Force and Translational Dynamics

A system is the object or group of objects you choose to study, and everything else is its environment; if the inside details don't matter, you can treat the whole system as a single object located at its center of mass. The center of mass is a mass-weighted average position (x_cm = Σmᵢxᵢ / Σmᵢ), and for a symmetric object it lies on the lines of symmetry.

A force is a push or pull that comes from an interaction between two objects, so nothing can push or pull on itself. Contact forces like the normal force, tension and friction come from electric forces between atoms. A free-body diagram shows each force on one object or system as a separate arrow starting from a dot; on the AP exam you draw whole forces, not their components.

When object A exerts a force on object B, B exerts a force of the same size in the opposite direction on A (F_A on B = −F_B on A), so these paired forces act on different objects and never cancel on one free-body diagram. Forces between parts inside a system can't change the motion of its center of mass, and an ideal (massless, non-stretching) string has the same tension all along it.

If the net force on a system is zero, its velocity stays constant: it stays at rest or keeps moving in a straight line at the same speed. This balanced state is called translational equilibrium, and forces can be balanced in one direction but not another, in which case the velocity changes only along the unbalanced direction.

When the net force on a system isn't zero, its center of mass accelerates in the direction of the net force, with a = F_net / m. To use this, draw a free-body diagram, pick one axis along the acceleration (for example, along the surface of an incline), and write ΣF = ma for each direction.

Any two masses attract each other with a force F = Gm₁m₂/r², where r is the distance between their centers. Near Earth's surface this force is your weight, F = mg, with g ≈ 9.8 N/kg (often rounded to 10 N/kg). Your apparent weight is the normal force (the scale reading) on you, so it changes when you speed up or slow down vertically, as in an elevator; experiments show that the mass that resists acceleration (inertial mass) equals the mass that feels gravity (gravitational mass).

Kinetic friction acts when two surfaces slide past each other, opposes that sliding, and has size F = μ_k F_N. Static friction acts when the surfaces don't slide: it takes whatever size and direction is needed to prevent slipping, up to a maximum of μ_s F_N, and μ_s is usually larger than μ_k for the same surfaces.

An ideal spring exerts a force proportional to how far it's stretched or compressed from its relaxed length, F = kΔx (Hooke's law), where the spring constant k is measured in N/m. The spring force always points back toward the equilibrium position, so a stiffer spring (bigger k) pushes or pulls harder for the same stretch.

An object moving in a circle at constant speed is still accelerating, because its direction keeps changing. This centripetal acceleration points toward the center with size a_c = v²/r, and some net inward force (tension, gravity, friction, the normal force or a mix) must cause it. The period T is the time for one lap (T = 1/f = 2πr/v), and for a circular orbit Kepler's third law links the period to the orbit's radius and the central mass (T² = 4π²r³/GM).