Newton’s laws and free-body diagrams

• Force is an interaction between bodies; forces are vectors with magnitude and direction.

• Newton’s first law: if the resultant force is zero, a body remains at rest or moves with constant velocity.

• Newton’s second law: resultant force equals rate of change of momentum: $F=\frac{\Delta p}{\Delta t}$; for constant mass, $F=ma$.

• Newton’s third law: forces occur in equal and opposite pairs acting on different bodies.

• A free-body diagram shows all external forces acting on one body only.

• In a free-body diagram, draw forces from the object, label clearly, and do not include forces the object exerts on other bodies.

• Resolve forces into components when needed: $F_x=F\cos\theta$, $F_y=F\sin\theta$.

• Equilibrium means resultant force = 0 in every relevant direction.

• In exam questions, start by choosing axes, drawing the free-body diagram, resolving components, then applying Newton’s laws.

This diagram shows a block on an inclined plane with the main forces labeled, making it ideal for practicing free-body diagrams. It helps you identify weight, normal reaction, and forces parallel/perpendicular to the surface. Use it to rehearse resolving forces on slopes. Source

Common forces you must recognize

• Weight / gravitational force: $F_g=mg$, always acts towards the centre of the Earth.

• Normal force $F_N$: contact force acting perpendicular to a surface.

• Friction acts parallel to the contact surface and opposes actual or impending motion.

• Static friction: $F_f\leq \mu_sF_N$.

• Dynamic (kinetic) friction: $F_f=\mu_dF_N$.

• Tension acts through a string / rope / cable and pulls away from the body.

• Elastic restoring force follows Hooke’s law: $F_H=-kx$.

• Viscous drag on a small sphere in a fluid: $F_d=6\pi\eta rv$.

• Buoyancy / upthrust: $F_b=\rho Vg$, equal to the weight of fluid displaced.

• Electric force and magnetic force are field forces; in this topic, know that they act without contact.

• Watch direction carefully: friction is not always opposite motion of the object overall; it is opposite the relative motion or tendency to move at the contact.

This diagram illustrates Archimedes’ principle and the origin of buoyant force from pressure differences in a fluid. It is useful for linking upthrust to the displaced fluid. Use it when reviewing floating and submerged-body force balances. Source

Momentum and impulse

• Linear momentum: $p=mv$.

• Momentum is a vector, so sign/direction matters in every conservation question.

• Conservation of momentum: total momentum remains constant unless a resultant external force acts on the system.

• Impulse: $J=F\Delta t$ where $F$ is the average resultant force during contact.

• Impulse equals change in momentum: $J=\Delta p$.

• A larger collision time gives a smaller average force for the same change in momentum.

• In isolated systems, use before = after for total momentum.

• Explosions also conserve momentum if external forces are negligible; initial momentum is often zero, so final momenta must balance vectorially.

• Elastic collision: momentum conserved and kinetic energy conserved.

• Inelastic collision: momentum conserved but kinetic energy not conserved.

• In perfectly inelastic collisions, bodies stick together after impact.

• Exam tip: momentum is always conserved in an isolated system; kinetic energy is only sometimes conserved.

• Syllabus guidance: SL collision calculations are limited to 1D; HL may also meet 2D collision calculations.

This diagram shows an elastic collision between two particles, highlighting the change in motion while preserving momentum and kinetic energy. It is helpful for visualizing before-and-after velocity directions. Use it to compare elastic and inelastic outcomes. Source

Circular motion

• A body moving in a circle has centripetal acceleration directed towards the centre.

• $a_c=\frac{v^2}{r}=\omega^2r=\frac{4\pi^2r}{T^2}$.

• The required centripetal force is the resultant inward force, not a new extra force.

• $F_c=m\frac{v^2}{r}=m\omega^2r$.

• The velocity is tangential to the circle; the centripetal acceleration is radially inward.

• A centripetal force changes the direction of velocity even when speed is constant.

• Relation between linear and angular quantities: $v=\omega r=\frac{2\pi r}{T}$.

• For circular-motion questions, identify the real inward force causing the motion: often tension, gravity, normal force, or friction.

• In vertical circles, speed may vary, so the resultant inward force changes around the path.

• No work is done by the centripetal force if it is always perpendicular to the instantaneous displacement.

This vector diagram shows that velocity is tangential while centripetal acceleration points toward the centre in circular motion. It is especially good for fixing the direction rules that students often mix up. Use it when interpreting motion in horizontal or vertical circles. Source

Exam method and traps

• Always define the system first before using momentum conservation.

• Check whether forces are internal or external to the chosen system.

• In force problems, write equations separately for each axis: for example, $\Sigma F_x=ma_x$ and $\Sigma F_y=ma_y$.

• In equilibrium questions, set acceleration = 0 and therefore resultant force = 0.

• Do not write centripetal force as an extra arrow on a free-body diagram; label the actual force(s) toward the centre.

• In collision questions, conserve momentum in all valid directions, but only conserve kinetic energy if the collision is explicitly elastic.

• In explosions, the momentum vectors after the explosion must add to the original total momentum.

• Distinguish mass constant form $F=ma$ from the more general form $F=\frac{\Delta p}{\Delta t}$.

This infographic gives a quick visual summary of Newton’s three laws of motion with everyday examples. It is useful for checking the meaning of each law and linking them to real situations. Use it as a final conceptual review before exam questions. Source