Forces are interactions, not possessions

A force is a vector interaction that can change the velocity of a body or deform it. It is not something an object “has”; it is something one body exerts on another. That small wording habit matters, because every force has an agent and a body experiencing the force.

A contact force is a force that acts only when bodies touch. A field force is a force that 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 water and 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. So 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, this is written as $F = ma$, where $F$ is the resultant force (N), $m$ is the mass (kg) and $a$ is the acceleration ($\mathrm{m\,s^{-2}}$). 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 $1\,\mathrm{m\,s^{-2}}$ in the direction of the force. In base units, $1\,\mathrm{N} = 1\,\mathrm{kg\,m\,s^{-2}}$.

The form $F = ma$ assumes that the mass of the object 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 useful test for third-law pairs is: 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 belongs to 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 a point of application matters, label every force with an accepted name or symbol, and keep the arrow directions physically meaningful.

For this syllabus, free-body diagrams and resultant-force analysis are limited to one- and two-dimensional situations. The 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. That can mean the body is stationary, but it can also mean it is 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 is applied 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, you can 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 and the ring is adjusted until stationary. Masses provide known weights and the angle of one string is measured. Repeating for different masses and plotting a processed quantity such as $\cos \theta$ 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 a contact-force component that acts perpendicular to the surface pressing on a body. It is written $F_N$. The normal force is not automatically equal to weight; it adjusts according to the situation. In a lift accelerating upward it is larger than the person’s weight, and on a slope it is usually smaller than the full weight.

Tension is a contact force transmitted through a stretched string, rope, cable or chain. In ideal IB problems, a light inextensible string over a frictionless pulley has the same tension throughout. I often label it $F_T$, where $F_T$ is the tension force (N), to avoid confusing it with the period symbol used later in circular motion.

Weight and other field forces

A gravitational force is a field force due to mass attracting mass. Close to Earth’s surface, the weight of a body is $F_g = mg$, 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. The motion of a mass in a gravitational field and the motion of a charged particle in an electric field are modelled in the same Newtonian way: find the field force, then use Newton’s second law to predict the acceleration. Magnetic forces are different in one important respect: 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 that 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, $F_f \le \mu_s F_N$, where $F_f$ is the surface frictional force (N), $\mu_s$ is the coefficient of static friction (unitless). The inequality is the key: static friction takes whatever value is needed up to a maximum. Once sliding occurs, $F_f = \mu_d F_N$, where $\mu_d$ is the coefficient of dynamic friction (unitless). Usually $\mu_d$ is smaller than $\mu_s$.

A simple way to measure $\mu_s$ is to place a block on a ramp and increase the angle until the block is just about to slip. Resolving the weight gives the limiting condition $\tan \theta = \mu_s$, where $\theta$ 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 $\mu_d$.

The friction equations are empirical. That means they summarize experimental behaviour of real surfaces rather than being derived from a detailed atomic model. Microscopically, even polished surfaces have tiny peaks and hollows; 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, $F_H = -kx$, 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 not decoration; 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 and check whether the two sets of readings agree; 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 $F_d = 6\pi \eta r v$, where $F_d$ is the viscous drag force (N), $\eta$ 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 its speed increases, the drag increases until the resultant force becomes zero; the sphere 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 $F_b = \rho V g$, where $F_b$ is the buoyancy force (N), $\rho$ is the density of the fluid (kg m$^{-3}$) and $V$ is the volume of fluid displaced (m$^3$).

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 equal to the product of mass and velocity. It is given by $p = mv$, where $p$ is linear momentum ($kg\,m\,s^{-1}$). Momentum has the same direction as velocity; mass only scales the vector.

A system’s total linear momentum remains constant unless a resultant external force acts on the system. Internal forces can redistribute momentum between parts of the system, but they cannot change the total momentum of the whole system.

Impulse

Impulse is a vector quantity equal to the product of average resultant force and the time interval over which it acts. It is given by $J = F\Delta t$, where $J$ is impulse (N s) and $\Delta t$ is the time interval or contact time (s). Since impulse equals change in momentum, $1\,\mathrm{N\,s}$ is equivalent to $1\,\mathrm{kg\,m\,s^{-1}}$.

The impulse-momentum theorem is $J = \Delta p$, where $\Delta p$ is the change in momentum ($kg\,m\,s^{-1}$). In words: the area under a resultant force-time graph is the change in momentum.

This is why seat belts, airbags, crash mats and sports follow-through matter. For the same change in momentum, increasing the stopping time reduces the average force. The momentum change has not disappeared; it has been made less brutal.

