Forces and momentum

A.1 described motion. A.2 explains it. Everything here comes from three laws and one conserved quantity, and almost every question begins the same way: draw one object, draw every force acting on it, and add them as vectors. Get that diagram right and the physics is usually one line of algebra.

🧲What a force is

A force is a push or a pull, measured in newtons, and you can recognise one by what it does: it deforms something, or it changes something’s velocity. That second half matters more than it looks. A change in velocity is an acceleration, so a resultant force causes an acceleration — and no resultant force is needed to keep an object moving at constant velocity.

A force is an interaction between two objects, never a property of one. To describe a force completely you need five things:

• its magnitude and its direction (it is a vector);
• the object it acts on;
• the object exerting it;
• the nature of the force.

So a proper description reads: “a 50 N push, at 20° to the horizontal, acting ON the football FROM the boot”. If you cannot name the object exerting a force, that force does not exist — which is the single most useful test in this whole topic.

The types you will meet. Every one of them is ultimately gravitational or electromagnetic in origin; almost everything in daily life is electromagnetic.

• Normal force, \( F_N \) — the part of the contact force between two surfaces that acts at right angles to them. Between smooth surfaces it is the only contact force there is.
• Surface friction, \( F_f \) — opposes relative motion of the surfaces, and acts along them.
• Tension and the elastic restoring force, \( F_H \) — a stretched spring pulls equally on the objects at both ends, in proportion to its extension (Hooke’s law).
• Viscous drag, \( F_d \) — opposes motion through a fluid, and grows with speed.
• Buoyancy, \( F_b \) — the upward force on anything immersed in a fluid.
• Gravitational force, \( F_g \) — between masses; near the Earth we call it weight.
• Electric, \( F_e \), and magnetic, \( F_m \) — between charges, and between currents or magnets.

One useful split is contact forces (normal, friction, tension, drag, buoyancy) against forces that act at a distance (gravitational, electric, magnetic). The first group needs surfaces touching; the second does not.

📐Forces as vectors, and free-body diagrams

Because forces are vectors, two of them at an angle do not add as numbers. Add them tip to tail, or resolve them into perpendicular components and add each direction separately — whichever makes the geometry easier.

Free-body diagrams are the discipline that makes all of this work. Choose one object, then draw every force acting on that object and nothing else.

⚖️Newton’s first law and equilibrium

An object continues at rest, or moving at constant velocity in a straight line, unless a resultant external force acts on it. Read carefully, that adds nothing to the definition of a force — it simply insists on the converse: no resultant force means no acceleration, and observing no acceleration tells you the forces balance.

An object with zero resultant force is in translational equilibrium:

Because force is a vector, that means zero resultant in every direction. In two dimensions it is enough to show the forces balance in any two non-parallel directions — usually the two you chose when you resolved them. For a mass hanging on a string and pulled sideways by a force \(P\), equilibrium gives \( T\sin\theta = P \) horizontally and \( T\cos\theta = W \) vertically, and those two equations solve the whole problem.

🧮Newton’s second law

The first law says a resultant force causes acceleration. The second says how much. In its most general form, the resultant force equals the rate of change of momentum:

Four things are worth pinning down. \(F\) is always the resultant force, so work that out first. The mass must be in kilograms for \(F = ma\) to give newtons. The law is experimental — it was not derived from anything. And when the mass changes (a rocket burning fuel, coal landing on a conveyor belt) or the force varies, go back to \( F = \Delta p / \Delta t \), which still holds.

🤝Newton’s third law

When two bodies A and B interact, the force A exerts on B is equal in magnitude and opposite in direction to the force B exerts on A. In symbols, \( F_{AB} = -F_{BA} \). Forces always come in pairs, which makes the third law a checklist: if you have drawn a force, something somewhere is feeling its partner.

Two properties settle every third-law question. The pair acts on two different objects — so two forces on the same free-body diagram are never a pair. And the pair is the same type of force: if A pulls B gravitationally, then B pulls A gravitationally too.

The law also explains things that look one-sided. Two skaters push apart and feel equal forces, but the lighter one accelerates more, because \( a = F/m \). A person pushing off a wall is accelerated by the wall’s push on them; the wall (attached to the Earth) has far too much mass for its acceleration to be noticed. And a car accelerates because the road pushes it forwards — the engine only turns the wheels, which push backwards on the road.

⚖️Mass, weight and Hooke’s law

Mass is the amount of matter in an object, measured in kilograms, and it does not change when you move the object. Weight is a force, measured in newtons. Take an object to the Moon and its mass is unchanged while its weight falls to about a sixth, because \( g \approx 1.6 \) N kg\(^{-1}\) there against 9.81 N kg\(^{-1}\) here.

