Forces and momentum

A free-body diagram isolates one object or chosen system and replaces every external interaction with a labelled force arrow. Draw the object as a simple dot or box. Arrows should begin on the object, point in the force direction, and be labelled by force type rather than by motion.

Common contact forces include normal force $N$, tension $T$, friction $f$, drag, buoyancy and spring force. Field forces act without direct contact: gravity produces weight $W=mg$, electric fields exert forces on electric charges, and magnetic fields exert forces on magnets, electric currents and moving charges. Include each relevant force on the chosen object's free-body diagram. Velocity and acceleration arrows belong on a separate motion sketch because they are not forces.

A system boundary can simplify a multi-object problem. When two blocks are treated as one system, their mutual contact forces are internal and cancel in the system equation. If one block is isolated, that contact force becomes external and must appear.

In an inertial reference frame, Newton's first law states that an object remains at rest or moves with constant velocity unless acted on by a resultant external force. Both rest and constant-velocity motion are states of translational equilibrium because acceleration is zero; zero resultant force alone does not establish rotational equilibrium for an extended body.

Newton's second law is vectorial. Apply it separately along chosen axes: $F_{\mathrm{net},x}=ma_x$ and $F_{\mathrm{net},y}=ma_y$. A zero resultant along one axis does not require zero resultant along another.

Newton's third law states that when object A exerts a force on object B, B simultaneously exerts an equal-magnitude, opposite-direction force of the same interaction type on A. The pair acts on different objects, so the forces do not cancel on either object's free-body diagram.

Choose axes that make the physics simple. On an inclined plane, take one axis parallel to the surface and the other perpendicular to it. The normal force then lies entirely on the perpendicular axis and friction lies on the parallel axis.

Weight remains vertical. Resolving $mg$ gives $mg\sin\theta$ down the plane and $mg\cos\theta$ into the plane. These are components of one weight force, not two extra forces. If there is no acceleration perpendicular to the surface, the normal force balances the net inward component.

Friction points opposite the actual or impending relative motion. A block sliding down a slope experiences friction up the slope; a block being pulled upward can experience friction down the slope.

Qualitative schematic: arrow lengths and drawn geometry are not to scale. The block is sliding, or tending to slide, down the plane, so friction acts up the plane. Purple arrows are real forces. Green dashed arrows resolve the single weight vector into $mg\sin\theta$ down the slope and $mg\cos\theta$ into it. Use these components instead of $mg$ in the chosen axes; do not count all three as forces.

The normal force is a contact force perpendicular to a surface. It prevents interpenetration and takes the value required by the perpendicular motion. It is not automatically equal to weight.

Static friction self-adjusts from zero up to $f_{s,\max}=\mu_sN$. It equals whatever value is needed to prevent slipping until this limit is reached. Once surfaces slide, the kinetic-friction model $f_k=\mu_kN$ is commonly used, with friction opposite relative motion.

Tension pulls along a taut string or cable. For an ideal massless string over an ideal frictionless pulley, the tension magnitude is the same throughout. Connected objects share a kinematic constraint, so their acceleration magnitudes are linked.

A spring exerts a force opposing its extension or compression from its natural (unloaded) length. Within its linear range, the force magnitude is proportional to this deformation. The spring constant $k$ measures stiffness: a larger $k$ means more force is required for the same extension or compression.

Along the spring axis, $\vec F_s=-k\vec x$, where $\vec x$ represents extension or compression relative to the natural length. The minus sign gives the opposing direction; the force magnitude is $F_s=k|\vec x|$. A spring can exert a non-zero force when a loaded system is in equilibrium: for a hanging mass, its upward force balances weight. Displacement from that loaded equilibrium describes the net restoring force of the combined forces, not the spring force alone.

Hooke's law is a model with a limit. Beyond the limit of proportionality the force–extension graph is no longer straight, and after the elastic limit the spring may not return to its original length.

Drag opposes an object's velocity relative to a fluid and usually increases with relative speed. It depends on shape, cross-sectional area and fluid properties, so acceleration is generally non-uniform. In slow laminar flow around a small sphere, Stokes' law provides a specific viscous-drag model.

