AS Physics: Forces, Density and Pressure — Revision Notes

Condensed for the final weeks. For the full explanation, use the Forces, Density and Pressure study guide.

Moments and equilibrium

moment = force x perpendicular distance from the pivot

“Perpendicular” is not optional. If the force is at an angle, use the perpendicular component or the perpendicular distance — a favourite trap.

Principle of moments: for a body in rotational equilibrium, the sum of clockwise moments about any point equals the sum of anticlockwise moments.

Two conditions for full equilibrium:

1. Resultant force = 0 (translational).
2. Resultant moment = 0 about any point (rotational).

Both are required. An object can have zero resultant force and still rotate — which is exactly what a couple does: two equal, opposite, parallel forces not in line.

torque of a couple = one force x perpendicular separation

Choosing the pivot cleverly is the exam technique: take moments about a point where an unknown force acts, and that unknown drops out of the equation.

Worked example. A 30 N weight hangs 0.8 m from a pivot. What distance must a 20 N weight hang on the other side to balance it? 30 × 0.8 = 20 × d, so d = 24 ÷ 20 = 1.2 m.

Three coplanar forces in equilibrium can be represented as a closed vector triangle — each force drawn as an arrow, tip-to-tail, returning to the starting point — a standard way to find an unknown force or angle.

Centre of gravity

The single point where the entire weight of the body appears to act. For a uniform body it is at the geometric centre.

An object topples when the line of action of its weight falls outside the base. Stability is improved by a lower centre of gravity and a wider base — this is why racing cars are built low and wide, and why a tall, narrow object topples more easily than a short, wide one of the same weight.

Density and pressure

density   rho = m / V
pressure  p = F / A
fluid     p = rho g h

Pressure in a fluid depends only on depth, density and g — not on the shape or volume of the container. That is why a narrow tube of water can exert the same pressure as a wide tank of the same depth, and it is examined regularly.

Upthrust and Archimedes’ principle

Upthrust equals the weight of fluid displaced.

The physical origin is worth stating: pressure increases with depth, so the upward pressure on the bottom of a submerged object exceeds the downward pressure on the top, and the difference gives a net upward force.

upthrust = rho g V        V = volume of fluid displaced

Note that V is the volume of fluid displaced, not necessarily the object’s total volume — for a partly submerged object, only the submerged fraction counts.

Floating: upthrust = weight, so the object displaces its own weight of fluid. Sinking: weight > maximum upthrust.

An object floats if its average density is less than the fluid’s — which is why a steel ship floats: its overall density including the enclosed air is below that of water, even though steel itself is denser. If the hull is breached and fills with water, the average density rises above that of water and the ship sinks.

Exam traps

• Omitting “perpendicular” from the definition of a moment.
• Giving only one condition for equilibrium.
• Thinking pressure in a liquid depends on the container’s width or total volume.
• Saying upthrust depends on the object’s weight rather than the fluid displaced.
• Forgetting that a couple gives rotation with zero resultant force.
• Using the diameter instead of the radius when finding an area.
• Forgetting that zero resultant force alone permits rotation, and zero resultant moment alone permits acceleration — both conditions are needed for full equilibrium.
• Not returning to the starting point when drawing a vector triangle for three coplanar forces.

Self-test

1. Define a moment precisely.
2. State both conditions for equilibrium.
3. Why does the shape of a container not affect the pressure at a given depth?
4. State Archimedes’ principle and explain where upthrust comes from.
5. Why does a steel ship float?
6. A 40 N weight hangs 0.5 m from a pivot. What distance must a 25 N weight hang on the other side to balance it?
7. How can three coplanar forces in equilibrium be represented graphically?

Answers: 1. The product of the force and the perpendicular distance from the pivot to the line of action of the force. 2. The resultant force must be zero and the resultant moment about any point must be zero. 3. Pressure in a fluid is given by ρgh, which depends only on depth, density and gravitational field strength. 4. Upthrust equals the weight of fluid displaced; it arises because pressure increases with depth, so the upward pressure on the underside exceeds the downward pressure on the top. 5. Its average density, including the enclosed air, is less than that of water, so it displaces its own weight before becoming fully submerged. 6. 40 × 0.5 = 25 × d, so d = 20 ÷ 25 = 0.8 m. 7. As a closed vector triangle, with each force drawn as an arrow tip-to-tail, returning to the starting point.