The word “weight” is genuinely ambiguous in physics: some people mean the gravitational force \( mg \), others mean the reading on a supporting scale. In equilibrium the two agree, and in an accelerating lift they do not — which is exactly why you feel heavy as a lift starts upwards, while the gravitational force on you has not changed at all. Say “gravitational force” when you mean \( mg \) and the ambiguity disappears.

Hooke’s law turns a spring into a force meter. Up to its elastic limit, the extension \(x\) of a spring is proportional to the force stretching it:

so a graph of force against extension is a straight line through the origin whose gradient is the spring constant. Measure the extension and you have measured the force — a newtonmeter is nothing more than this.

🧱Solid friction

Friction opposes the relative motion of two surfaces, and it arises because no surface is smooth at the microscopic scale. It comes in two varieties, and the difference matters.

Static friction acts between surfaces that are not sliding, and it takes whatever value is needed to prevent sliding — up to a maximum. Dynamic (or kinetic) friction acts once they are sliding, and is roughly constant:

where \(R\) is the normal reaction. The coefficients are ratios of two forces, so they have no units, and they are usually less than 1. Two consequences surprise people: the maximum friction does not depend on the area of contact when the surfaces are held together by gravity, and friction drops at the moment the object starts to move — which is why a heavy box lurches forward as it finally gives way.

Tilting a slope until an object just slides is the neatest way to measure \( \mu_s \), because everything else cancels.

💧Viscosity, drag and buoyancy

An ideal fluid slides past itself freely. A real fluid does not: its layers rub, and that internal friction is viscosity, \( \eta \). It is defined as the ratio of the tangential stress applied to a fluid to the velocity gradient it produces,

which gives units of Pa s (equivalently N s m\(^{-2}\) or kg m\(^{-1}\) s\(^{-1}\)). Water is about \( 1.0 \times 10^{-3} \) Pa s at room temperature and thick syrup about \( 1.0 \times 10^{2} \) Pa s — five orders of magnitude apart. Viscosity is extremely sensitive to temperature, which is why engine oil is graded for it and why syrup pours so much better warm.

For a sphere moving slowly through a fluid, the drag force is given by Stokes’ law, and an immersed object also feels an upthrust given by Archimedes’ principle.

In the buoyancy equation \( \rho \) is the density of the fluid and \(V\) the volume displaced, so the upthrust equals the weight of fluid pushed aside. A floating object displaces exactly its own weight of fluid, which is the whole of why ships float.

Stokes’ law is not universal. It assumes the sphere moves slowly enough for the flow around it to stay streamlined, that the fluid is effectively infinite (no nearby container walls), and that the object really is a sphere. Push any of those and it fails.

🎯Momentum and impulse

Linear momentum is mass times velocity,

and because velocity is a vector, momentum is a vector: signs matter, and you must fix a positive direction before substituting anything. The impulse of a force is the change of momentum it produces,

which is just Newton’s second law rearranged. For a force that varies during an impact — and every real impact does — the impulse is the area under the force–time graph.

🔒Conservation of momentum

The total momentum of a system of interacting particles is constant provided no resultant external force acts. This is not a new law — it follows from the second and third laws together. When A and B collide, the force on B from A is equal and opposite to the force on A from B, and they act for exactly the same time. So the impulses are equal and opposite, the momentum changes are equal and opposite, and the total is unchanged.

Kinetic energy is a different question entirely, and this is where marks are lost.

• Elastic: no kinetic energy is lost at all. Real everyday objects never quite manage it — collisions between molecules are the honest example. A useful test: in an elastic collision the relative velocity of approach equals the relative velocity of separation.
• Totally inelastic: the objects stick together, so their relative velocity of separation is zero. This loses the most kinetic energy of any collision with that momentum — to deformation, heat and sound.
• Inelastic: everything in between, which is almost every collision you will ever meet.
• Explosions run the same law backwards: the total momentum stays whatever it was (usually zero), while kinetic energy increases, released from stored chemical or elastic energy.

🎡Uniform circular motion

An object going round a circle at constant speed is accelerating, because its velocity is constantly changing direction. That is the whole idea, and everything else follows from it.

Angles in radians make the algebra work. An angle in radians is the arc length divided by the radius, \( \theta = s/r \), so a full circle is \( 2\pi \) rad. The angular velocity \( \omega \) is the angle turned per second, in rad s\(^{-1}\):

and the centripetal acceleration and force follow, each written three equivalent ways:

Because the centripetal force is always perpendicular to the velocity, it does no work and cannot change the speed. That is why a conical pendulum keeps a steady speed while its direction changes continuously, and why the horizontal component of the normal force on a banked track can turn a car with no help from friction at all.

A vertical circle is the interesting case, because now gravity has a component along the path for most of the journey and the speed is no longer constant.