Buoyancy is the upward force caused by fluid pressure increasing with depth. Archimedes' principle gives $F_b=\rho_{\mathrm{fluid}}V_{\mathrm{displaced}}g$: the buoyant force equals the weight of displaced fluid.

For steady vertical falling through a still fluid, with weight, buoyancy and drag as the only forces, the upward drag and buoyancy balance weight at terminal velocity. The resultant and acceleration are then zero while the object continues downward at constant non-zero velocity. If the fluid itself moves, determine the drag direction from velocity relative to the fluid before writing the force balance.

Momentum $\vec p=m\vec v$ combines inertia and motion. Because velocity is a vector, momentum signs and directions matter. In one dimension, choose a positive direction before calculating $\Delta p=p_f-p_i$.

The impulse of an individual force is its time integral, or the signed area under its force–time graph in one dimension. The net external impulse equals the system's change in momentum: $\vec J_{\mathrm{net}}=\Delta\vec p=\vec F_{\mathrm{net,avg}}\Delta t$. Add the impulses of all external forces, or use the resultant-force graph directly. Net impulse has the direction of the momentum change.

The same momentum change can be produced by a large force for a short time or a smaller force for a longer time. This is why airbags, helmets and crumple zones reduce average collision force by increasing stopping time.

For an isolated system with negligible external impulse, total momentum is conserved. Internal forces can be very large during a collision, but their impulses are equal and opposite, so they do not change the system's total momentum.

Choose a sign convention and keep every velocity signed. For SL, quantitative collision and explosion problems are one-dimensional. In two dimensions the same physical conservation law applies separately in perpendicular components; quantitative two-dimensional applications are required at HL. Include all objects in the chosen system both before and after the interaction.

Momentum is conserved in both elastic and inelastic collisions when net external impulse is negligible. Total kinetic energy is conserved in an elastic collision. When objects stick together, the collision is perfectly inelastic and kinetic energy is transferred to deformation, thermal energy and sound. In an explosion, stored internal energy can become kinetic energy of the separating parts, even when their total momentum remains unchanged; total energy is conserved throughout.

In uniform circular motion, speed is constant but velocity changes direction. Acceleration is then purely radial, toward the centre, and perpendicular to the tangential velocity. If the speed changes, there is also a tangential acceleration $a_t=\frac{\mathrm{d}v}{\mathrm{d}t}$. The radial component remains $a_c=\frac{v^2}{r}$ at the instantaneous speed, but the total acceleration is no longer purely centripetal.

The required inward resultant is $F_{\mathrm{radial}}=\frac{mv^2}{r}$. “Centripetal force” is not a new type of force: it is the inward resultant supplied by real forces such as tension, gravity, friction or normal contact.

Choose the inward radial direction as positive and add the components of real forces along it. For a mass on a taut string in a vertical circle, at the top $T+mg=\frac{mv_{\mathrm{top}}^2}{r}$; at the bottom $T-mg=\frac{mv_{\mathrm{bottom}}^2}{r}$. The speeds need not be equal. Use the instantaneous speed at each point and check that the calculated tension is non-negative; a string cannot push. These top and bottom balances apply whether the circular motion is uniform or non-uniform.

Safety systems apply impulse physics. For the same initial and final momentum, increasing stopping time lowers average force. Seat belts spread force across stronger body regions, airbags increase stopping time and area, helmets deform, and vehicle crumple zones absorb energy while extending the collision.

Force investigations should connect measurements to a model. A force sensor and motion sensor can test $F=ma$; a force–extension graph can determine $k$; a force–time graph can be integrated for impulse; and colliding carts can test momentum conservation. Record uncertainty, use best-fit relationships and identify systematic effects such as track tilt or friction.

For every problem: define the system, draw a force or before/after momentum diagram, choose axes, list signed values, write one equation per independent direction, solve, then interpret direction and units. Check whether assumptions such as constant mass, ideal string, negligible external impulse or uniform circular motion are justified.