Forces are interactions, not possessions

A force is a vector interaction that can change a body’s velocity or deform it. An object doesn’t “have” a force; one body exerts a force on another. That wording is worth getting right, because every force needs an agent and a body that experiences the force.

A contact force acts only when bodies touch. A field force acts between bodies without contact, through a physical field such as a gravitational, electric or magnetic field. A swimmer, for example, has contact forces from the water and a field force from Earth; a charged particle in space can be deflected without touching anything.

Newton’s three laws

Newton’s first law is a law of motion stating that a body remains at rest or moves with constant velocity unless acted on by a resultant external force. Constant velocity means constant speed in a constant direction. If a hockey puck slows down, a force is acting. If a spacecraft coasts at constant velocity, no resultant force is needed to keep it going.

Newton’s second law is a law of motion stating that the resultant force on a body is proportional to the rate of change of its velocity. For constant mass, we write this as F = m**a, where F is the resultant force (N), m is the mass (kg) and a is the acceleration (m s⁻²). The force and acceleration point in the same direction.

A newton is an SI derived unit of force equal to the force required to give a 1 kg mass an acceleration of 1 m s⁻² in the direction of the force. In base units, 1 N = 1 kg m s⁻².

The form F = m**a assumes the object’s mass is constant. Later in this topic, we use the momentum form of the same law, which is more general.

Newton’s third law is a law of interaction stating that when body A exerts a force on body B, body B simultaneously exerts an equal-magnitude opposite-direction force of the same type on body A. The two forces act on different bodies, so they do not cancel each other.

A good test for third-law pairs is this: same interaction, same type of force, different bodies. The weight of a book is Earth pulling on the book; the partner force is the book pulling gravitationally on Earth. The normal force from the table on the book is not the partner force to the book’s weight. It comes from a different contact interaction.

Free-body diagrams and resultants

A free-body diagram is a simplified force diagram that shows only the forces acting on one chosen body. Draw the body as a dot or small block. Put force arrows from the centre of mass unless the point of application matters, label every force with an accepted name or symbol, and make sure the arrow directions make physical sense.

For this syllabus, free-body diagrams and resultant-force analysis are limited to one- and two-dimensional situations. A sensible routine is:

• choose the body, not the whole story;
• include only forces on that body;
• label forces, for example weight, normal force, tension or friction;
• resolve forces into perpendicular components when the directions are awkward;
• add the vector components to find the resultant force.

A resultant force is the vector sum of all forces acting on a body. If the resultant force is zero, the acceleration is zero. The body might be stationary, or it might be moving at constant velocity.

Translational equilibrium

Translational equilibrium is a state of a body in which the resultant force is zero, so the body has no linear acceleration. Newton’s first law applies directly here: a body in translational equilibrium is either at rest or moving with constant velocity.

For two forces to produce equilibrium, they must be equal in magnitude, opposite in direction and along the same line of action. For three forces, resolve components horizontally and vertically, or draw the vectors head-to-tail. If the force vectors form a closed triangle, the resultant is zero.

In a practical triangle-of-forces experiment, three strings meet at a small ring, then the ring is adjusted until it is stationary. Masses provide known weights, and the angle of one string is measured. Repeating for different masses and plotting a processed quantity such as cos θ is a good way to connect a vector diagram with data. The quiet experimental lesson is that equilibrium is not a slogan; it is a component-by-component test.

Normal force and tension

A normal force is the component of a contact force that acts perpendicular to the surface pressing on a body. We write it as F_N. Don’t assume the normal force equals the weight. It changes with the situation: in a lift accelerating upward it is larger than the person’s weight, while on a slope it is usually smaller than the full weight.

Tension is a contact force carried through a stretched string, rope, cable or chain. In ideal IB questions, a light inextensible string passing over a frictionless pulley has the same tension throughout. I often label it F_T, where F_T is the tension force (N), so it doesn’t get mixed up with the period symbol used later in circular motion.

Weight and other field forces

A gravitational force is a field force caused by mass attracting mass. Close to Earth’s surface, the weight of a body is F_g = m**g, where F_g is the gravitational force or weight (N) and g is the gravitational field strength near Earth (N kg⁻¹), numerically equal to the acceleration of free fall (m s⁻²). In free fall with no resistance, all bodies have the same acceleration because gravitational and inertial mass are experimentally equivalent.