A neat practical method for estimating the force of a kick is to measure the contact time with foil contacts and a timer, then determine the ball’s launch speed using projectile motion from a bench. The ball’s change in momentum divided by the contact time gives 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 $F = \Delta p/\Delta t$. This form works even when mass is changing, whereas $F = ma$ is the constant-mass special case.

For a rocket, the exhaust gains momentum in one direction and 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 $\Delta m/\Delta t$, the thrust magnitude is approximately $F_{\text{thrust}} = u_e(\Delta m/\Delta t)$, where $F_{\text{thrust}}$ is the thrust force (N), $u_e$ is the exhaust speed relative to the rocket ($m\,s^{-1}$) and $\Delta m/\Delta t$ is the mass ejection rate ($kg\,s^{-1}$).

For water leaving a hose, the same idea appears as a force on the nozzle. If water speeds up from the hose to the nozzle, its momentum changes each second; the hose experiences 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 masses and velocities immediately before and after collision. Reduce external forces: level or slightly incline the track 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 include uncertainty. If the initial and final total momentum values agree within experimental uncertainty, the result supports conservation of momentum. If they do not, the first suspicion should be external forces, calibration, timing, alignment or rotation — not that the universe has quietly abandoned momentum conservation.

Isolated systems and conservation

An isolated system is a chosen collection of bodies for which the resultant external force is zero or negligible during the interaction being studied. In such a system, total linear momentum is conserved.

In one dimension, conservation of momentum is written $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$, where $m_1$ is the mass of body 1 (kg), $u_1$ is the initial velocity of body 1 (m s$^{-1}$), $m_2$ is the mass of body 2 (kg), $u_2$ is the initial velocity of body 2 (m s$^{-1}$), $v_1$ is the final velocity of body 1 (m s$^{-1}$) and $v_2$ is the final velocity of body 2 (m s$^{-1}$). Choose a positive direction before substituting numbers; signs carry the physics.

Elastic and inelastic collisions

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

For kinetic-energy checks, $E_k = \frac{1}{2}mv^2$, where $E_k$ is kinetic energy (J). If the total kinetic energy before and after is the same, the collision is elastic; if it decreases, the collision is inelastic. You are not required to solve simultaneous momentum-and-energy equations for collision speeds, but you should be able to use energy to classify or compare collisions.

In a perfectly inelastic collision where body 2 is initially stationary and the bodies stick, $v_c = \frac{m_1u_1}{m_1 + m_2}$, where $v_c$ is the common final velocity (m s$^{-1}$). Kinetic energy is not conserved, even though momentum is.

Explosions

An explosion is an interaction in which 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. Two fragments move in opposite directions with equal-magnitude momenta if no external impulse acts.

This explains recoil. A gun and shell start 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 it has much larger speed; equal momentum does not mean equal kinetic energy.

Momentum conservation also matters in nuclear power stations. Fast neutrons transfer momentum and energy in collisions with moderator atoms, slowing the neutrons so they are more likely to cause further fission. Steam striking turbine blades is another momentum-transfer process, converting kinetic energy of moving fluid into rotation.

Momentum, energy and uncertainty

Conservation laws are powerful because they keep working 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. Laws are developed because repeated measurements, with uncertainties included, show stable patterns. When momentum appears 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

In two-dimensional interactions, momentum conservation is applied independently along two perpendicular axes. Choose axes that simplify the situation, usually one axis along an initial velocity and the other at right angles.

For an off-centre collision where body 2 is initially stationary, the component equations are $m_1u_1 = m_1v_1\cos\theta_1 + m_2v_2\cos\theta_2$ and $0 = m_1v_1\sin\theta_1 - m_2v_2\sin\theta_2$, where $\theta_1$ is the angle made by body 1’s final velocity to the original direction (rad or degrees as stated) and $\theta_2$ is the corresponding angle for body 2.

If the collision is elastic, kinetic energy is also conserved, but the syllabus does not require solving 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 quantities are already accessible.

For two identical masses where one is initially stationary and the collision is elastic, the final velocity directions are at right angles, unless there is no real collision or the collision is exactly head-on. This is a lovely result on an air table, but in real data you will see small deviations 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; intermolecular forces are negligible except during collisions; and collision times are tiny compared with the time between collisions. Those assumptions are what allow the microscopic momentum changes of many particles to produce macroscopic pressure laws.

Describing circular motion

Uniform circular motion is motion around a circular path at constant speed. The velocity is not constant, because its direction changes continuously. That single point is the source of nearly all circular-motion questions.

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 $2\pi\ \text{rad}$.

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 $\omega = \Delta\theta/\Delta t$, where $\omega$ is angular velocity $(\text{rad s}^{-1})$ and $\Delta\theta$ is angular displacement $(\text{rad})$. The angular-velocity vector, if treated fully, lies along the axis of rotation; but IB calculations here use its magnitude.

