Forces

October 16, 2026

Almost every engineering question starts as a force question. Will it hold? Will it move? At what speed? What breaks first? This post covers forces from the ground up: what a force is, how to draw and add them, and every kind you'll meet day to day. Each one has a diagram, the maths, and a worked example with real numbers.

1. What a force is

A force is a push or a pull. It has a size, measured in newtons (N), and a direction. One newton is about the weight of a small apple. Newton's three laws say everything else:

An object keeps doing what it's doing, sitting still or moving in a straight line at a steady speed, unless a net force acts on it.
A net force accelerates it: F = m·a. Twice the force, twice the acceleration. Twice the mass, half the acceleration.
Forces come in pairs. If A pushes on B, B pushes back on A just as hard, in the opposite direction.

The tool that makes all of this usable is the free-body diagram: draw the object on its own, and every force acting on it as an arrow.

Then add the forces up in each direction. If they cancel, the object is in equilibrium (still, or moving steadily). If they don't, the leftover force sets the acceleration.

2. Forces are vectors

Because a force has a direction, you can't add forces like plain numbers. 3 N east plus 4 N north is 5 N toward the north-east, not 7 N.

The practical method is to split every force into components along two perpendicular axes:

F_{x} = F cos θ F_{y} = F sin θ (θ measured from the x axis)

Add all the x components, then all the y components, as ordinary numbers. Then rebuild the result:

∣ F ∣ = F_{x}^{2} + F_{y}^{2} direction = arctan \frac{F _{y}}{F _{x}}

Choose axes that make the problem easy. On a ramp, tilt them to run along and across the slope.

3. Gravity and weight

Every mass attracts every other mass:

F = \frac{G m _{1} m _{2}}{r ^{2}} G = 6.674 \times 1 0^{- 11} N m^{2} /k g^{2}

G is so small that gravity between everyday objects is undetectable, but a planet's mass makes up for it. At Earth's surface it all simplifies to one number, g = 9.81 m/s², and an object's weight is:

W = m g

A 70 kg person weighs 70 × 9.81 ≈ 687 N. Mass (kg) is the amount of stuff, and weight (N) is the force of gravity on it. On the Moon, g = 1.62 m/s², so the same person weighs 113 N with the same mass.

4. The normal force, and ramps

When an object rests on a surface, the surface pushes back, perpendicular to itself, exactly as hard as it needs to so the object doesn't sink in. That's the normal force N. On flat ground N = mg. On a slope, only part of the weight presses into the surface.

Tilt the axes along the slope and the weight splits cleanly:

along the slope: into the slope: m g sin θ m g cos θ = N pulls it down the ramp presses it on

Worked example. A 10 kg crate sits on a 20° ramp. Down the slope: 98.1 × sin 20° = 33.6 N. Normal force: 98.1 × cos 20° = 92.2 N. With μs = 0.5, friction can hold up to 46.1 N, so it stays put. At 30° the pull down the slope is 49.1 N and friction can hold only 42.5 N, so it slides.

5. Friction

Friction opposes sliding between surfaces. It isn't one fixed value: it pushes back exactly as hard as the applied force, up to a limit.

not sliding: sliding: f \leq μ_{s} N f = μ_{k} N whatever it takes, up to the limit μ_{k} is a bit smaller than μ_{s}

The coefficient μ depends on the two materials and their surfaces, and surprisingly not on the contact area. Some typical values:

Surfaces	μ (static)
rubber on dry concrete	0.7–0.9
rubber on wet concrete	0.4–0.6
steel on steel, dry	~0.6
steel on steel, oiled	~0.1
PTFE on steel	~0.04
ice on ice	~0.03

Worked example. How hard do you have to push a 50 kg crate to get it moving, with μs = 0.6? N = 50 × 9.81 = 490 N, so it takes 0.6 × 490 = 294 N. Once it slides, with μk = 0.45, you only need 221 N to keep it moving.

6. Tension

A rope, cable or chain can only pull, and only along its own length. That pull is tension, and in a light rope it's the same all the way along.

Worked example. A 5 kg lamp (49.1 N) hangs from two cables at 37° and 47° to the ceiling. Treat the knot as the object. Horizontally the cables must cancel:

T_{1} cos 3 7^{\circ} = T_{2} cos 4 7^{\circ} ⟹ T_{1} = 0.854 T_{2}

Vertically they hold the weight:

T_{1} sin 3 7^{\circ} + T_{2} sin 4 7^{\circ} = 49.1 ⟹ T_{2} = 39.4 N, T_{1} = 33.6 N

The two tensions add up to more than the weight, because each cable spends part of its pull fighting the other one sideways. Flatten the cables to 5° and each needs about 280 N to hold the same lamp.