An electric force is a field force exerted on a charged body by an electric field. A magnetic force is a field force exerted on magnets, currents or moving charged particles by a magnetic field. We model the motion of a mass in a gravitational field and the motion of a charged particle in an electric field in the same Newtonian way: find the field force, then use Newton’s second law to predict the acceleration. Magnetic forces have one important difference. They often act perpendicular to velocity, so they can bend a charged particle’s path without changing its speed.

Friction

A surface frictional force is a contact force parallel to the plane of contact. It opposes relative motion, or the tendency for relative motion, between two surfaces. Static friction acts before sliding; dynamic friction acts during sliding.

For a stationary body, F_f ≤ μ_sF_N, where F_f is the surface frictional force (N), μ_s is the coefficient of static friction (unitless). The inequality matters: static friction takes whatever value is needed, up to a maximum. Once sliding occurs, F_f = μ_dF_N, where μ_d is the coefficient of dynamic friction (unitless). Usually μ_d is smaller than μ_s.

A simple way to measure μ_s is to place a block on a ramp, then increase the angle until the block is just about to slip. Resolving the weight gives the limiting condition tan θ = μ_s, where θ is the angle of the ramp to the horizontal (rad or degrees as stated). For dynamic friction, a force sensor pulling a block at constant speed gives F_f directly; a graph of friction force against normal force has gradient μ_d.

The friction equations are empirical. They summarize the experimental behaviour of real surfaces rather than coming from a detailed atomic model. Even polished surfaces have tiny peaks and hollows under a microscope; static friction involves deforming or breaking these contacts, and lubrication reduces friction by separating the surfaces.

Elastic restoring force

An elastic restoring force is a contact force exerted by a deformed elastic object, in a direction that tends to restore its original shape. For an ideal spring obeying Hooke’s law, F_H = −k**x, where F_H is the elastic restoring force (N), k is the spring constant (N m⁻¹) and x is the displacement from the spring’s natural length (m). The minus sign is doing work here: it says the force is opposite to the displacement.

A spring constant is a proportionality constant that measures the force needed per unit extension of a spring. On a graph of force against extension, the gradient is k while Hooke’s law is valid. In the lab, load and unload the spring, then check whether the two sets of readings agree; any disagreement hints at hysteresis or measurement problems.

The link to simple harmonic motion is direct. A restoring force that is proportional to displacement and always directed toward equilibrium produces an acceleration back toward the equilibrium position. With little damping, the mass overshoots, reverses, and oscillates.

Viscous drag and fluid resistance

A fluid resistance force is a resistive force exerted by a gas or liquid on a body moving through it. A viscous drag force is a fluid resistance force caused by viscosity, acting opposite the relative motion through the fluid.

For a small smooth sphere moving slowly through a uniform fluid, Stokes’ law gives F_d = 6πηrv, where F_d is the viscous drag force (N), η is the dynamic viscosity of the fluid (Pa s), r is the radius of the sphere (m) and v is the velocity of the sphere relative to the fluid (m s⁻¹). This model assumes laminar flow, a smooth sphere, a homogeneous fluid and no interaction between particles. At higher speeds, turbulent drag is often closer to being proportional to speed squared, so do not use Stokes’ law blindly.

A falling sphere in a liquid has weight downward, buoyancy upward and drag upward. As the sphere speeds up, the drag increases until the resultant force becomes zero; it then moves at terminal velocity. A viscosity experiment uses timing marks or light gates after the sphere has already reached terminal speed, because timing the acceleration phase would not test Stokes’ terminal-speed relationship.

Buoyancy

A buoyancy force is an upward force exerted by a fluid on a body because fluid pressure is greater at lower depth than at higher depth. It is given by F_b = ρVg, where F_b is the buoyancy force (N), ρ is the density of the fluid (kg m⁻³) and V is the volume of fluid displaced (m³).

Archimedes’ principle says that the buoyancy force on a completely or partly submerged body is equal to the weight of the displaced fluid. A floating body is in equilibrium, so its weight equals the buoyancy force. That is why the fraction of a floating object below the surface equals the ratio of the object’s density to the fluid’s density.