The period is the time taken for one complete revolution. The circular-motion equations are $v = 2\pi r/T = \omega r$, where $r$ is the radius of the circular path $(\text{m})$ and $T$ is the period $(\text{s})$. These equations are the rotational cousins of linear motion: 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 connection is geometry. The circumference of a circle is $2\pi r$, so one revolution in time $T$ gives speed $2\pi r/T$. Also, one revolution is angular displacement $2\pi$, so $\omega = 2\pi/T$. Combining the two gives $v = \omega r$.

This is the answer to the common linking question about rotational and linear equations: they are parallel descriptions of the same motion, with radius acting as the conversion factor between angular and linear quantities.

Centripetal acceleration

A body moving at constant speed around a circle has an acceleration directed radially toward the centre. This is centripetal acceleration, an acceleration caused by the changing direction of velocity.

Its magnitude is $a_c = \frac{v^2}{r} = \omega^2 r = \frac{4\pi^2 r}{T^2}$, where $a_c$ is centripetal acceleration ($\mathrm{m\,s^{-2}}$). The acceleration is perpendicular to the velocity at every instant. It changes the direction of motion, not necessarily the speed.

Centripetal force

A centripetal force is the resultant inward force required to produce centripetal acceleration. It is not a new type of force. It is the role played by a real force or combination of forces: tension for a ball on a string, gravity for a satellite, friction for a car on a flat bend, normal force for a rotor ride, or lift for an aircraft turning.

The magnitude is $F_c = ma_c = \frac{mv^2}{r} = m\omega^2 r$, where $F_c$ is centripetal force (N). The force is perpendicular to the velocity, so for ideal uniform circular motion it does no work: the displacement at each instant is tangential, while the centripetal force is radial. No component of the centripetal force acts along the motion.

The word centrifugal is best treated carefully. In an inertial frame, there is no outward force flinging the object away; the object tends to continue in a straight line while an inward force keeps changing its direction. In a rotating frame, an apparent outward force can be introduced as a useful model, but it is not an interaction force like tension or friction.

Horizontal circular motion

On a horizontal road, static friction provides the centripetal force for a turning car. The no-skid condition is $\frac{mv^2}{r} \le \mu_s mg$, giving $v_{\max} = \sqrt{\mu_s g r}$, where $v_{\max}$ is the maximum speed before skidding ($\mathrm{m\,s^{-1}}$). The mass cancels, which is why the safe speed depends on the road, bend radius and tyres, not on the car’s mass in this simple model.

In a rotor ride, the wall’s normal force provides the inward centripetal force and friction prevents the rider from sliding down. These two directions must not be muddled: normal force is horizontal inward; friction is vertical upward.

A rotating bung experiment can test $F_c = \frac{mv^2}{r}$. A hanging mass supplies a known tension, the rotating radius is fixed by a marker, and the speed is found from timing several revolutions. A graph of $v^2$ against $r$ should be linear when the centripetal force and rotating mass are held constant. The main systematic issue is that the string is not perfectly horizontal, partly because it has weight.

Banking and vertical circles

A banked surface is a curved surface tilted so that the normal force has a horizontal component toward the centre of the turn. Qualitatively, banking reduces the reliance on friction. Quantitative banked-surface calculations are not required, but you should understand the principle: the normal reaction helps provide the centripetal force.

Circular-motion situations may be uniform or non-uniform, and may be in horizontal or vertical planes. In a vertical circle, weight changes its component along the radial direction as the object moves. At the top of a vertical circle, weight acts toward the centre; at the bottom, weight acts away from the centre. For a mass on a string moving at speed $v_{\text{top}}$ at the top, $F_{T,\text{top}} + mg = \frac{mv_{\text{top}}^2}{r}$, where $F_{T,\text{top}}$ is the string tension at the top (N) and $v_{\text{top}}$ is the speed at the top ($\mathrm{m\,s^{-1}}$). At the bottom, $F_{T,\text{bottom}} - mg = \frac{mv_{\text{bottom}}^2}{r}$, where $F_{T,\text{bottom}}$ is the string tension at the bottom (N) and $v_{\text{bottom}}$ is the speed at the bottom ($\mathrm{m\,s^{-1}}$). The bottom is where the string is most likely to break.

For non-uniform vertical circular motion, quantitative force analysis is limited to the top and bottom of the path. At other points the radial and tangential components both matter, so the syllabus does not require that analysis.

Gravity providing centripetal force is the basic reason orbital motion is possible. A satellite is continuously falling toward Earth while its tangential motion carries it around Earth. This links circular motion to gravitational fields: the gravitational force changes the direction of the satellite’s velocity, producing an orbit rather than a straight fall.