modifies the acceleration formula to a = g(sin θ − μ cos θ).
• Connected Particles — Problems with two masses linked by a string often involve one mass on a rough surface. Friction on that mass must be included when writing equations of motion for the system.