Themes › Theme A Space, time and motion

A.2Forces and momentum

A.1 described motion. A.2 explains it: forces change motion, Newton's three laws tell us how, and momentum lets us predict what happens when objects collide or explode. The topic ends with circular motion, where a force changes an object's direction rather than its speed.

1. Forces and free-body diagrams

A force is an interaction between two bodies: a push or a pull. Forces are vectors and are measured in newtons (N). Every force has a cause: something else must be doing the pushing or pulling.

A free-body diagram shows a single object, often drawn as a dot or a box, with every force acting on it drawn as a labelled arrow. The length of each arrow shows the size of the force.

• Draw only forces acting on the object, never forces the object exerts on other things.
• Weight acts from the centre of mass, straight down.
• The normal force acts perpendicular to the surface. Friction acts parallel to the surface.
• Use standard labels: $F_g$ or $mg$ (weight), $F_N$ (normal), $F_f$ (friction), $T$ (tension), $F_d$ (drag), $F_b$ (buoyancy).

Finding the resultant force

The resultant (net) force is the vector sum of all the forces. In two dimensions, resolve each force into perpendicular components, add the components in each direction, and then combine them:

Choose axes that make the problem easy. On a slope, use axes parallel and perpendicular to the slope. The weight then has components $mg\sin\theta$ down the slope and $mg\cos\theta$ into the slope.

2. Newton's first law and translational equilibrium

This means a force is needed to change motion, not to keep it going. A hockey puck slides a long way on ice because there is very little friction to slow it. The tendency of a body to resist changes to its motion is called inertia, and mass is a measure of it.

A body with zero resultant force is in translational equilibrium. It is either at rest or moving at constant velocity, and the forces in every direction balance: $\sum F_x = 0$ and $\sum F_y = 0$.

3. Newton's second law

One newton is the resultant force that gives a 1 kg mass an acceleration of 1 m s$^{-2}$. The first law is just the special case $F = 0 \Rightarrow a = 0$.

4. Contact forces

Normal force

The normal force $F_N$ is the component of the contact force perpendicular to a surface. It pushes back on a body that presses on the surface. It adjusts its size to whatever is needed, up to the point where the surface breaks. It is not always equal to the weight: think of the lift, a slope, or a rope pulling upwards.

Tension

Tension $T$ is the pulling force in a stretched string, rope or cable. It acts along the string, away from the body it is attached to. A string can only pull, never push. For a light (massless) string over a frictionless pulley, the tension is the same all the way along.

Elastic restoring force: Hooke's law

When a spring is stretched or compressed by $x$, it pulls or pushes back with a restoring force:

$k$ is the spring constant (N m$^{-1}$): a stiff spring has a large $k$. The minus sign shows the force is opposite to the displacement, always trying to return the spring to its natural length. Hooke's law only holds up to the limit of proportionality. In C.1 this restoring force leads to simple harmonic motion.

Example: a spring with $k = 40\ \text{N m}^{-1}$ stretched by 0.15 m pulls back with $40 \times 0.15 = 6.0$ N.

Density, pressure and buoyancy

Density is mass per unit volume: $\rho = \dfrac{m}{V}$, in kg m$^{-3}$. Water has a density of about 1000 kg m$^{-3}$, so 1 litre of water has a mass of 1 kg.

A fluid presses on everything in it, and the pressure increases with depth because there is more fluid above pushing down. At depth $h$, the extra pressure is $P = \rho g h$. (You'll meet pressure properly in B.3. Here it simply explains where buoyancy comes from.)

For a block of height $h$ and top area $A$, the pressure difference between its bottom and top is $\rho g h$. So the net upward force is $\rho g h \times A = \rho g V$, which is the weight of the fluid the block displaces (Archimedes' principle):

Here $\rho$ is the density of the fluid (not the object) and $V$ is the volume of fluid displaced. That is the volume of the object below the surface.

• If $F_b$ is less than the weight when the body is fully submerged, it sinks. This happens when the body is denser than the fluid.
• A floating body sinks until it has displaced just enough fluid for $F_b = mg$.

Example: a 2.0 × 10$^{-3}$ m$^3$ block held fully under water feels $F_b = 1000 \times 2.0\times10^{-3} \times 9.8 = 20$ N upwards.

Friction

Friction $F_f$ acts parallel to the surface and opposes sliding, or the tendency to slide.

