What is the difference between force and pressure?

Pressure is force spread over an area: pressure = force / area. The same 240 N pushing on 0.04 m² gives 6,000 Pa, but on 0.12 m² it gives only 2,000 Pa. Pressure is why a sharp knife cuts and a wide snowshoe does not sink.

Does the shape of a container change the pressure at the bottom?

No. In a liquid at rest, pressure depends on depth, the liquid's density and gravity, not on the width or shape of the vessel. Tall and narrow or short and wide, the same depth gives the same pressure.

How does a hydraulic jack multiply force?

A small piston pushes on a sealed liquid, which passes the same pressure to a larger piston. Force equals pressure times area, so the larger piston delivers a larger force. The larger piston moves proportionally less, so no work is gained.

What is the difference between springs in series and in parallel?

In parallel, springs share the load, so the combination is stiffer. In series, each spring carries the whole load and the extensions add up, so the combination is softer.

Pressure, Hydraulics and Springs Explained

Four rules cover almost every fluid and spring question on a mechanical reasoning test: pressure is force over area, pressure in a sealed liquid is passed on, squeezing a gas raises its pressure, and a spring stretches in proportion to the pull.

Pressure Is Force Over Area

Pressure = force / area. In SI units the force is in newtons, the area in square metres, and the answer in pascals (Pa), where 1 Pa is 1 N on every square metre.

A crate pushes down with 240 N. Resting on a face of 0.04 m², it exerts 240 / 0.04 = 6,000 Pa. Turn it onto a face of 0.12 m² and the same weight exerts 240 / 0.12 = 2,000 Pa. The force is unchanged; the pressure is a third as large because the area is three times as big.

Pressure in a Liquid Depends on Depth

At a depth h in a liquid of density d, pressure from the liquid is d x g x h, where g is about 9.8 N per kilogram. For water, d is 1,000 kg per cubic metre. At 3 m down, the pressure is 1,000 x 9.8 x 3 = 29,400 Pa.

Shape and width do not appear in that formula. A narrow tube and a wide tank, filled to the same depth, give the same pressure at the bottom. It also explains why liquid in connected vessels settles at the same level: if the levels differed, the pressure at the bottom would differ and the liquid would flow until it did not.

To find the force on a hatch, multiply the pressure by the hatch area. A hatch of 0.02 m² at 2 m depth in water feels 1,000 x 9.8 x 2 x 0.02 = 392 N.

Hydraulics: Pressure Passed On, Distance Paid

In a sealed liquid, a pressure applied at one point is passed on equally to every other point. A hydraulic system uses that with two pistons of different areas.

A small piston of 6 cm² is pushed with 40 N, and it connects to a large piston of 90 cm². The pressure is 40 / 6, about 6.67 N per cm², and the large piston delivers 6.67 x 90 = 600 N. The force is multiplied by the area ratio, 90 / 6 = 15.

Now the price. Liquid is effectively incompressible, so the volume pushed out of the small cylinder equals the volume entering the large one. If the small piston moves 15 cm, the large one moves 15 x 6 / 90 = 1 cm. Force in times distance in is 40 N x 0.15 m = 6 N m, and force out times distance out is 600 N x 0.01 m = 6 N m. Same trade as a lever.

Gases: Squeeze It and the Pressure Rises

For a fixed amount of gas at a steady temperature, pressure times volume stays constant: P1 x V1 = P2 x V2. This is Boyle's law. Halve the volume and the pressure doubles.

Air at 100 kPa fills 80 cm³. It is compressed slowly to 50 cm³. The new pressure is 100 x 80 / 50 = 160 kPa. Test questions normally say "slowly" or "constant temperature" so that this simple rule applies. Pressures in this rule are absolute, measured from a vacuum, not gauge readings.

Springs: Stretch in Proportion to the Pull

Within its limit, a spring stretches in proportion to the force: F = k x extension (Hooke's law). The constant k is the stiffness. If 8 N stretches a spring by 4 cm, then k = 2 N per cm, and 14 N stretches it 7 cm.

Watch the wording. The extension is the increase over the natural length, not the total length. A spring of natural length 20 cm that is stretched by 8 N at 2 N per cm is 24 cm long, not 4 cm.

In parallel (side by side), the springs share the load. Two 2 N-per-cm springs carrying 6 N take 3 N each, so each stretches 1.5 cm, and the pair is twice as stiff.
In series (end to end), each spring carries the full load and the stretches add. Two 2 N-per-cm springs under 6 N each stretch 3 cm, so the chain stretches 6 cm, and the pair is half as stiff as one spring.

Try These Four

A force of 180 N presses on 0.06 m². What is the pressure? Answer: 180 / 0.06 = 3,000 Pa.
A hydraulic system has a 3 cm² input piston and a 45 cm² output piston. What input force gives an output of 300 N, and what is the output piston's movement if the input moves 30 cm? Answer: the area ratio is 15, so the input force is 300 / 15 = 20 N, and the output moves 30 / 15 = 2 cm.
A spring stretches 3 cm under 12 N. How far does it stretch under 10 N? Answer: k = 12 / 3 = 4 N per cm, so 10 / 4 = 2.5 cm.
Gas at 150 kPa fills 60 cm³. It is compressed slowly to 45 cm³. What is the new pressure? Answer: 150 x 60 / 45 = 200 kPa.

Practise all four rules with the JobCannon mechanical reasoning test, and see the force-for-distance idea again in pulleys and mechanical advantage. For the slips that cost points here, read common mistakes on mechanical reasoning tests.

Pressure, Hydraulics and Springs: The Rules Behind the Questions