7. Pulleys

A pulley redirects a rope. With movable pulleys, several strands of the same rope share a load, and since every strand carries the same tension, the force you need drops.

force needed rope pulled = W / n = n \times height lifted n = rope strands holding up the moving block

The work is the same either way: force × distance. Real pulleys lose a few percent per sheave to friction.

Worked example. Lifting a 200 kg engine (1,962 N) with a 4-strand block and tackle takes about 490 N (a 50 kg pull) plus friction. To raise it 0.5 m you pull 2 m of rope.

8. Springs and elasticity

A spring pushes back in proportion to how far it's stretched or squeezed:

F = k x energy stored = \frac{1}{2} k x^{2}

k is the stiffness in N/m. Every solid material behaves like this for small deformations, and that's the foundation of structural engineering. Beyond the elastic limit it stops springing back.

Worked example. A 10 kg mass hangs on a spring with k = 2,000 N/m. It stretches until k·x = mg: x = 98.1 / 2,000 = 0.049 m, about 5 cm.

9. Moments: forces that turn

A force applied off-centre makes things rotate. Its turning effect about a point is its moment (or torque):

M = F d d = perpendicular distance from the point to the force’s line

For anything that stays still, two conditions must both hold:

\sum F \sum M = 0 = 0 no net push, so it doesn’t slide no net twist, about any point you choose, so it doesn’t rotate

The trick is to take moments about a point where an unknown force acts. That force then has zero lever arm, drops out, and you can solve for the other one directly.

Worked example. A 4 m beam on two supports carries 1,000 N at 3 m from the left support. Moments about the left support: R₂ × 4 = 1,000 × 3, so R₂ = 750 N. Then ΣF = 0 gives R₁ = 1,000 − 750 = 250 N. The support nearer the load carries more of it.

10. Pressure

Pressure is force spread over an area:

p = \frac{F}{A} 1 Pa = 1 N/ m^{2} 1 bar = 100 kPa \approx atmospheric pressure

A sealed fluid carries the same pressure everywhere (Pascal's principle). That's why a small force on a small piston becomes a big force on a big one, and it's how brakes, jacks and excavators work. In a liquid, pressure also grows with depth: p = ρ·g·h, about 1 bar for every 10 m of water.

Worked example. A hydraulic jack has a 10 mm diameter pump piston and a 50 mm lifting piston, a 25× area ratio. 200 N on the pump makes 2.55 MPa in the oil, which lifts with 200 × 25 = 5,000 N, about half a tonne. The pump has to travel 25 times as far as the lift.

11. Buoyancy

A fluid pushes up on anything in it with a force equal to the weight of the fluid it displaces (Archimedes):

F_{b} = ρ_{fluid} g V_{submerged}

The cause is pressure: the fluid pushes harder on the bottom of the object, which is deeper, than on the top. If the object is denser than the fluid, it sinks. If it's less dense, it floats, with just enough of its volume underwater to displace its own weight.

Worked example. A pine block with density 600 kg/m³ floats in water (1,000 kg/m³) with 600/1,000 = 60% of its volume submerged. A 1 m³ steel block (7,850 kg) sinks, but shaped into a hull that displaces 7.85 m³ of water, the same steel floats.

12. Drag

Moving through air or water costs a force that grows with the square of speed:

F = \frac{1}{2} ρ v^{2} C_{d} A

Here ρ is the fluid density (air ≈ 1.2 kg/m³), Cd is the drag coefficient set by shape, and A is the frontal area.

A falling object speeds up until drag equals its weight. That's its terminal velocity:

v_{terminal} = \frac{2 m g}{ρ C _{d} A}

Worked example. A car with Cd = 0.3 and A = 2.2 m² at 30 m/s (108 km/h) feels ½ × 1.2 × 30² × 0.3 × 2.2 ≈ 356 N of drag, which takes 356 × 30 ≈ 10.7 kW to overcome. At 40 m/s the drag is 633 N and the power 25 kW. Power goes with speed cubed, which is why fuel use climbs so steeply on the motorway.