• Static friction acts when the surfaces are not sliding. It matches the applied force, up to a maximum value: $F_f \le \mu_s F_N$.
• Dynamic (kinetic) friction acts once sliding starts, and is roughly constant: $F_f = \mu_d F_N$.
• $\mu_s$ and $\mu_d$ are the coefficients of static and dynamic friction. They have no units, and usually $\mu_d < \mu_s$. That's why it is harder to start something sliding than to keep it sliding.

Measuring the coefficient of friction

A neat way to measure $\mu_s$: put a block on a board and slowly raise one end until the block just starts to slip, at angle $\theta$. At that moment, static friction is at its maximum and balances the component of the weight down the slope:

For example, a block that starts to slip at 22° gives $\mu_s = \tan 22° = 0.40$. Lower the board until the block slides down at constant velocity (zero acceleration, so the forces balance again). The tangent of that smaller angle gives $\mu_d$.

Viscous drag

A body moving through a fluid (a liquid or gas) experiences a drag force opposing its motion. A fluid's resistance to flowing is its viscosity $\eta$ (unit Pa s). Honey has a high viscosity and air a very low one. For a small sphere moving slowly (smooth, non-turbulent flow), Stokes' law gives:

So the drag is proportional to the radius and to the speed. A small sphere falling through oil speeds up until $F_d + F_b = mg$, and then falls at terminal speed. Measuring that speed is a standard way to find a fluid's viscosity.

5. Field forces

Some forces act at a distance, with no contact needed:

• Gravitational force (weight): $F_g = mg$. Mass (kg) is the amount of matter and is the same everywhere. Weight (N) is a force and depends on $g$. On the Moon your mass is unchanged, but your weight is about one-sixth.
• Electric force $F_e$: between charged bodies. Like charges repel and unlike charges attract.
• Magnetic force $F_m$: on magnets, and on moving charges and currents in a magnetic field.

You only need to recognise electric and magnetic forces and include them in free-body diagrams here. You'll calculate them in Theme D (Fields).

6. Newton's third law

The two forces in a third-law pair:

• act on different bodies, so they never cancel each other out;
• are always the same type of force (both gravitational, both contact, and so on);
• act at the same time, for the same length of time.

A classic trap: a book resting on a table has its weight (Earth pulls book down) and a normal force (table pushes book up). These are equal and opposite, but they are not a third-law pair. They act on the same body and are different types of force. They balance because of the first law. The real pairs are:

• Earth pulls book down ↔ book pulls Earth up (gravitational);
• table pushes book up ↔ book pushes table down (contact).

7. Momentum and impulse

The linear momentum of a body is its mass times its velocity. It is a vector, in the direction of the velocity:

Newton actually wrote his second law in terms of momentum: the resultant force equals the rate of change of momentum.

Impulse

A resultant force acting for a time gives an impulse, which equals the change in momentum:

Here $F$ is the average resultant force during the contact time $\Delta t$. In a real collision the force changes, and the impulse is the area under the force–time graph.

Why it matters for safety: for a given change in momentum, making the collision last longer makes the average force smaller ($F = \Delta p / \Delta t$). That is how airbags, crumple zones, bike helmets, crash mats and bending your knees when you land all protect you.

Rockets and jet engines

A rocket pushes exhaust gas backwards, giving the gas momentum. By Newton's third law, the gas pushes the rocket forwards with an equal force, called the thrust. If gas of mass $\Delta m$ is ejected in time $\Delta t$ at speed $v$ relative to the rocket:

Example: an engine that ejects 250 kg of gas per second at 2.0 km s$^{-1}$ produces a thrust of $2000 \times 250 = 5.0 \times 10^{5}$ N. The rocket needs no air to push against, which is why rockets work in space. The nozzle is shaped so that the gas leaves in one direction, straight backwards, to make the thrust as large as possible.

A jet engine is similar, but it takes in air at the plane's speed $u$ and blasts it out faster, at $v$. The thrust is $\dfrac{\Delta m}{\Delta t}(v - u)$.

8. Conservation of momentum, collisions and explosions

Why? In a collision, the forces the bodies exert on each other are a Newton's third-law pair: equal, opposite, and acting for the same time. So the impulses are equal and opposite. Whatever momentum one body gains, the other loses. These internal forces always cancel for the system as a whole.

