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You have used a lever with a pivot. The mechanic's spanner turns about the bolt. She pushes at right angles to it, 0.3 m from the bolt.
What perpendicular distance should she use to calculate the moment?
Moments, Pressure & Hooke's Law
Calculate moments using force × perpendicular distance, explain pressure transmission in enclosed liquids and atmospheric pressure, and describe spring extension, Hooke's Law, the limit of proportionality and the elastic limit.
What a learner can do afterwards
- Calculates the moment of a force and uses the principle of moments to solve lever problems
- Explains why hydraulic systems can multiply force (pressure transmitted equally)
- States Hooke's Law, plots force-extension data and distinguishes the limit of proportionality from the elastic limit
- Calculates spring constant k from a force-extension graph (k = F/x)
1 · Read
A lever turns about a pivot. Its turning effect is a moment: moment = force × perpendicular distance, measured in newton metres (Nm). The line of action follows the force's direction through its push point. Use the shortest distance from the pivot to that line. For a push at right angles to a straight lever, this equals the distance along the lever. A force whose line passes through the pivot has zero moment.
Try it together
Decimals name parts of a unit: 0.3 m is 3 tenths of a metre. A mechanic pushes at right angles with 40 N, 0.3 m from a bolt: 40 × 0.3 = 40 × 3 ÷ 10 = 12 Nm. At 0.6 m, force = 12 ÷ 0.6 = 120 ÷ 6 = 20 N. Multiply both numbers in a division by 10 to keep its value. Force = moment ÷ perpendicular distance; perpendicular distance = moment ÷ force. Use metres for Nm: divide centimetres by 100, so 15 cm = 0.15 m.
Pressure is force per area: p = F ÷ A and F = p × A. With N and m², pressure is in pascals (Pa). Pascal's principle says an added pressure in an enclosed liquid is transmitted unchanged throughout it. Total pressure can vary with depth. For ideal hydraulic pistons at the same height, ignore friction: F1 ÷ A1 = F2 ÷ A2. The larger area gives a larger force.
Extension x is loaded length minus unloaded length. In the proportional range, Hooke's Law is F = k × x. Doubling force doubles extension. Find k with F ÷ x, or x with F ÷ k. A larger k means less extension for the same force. Use N/cm with centimetres or N/m with metres: 1 cm = 0.01 m, so 1 N/cm = 100 N/m.
The limit of proportionality marks where force and extension stop being proportional. Beyond the elastic limit, a spring has a permanent extension and no longer returns to its original length when unloaded. These limits need not coincide: a spring can stop obeying Hooke's Law while still returning to its original length.
For a lever at rest, total clockwise moment equals total anticlockwise moment. Ignore other moments only when the question tells you to. Add moments turning the same way; subtract opposing moments to find the net moment. For example, 18 Nm clockwise against 6 Nm anticlockwise leaves 12 Nm clockwise. For balance, find the known moment, then divide it by the other side's perpendicular distance to find the missing force.
Try it together
An ideal hydraulic lift has pistons at the same height. A 12 N input acts on 0.004 m². For p = F ÷ A, multiply both numbers by 1000: 12 ÷ 0.004 = 12000 ÷ 4 = 3000 Pa. An output area of 0.02 m² gives F = 3000 × 0.02 = 3000 × 2 ÷ 100 = 60 N. Both pistons displace equal volumes: area × travel is equal on both sides. An output area five times larger gives one fifth the travel. Ignore leaks, friction and liquid compression.
Air molecules collide with surfaces from all directions, producing atmospheric pressure. Gravity holds the atmosphere near Earth. Higher up, pressure is generally lower because less air lies above. If air is removed from a sealed thin can, pressure inside falls. The greater pressure outside can crush it. This is a pressure difference, not a change in the metal's stiffness.
Try it together
A spring is 11 cm long without a load and 14 cm long under 9 N. Its extension is 14 - 11 = 3 cm, not 14 cm. Within the proportional range, k = 9 ÷ 3 = 3 N/cm. In metres, x = 0.03 m and k = 9 ÷ 0.03 = 300 N/m. The spring constant is the same physical stiffness expressed in different units.
For a force-extension graph, put extension on the horizontal axis and force on the vertical axis. Label both axes with units, choose even scales and plot each measured pair. Hooke's Law gives a straight line through the origin. Its slope is k: change in force ÷ change in extension. A curve shows loss of proportionality. To test the elastic limit, unload and check whether the spring returns to its original length.
Use perpendicular distance for moments. Added liquid pressure is transmitted unchanged. Hooke's Law ends at the limit of proportionality; permanent stretching begins beyond the elastic limit.
The twelve questions and five readings were written for this dataset. Physics references were consulted for facts and assessment design. Video references are credited individually in the file; their publisher terms have not been verified.
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
Then practise
36 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.