13. Going round in circles

An object moving in a circle is always changing direction, so it's always accelerating toward the centre, even at a constant speed. That takes a net inward force:

F = \frac{m v ^{2}}{r} centripetal: toward the centre

This isn't a separate kind of force. Something real has to provide it: tyre friction, string tension, gravity. The "outward" force you feel in a turning car is your body trying to carry on in a straight line.

Worked example. What is the top speed for a car on a flat 50 m radius curve, with μ = 0.8? Friction supplies at most μ·m·g, so m·v²/r = μ·m·g and v = √(μ·g·r) = √(0.8 × 9.81 × 50) ≈ 19.8 m/s, about 71 km/h. The mass cancels. On a wet road with μ = 0.5 it drops to 56 km/h.

14. Electric and magnetic forces

Charges push and pull on each other with an inverse-square law just like gravity, only vastly stronger:

F = \frac{k q _{1} q _{2}}{r ^{2}} k = 8.99 \times 1 0^{9} N m^{2} / C^{2}

Moving charges also feel magnetic forces. A magnetic field B pushes sideways on a wire of length L carrying a current I:

F = B I L

That one equation runs every electric motor (How electric motors work builds on it).

15. Where forces come from

Physics recognises only four fundamental forces. Everything above is one of them, or many of them added up:

Force	Rough relative strength	Range	What it does
Strong	1	inside nuclei	holds protons and neutrons together
Electromagnetic	1/137	infinite	chemistry, materials, light, motors
Weak	~10⁻⁶	inside nuclei	some radioactive decay, powers the Sun's fusion
Gravity	~10⁻³⁹	infinite	planets, weight, orbits

Normal force, friction, tension, spring force, pressure, drag and buoyancy are all electromagnetic. They are the electrons in one object's atoms repelling the electrons in another's. When you lean on a wall, nothing touches. The atoms just refuse to get closer.

16. Forces inside things

Forces don't stop at an object's surface. Inside a part, every slice pushes or pulls on its neighbours, and how hard per unit area is the stress:

σ = \frac{F}{A} (stress, Pa) ε = \frac{Δ L}{L} (strain) σ = E ε (while elastic)

There are three basic ways to load material: tension (pulled apart), compression (squeezed), and shear (sliding layers). Bending is tension and compression at once, across the section. E, Young's modulus, is the material's stiffness: about 200 GPa for steel, 70 GPa for aluminium, 3 GPa for nylon.

Worked example. A 10 mm steel rod hangs a 1 tonne load (9,810 N). Its area is π × 5² = 78.5 mm², so the stress is 9,810 / 78.5 ≈ 125 MPa. That's safe against mild steel's ~250 MPa yield, with a margin of 2. It stretches ε = 125 MPa / 200 GPa = 0.06%, about 0.6 mm per metre.

17. Force over time: impulse

You can also write Newton's second law in terms of momentum (p = m·v):

F Δ t = Δ (m v) impulse = change in momentum

The same change in momentum can come from a big force for a short time, or a small force for a long time. That's the whole idea behind airbags, crumple zones, helmets, and bending your knees when you land.

Worked example. A 1,500 kg car at 15 m/s (54 km/h) has to lose 22,500 kg·m/s of momentum to stop. Stopping against a wall in 0.1 s takes an average force of 225 kN. Stopping over 1 s with crumple zones takes 22.5 kN, ten times less.

18. Work and power

A force that moves something does work, and work per second is power:

W P = F d = F v joules, J = N\cdotm; only the part of F along the motion counts watts, W = J/s

Worked example. Lifting a 20 kg box 1.5 m onto a shelf takes 20 × 9.81 × 1.5 ≈ 294 J, however you do it: straight up, up a ramp, or with pulleys. Do it in 2 s and you're putting out 147 W.

How to solve any force problem

Pick the object. Draw it on its own.
Draw every force on it: weight, every contact (normal force and friction), every rope, and any applied load. Nothing else.
Choose axes that line up with the motion or the slope.
Split the forces into components and write ΣFx, ΣFy, and ΣM about a smart point.
Set them equal to zero (equilibrium) or to m·a.
Solve, then check the units and whether the answer is sensible: does the support near the load carry more?

Sources

Halliday, Resnick and Walker, Fundamentals of Physics
The Feynman Lectures on Physics, Volume I
Hibbeler, Engineering Mechanics: Statics
Gordon, Structures: Or Why Things Don't Fall Down