A.2 Forces and momentumIB Physics HL: Revision notes

A force is an interaction between two bodies.

• Newton's first law: a body remains at rest or moves with constant velocity unless a resultant external force acts on it.
• Newton's second law: the resultant force equals the rate of change of momentum, $F = \frac{\Delta p}{\Delta t}$. For constant mass this becomes $F = ma$.
• Newton's third law: if body A exerts a force on body B, then B exerts a force of equal magnitude and opposite direction on A. The pair act on different bodies and are of the same type.

A free-body diagram shows all the forces on one body as arrows from the body; adding them as vectors gives the resultant force. Resolve forces into perpendicular components when they are not in line.

Contact forces:

• Normal force $F_N$: the component of the contact force perpendicular to the surface.
• Friction $F_f$: parallel to the surface. Static: $F_f \le \mu_s F_N$ (it takes whatever value is needed, up to a maximum). Dynamic (sliding): $F_f = \mu_d F_N$, usually with $\mu_d < \mu_s$.
• Tension: the pulling force in a string or cable.
• Elastic restoring force (Hooke's law): $F_H = -kx$, where k is the spring constant and the minus sign shows it opposes the extension.
• Viscous drag on a small sphere: $F_d = 6\pi\eta r v$.
• Buoyancy: $F_b = \rho V g$, where ρ is the fluid density and V the volume of fluid displaced.

Field forces act without contact:

• Gravitational force (weight) $F_g = mg$
• Electric force $F_e$ between charges
• Magnetic force $F_m$ on moving charges and currents

They appear on free-body diagrams in the same way as contact forces.

Linear momentum $p = mv$ is a vector (unit kg m s⁻¹ or N s). The total momentum of a system is constant unless a resultant external force acts — the conservation of momentum.

An impulse $J = F\Delta t$ (F the average resultant force) equals the change in momentum. The same change in momentum over a longer contact time means a smaller average force — the principle behind airbags, crumple zones and bending your knees when landing.

$F = ma$ assumes constant mass; $F = \Delta p/\Delta t$ also works when the mass changes (for example a rocket ejecting gas, or sand dropping onto a conveyor belt).

In every collision and explosion momentum is conserved (for an isolated system). Kinetic energy is a different matter:

• Elastic collision: total kinetic energy is conserved.
• Inelastic collision: some kinetic energy is transferred to other forms (internal energy, sound, deformation). If the bodies stick together, the loss of kinetic energy is the maximum possible.
• Explosion: bodies initially at rest (total momentum zero) push apart; they move off with equal and opposite momenta, and kinetic energy increases, supplied from stored energy (chemical or elastic).

The lighter fragment of an explosion gets the larger speed and the larger share of kinetic energy, since $E_k = p^2/2m$.

A body moving in a circle at constant speed is accelerating, because its direction (and so its velocity) changes. The centripetal acceleration points towards the centre:

$a = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2 r}{T^2}$

The angular velocity ω is linked to the speed by $v = \frac{2\pi r}{T} = \omega r$.

The acceleration is caused by a centripetal force — the resultant force, directed towards the centre and perpendicular to the velocity. It changes the direction of motion but not the speed. It is always provided by a real force: friction for a car on a flat bend, tension for a ball on a string, gravity for a satellite.