Ships carry load-line marks because seawater density changes with salinity and temperature. Submarines change their average density by taking water into, or pushing water out of, ballast tanks. Hot-air balloons rise for the same underlying reason: the air displaced weighs more than the balloon system.

Momentum

Linear momentum is a vector quantity found by multiplying mass by velocity. It is given by p = m**v, where p is linear momentum (kg m s⁻¹). Momentum points in the same direction as velocity; mass just scales the vector.

The total linear momentum of a system stays constant unless a resultant external force acts on that system. Internal forces may move momentum from one part of the system to another, but they cannot change the total momentum of the whole system.

Impulse

Impulse is a vector quantity found by multiplying the average resultant force by the time interval for which it acts. It is given by J = FΔt, where J is impulse (N s) and Δt is the time interval or contact time (s). Since impulse equals change in momentum, 1 N s is equivalent to 1 kg m s⁻¹.

The impulse-momentum theorem is J = Δp, where Δp is the change in momentum (kg m s⁻¹). Put simply: the area under a resultant force-time graph gives the change in momentum.

That is why seat belts, airbags, crash mats and sports follow-through matter. For the same change in momentum, a longer stopping time gives a smaller average force. The momentum change hasn’t vanished; it has been spread out and made less brutal.

A neat practical way to estimate the force of a kick is to measure the contact time using foil contacts and a timer, then find the ball’s launch speed using projectile motion from a bench. Divide the ball’s change in momentum by the contact time to get the average force. Realistic improvements include video analysis, repeated trials, a stiffer timing circuit contact, and careful measurement of the launch height and horizontal range.

Newton’s second law in momentum form

Newton’s second law can be written more generally as F = Δp/Δt. This version still works when mass is changing, while F = m**a is the constant-mass special case.

For a rocket, the exhaust gains momentum in one direction, so the rocket gains momentum in the opposite direction. The rocket does not push on the atmosphere; it can accelerate in space. If exhaust of relative speed u_e is ejected at a mass rate Δm/Δt, the thrust magnitude is approximately F_thrust = u_e(Δm/Δt), where F_thrust is the thrust force (N), u_e is the exhaust speed relative to the rocket (m s⁻¹) and Δm/Δt is the mass ejection rate (kg s⁻¹).

The same idea shows up when water leaves a hose, as a force on the nozzle. If water speeds up from the hose to the nozzle, its momentum changes each second; the hose feels an equal and opposite force. Firefighters know this without needing the equation.

Measuring momentum conservation

A school momentum experiment usually uses dynamics carts, air tracks or video analysis. Measure the masses and the velocities immediately before and after the collision. To reduce external forces, level the track or give it a slight incline to compensate friction, use low-friction carts or air pucks, and measure velocities close to the collision so small resistive forces have little time to act.

The conclusion should always mention uncertainty. If the initial and final total momentum values agree within experimental uncertainty, the result supports conservation of momentum. If they don’t, suspect external forces, calibration, timing, alignment or rotation first — not that the universe has quietly abandoned momentum conservation.

Isolated systems and conservation

An isolated system is a chosen collection of bodies where the resultant external force is zero, or small enough to ignore, during the interaction being studied. In this case, total linear momentum is conserved.

In one dimension, conservation of momentum is written m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂, where m₁ is the mass of body 1 (kg), u₁ is the initial velocity of body 1 (m s⁻¹), m₂ is the mass of body 2 (kg), u₂ is the initial velocity of body 2 (m s⁻¹), v₁ is the final velocity of body 1 (m s⁻¹) and v₂ is the final velocity of body 2 (m s⁻¹). Pick a positive direction before putting in values; the signs carry the physics.

Elastic and inelastic collisions

A collision is a short interaction in which bodies exert large forces on each other and exchange momentum. In a perfectly elastic collision, total kinetic energy is conserved as well as momentum. An inelastic collision conserves momentum, but kinetic energy is transferred to other forms such as internal energy, deformation or sound. A perfectly inelastic collision is an inelastic collision where the bodies stick together and move with one common final velocity.

For kinetic-energy checks, E_k = ½m**v², where E_k is kinetic energy (J). If the total kinetic energy before and after matches, the collision is elastic; if it falls, the collision is inelastic. You don't need to solve simultaneous momentum-and-energy equations for collision speeds, but you should be able to use energy to classify or compare collisions.