Types of collision

Total energy is always conserved. In an inelastic collision, kinetic energy is transferred to other stores, mainly thermal. Momentum is conserved in every type of collision, as long as there is no external force.

A useful link: $E_k = \dfrac{p^2}{2m}$. When two bodies fly apart from rest with equal and opposite momenta, the lighter one gets more of the kinetic energy.

Collisions in two dimensions HL

Momentum is a vector, so it is conserved separately in each direction. Resolve every momentum into $x$ and $y$ components and apply conservation to each.

9. Circular motion

An object moving in a circle at constant speed is still accelerating, because the direction of its velocity is always changing. The velocity is always along the tangent to the circle, and the acceleration points towards the centre. This is called centripetal acceleration ("centre-seeking").

Describing circular motion

• Period $T$: the time for one complete revolution (s). Frequency $f = 1/T$ (Hz).
• Radians: in circular motion, angles are measured in radians. An angle $\theta$ in radians is the arc length $s$ divided by the radius $r$, so $s = r\theta$. A full circle has arc length $2\pi r$, so $360° = 2\pi$ rad, and 1 rad ≈ 57.3°.
• Angular velocity $\omega$: the angle swept out per second, $\omega = \dfrac{\Delta\theta}{\Delta t}$, in rad s$^{-1}$. One revolution is $2\pi$ rad, so $\omega = \dfrac{2\pi}{T} = 2\pi f$. Example: a fairground ride that turns through 120° in 4.0 s has $\theta = \frac{2\pi}{3} = 2.09$ rad and $\omega = 2.09 / 4.0 = 0.52\ \text{rad s}^{-1}$.
• Linear speed: one circumference $2\pi r$ per period, so:

Every point on a rotating rigid object, such as a wheel or a turntable, has the same $\omega$. But points further from the centre have a larger speed $v$.

Centripetal acceleration and force

The centripetal force is not a new kind of force. It is the name for the resultant force towards the centre, and some real force has to provide it:

• a car on a flat bend: friction between the tyres and the road;
• a ball whirled on a string: tension;
• a planet or satellite in orbit: gravity;
• an electron moving in a magnetic field: the magnetic force;
• a plane turning, or a car on a banked track: a horizontal component of the lift or normal force. You only need to describe banking, not calculate it.

The centripetal force is always perpendicular to the velocity, so it changes the direction of motion but not the speed, and it does no work. If it suddenly disappears (the string breaks), the object carries on in a straight line along the tangent (Newton's first law). It does not fly outwards along the radius.

Circles in a vertical plane

For a ball on a string, a bucket swung overhead or a roller-coaster loop, gravity helps at the top and works against you at the bottom. The resultant force towards the centre must always equal $\dfrac{mv^2}{r}$. The speed usually changes around a vertical circle (non-uniform circular motion), but you only need to analyse the top and bottom.

• At the top: $T + mg = \dfrac{mv^2}{r}$. Tension is smallest here. The slowest possible speed is when $T = 0$ and gravity alone provides the centripetal force: $mg = \dfrac{mv^2}{r}$, so $v_{\min} = \sqrt{gr}$. Any slower and the string goes slack (or the water falls out of the bucket).
• At the bottom: $T - mg = \dfrac{mv^2}{r}$. Tension is largest here. The string is most likely to snap at the bottom, and on a roller coaster you feel heaviest at the bottom of a dip.

10. Common mistakes

• Thinking a force is needed to keep something moving. A force is needed to change its velocity.
• Calling weight and the normal force a third-law pair. Third-law pairs act on different bodies and are the same type of force.
• Assuming $F_N = mg$. That's only true on a level surface with no other vertical forces and no vertical acceleration.
• Using $F_f = \mu_s F_N$ for something that isn't moving. Static friction is only as big as it needs to be, up to $\mu_s F_N$.
• Forgetting the direction change in rebounds. $\Delta p = m(v - u)$ with signs, so a bounce gives a bigger change than stopping.
• Saying kinetic energy is conserved in every collision. Momentum always is (with no external force); kinetic energy is only conserved in elastic collisions.
• Drawing "centrifugal force" outwards on a free-body diagram, or adding centripetal force as an extra force. Centripetal force is the resultant of the real forces, and it points inwards.
• Using degrees instead of radians in $v = \omega r$.

11. Check your understanding

Practise A.2 questions