For a perfectly inelastic collision where body 2 is initially stationary and the bodies stick, v_c = m₁u₁/(m₁ + m₂), where v_c is the common final velocity (m s⁻¹). Momentum is conserved, but kinetic energy is not.

Explosions

An explosion is an interaction where internal energy is converted into kinetic energy of separating parts while total momentum remains conserved. If the system starts from rest, the final momenta of the parts add to zero. With no external impulse, two fragments move in opposite directions with equal-magnitude momenta.

This is recoil. A gun and shell begin with zero total momentum. After firing, the shell has forward momentum, and the gun has equal-magnitude backward momentum. The shell has much more kinetic energy because its speed is much larger; equal momentum does not mean equal kinetic energy.

The same conservation idea appears in nuclear power stations. Fast neutrons transfer momentum and energy when they collide with moderator atoms, slowing the neutrons so they are more likely to cause further fission. Steam striking turbine blades is another momentum-transfer process, turning the kinetic energy of moving fluid into rotation.

Momentum, energy and uncertainty

Conservation laws are useful because they work across scales: carts in the lab, gas molecules, nuclear particles and astronomical systems. Equilibrium and conservation are the two quiet workhorses of mechanics.

No experiment gives perfect equality. We develop laws because repeated measurements, with uncertainties included, show stable patterns. When momentum seems not to be conserved, we ask whether the system was really isolated, whether hidden particles or hidden forms of energy are involved, and whether the uncertainty budget is honest.

Momentum is conserved in every direction

For two-dimensional interactions, apply momentum conservation separately along two perpendicular axes. Pick axes that make the algebra easier, usually one along an initial velocity and the other at right angles.

In an off-centre collision with body 2 initially stationary, the component equations are m₁u₁ = m₁v₁ cos θ₁ + m₂v₂ cos θ₂ and 0 = m₁v₁ sin θ₁ − m₂v₂ sin θ₂, where θ₁ is the angle between body 1’s final velocity and the original direction (rad or degrees as stated), and θ₂ is the corresponding angle for body 2.

If the collision is elastic, kinetic energy is conserved as well. The syllabus, though, does not expect you to solve simultaneous conservation-of-momentum and conservation-of-energy equations as a routine method. Use the energy comparison to decide whether kinetic energy was conserved, or to check a result when the needed quantities are already available.

For two identical masses, with one initially stationary, an elastic collision gives final velocity directions at right angles, unless there is no real collision or the collision is exactly head-on. It’s a neat air-table result. In real data, small deviations show up because of rotation, friction, imperfect elasticity and measurement uncertainty.

Ideal-gas link

The kinetic model of an ideal gas assumes that collisions between molecules, and between molecules and container walls, are elastic; that intermolecular forces are negligible except during collisions; and that collision times are tiny compared with the time between collisions. These assumptions let the microscopic momentum changes of many particles produce macroscopic pressure laws.

Describing circular motion

Uniform circular motion is motion around a circular path at constant speed. The velocity is not constant, since its direction keeps changing. That one idea sits behind almost every circular-motion question.

Angular displacement is the angle through which a body moves about the centre of a circular path. It is usually measured in radians. One complete revolution is 2π rad.

Angular velocity in this course means the rate of change of angular displacement, treated by magnitude unless direction is explicitly discussed. It is given by ω = Δθ/Δt, where ω is angular velocity (rad s⁻¹) and Δθ is angular displacement (rad). If treated fully, the angular-velocity vector lies along the axis of rotation; IB calculations here use its magnitude.

The period is the time taken for one complete revolution. The circular-motion equations are v = 2πr/T = ωr, where r is the radius of the circular path (m) and T is the period (s). Think of these as the rotational versions of the linear-motion ideas: angular displacement corresponds to displacement, angular velocity corresponds to velocity, and multiplying by radius converts angular quantities into linear quantities along the arc.

Rotational and linear equations

The link comes from geometry. The circumference of a circle is 2πr, so one revolution in time T gives speed 2πr/T. One revolution also has angular displacement 2π, so ω = 2π/T. Put these together and you get v = ωr.

This answers the common linking question about rotational and linear equations: they describe the same motion in parallel ways, with radius acting as the conversion factor between angular and linear quantities.