The universal law of gravitation
The force that pulls an apple to the ground is the same force that keeps the Moon in orbit around the Earth. Isaac Newton expressed it as the universal law of gravitation:
Every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
F = G M m ⁄ d²
where M and m are the masses, d the distance between their centres, and G the universal gravitational constant:
G = 6.67 × 10⁻¹¹ N m² kg⁻²
G is very small, so the force between everyday objects is tiny. It becomes large only when at least one mass is huge, like the Earth.
Effects of changes:
Doubling one mass doubles the force.
Doubling the distance makes the force one-quarter (1/2²).
Halving the distance makes it four times as large.
The law explains the motion of the Moon and planets, the tides (caused mainly by the Moon's pull), and the force that binds us to the Earth.
Free fall and g
When an object falls under gravity alone, it is in free fall. Its velocity changes, so it has an acceleration, called the acceleration due to gravity, g.
From F = mg and F = GMm/R²:
g = G M ⁄ R²
where M is the mass of the Earth and R its radius. On the Earth's surface, g ≈ 9.8 m/s².
Notice that g does not depend on the mass of the falling object. Without air resistance, a feather and a stone fall together. (In air the feather is slowed more by air resistance, which is why it seems to fall slowly.)
Because the Earth is slightly flattened at the poles, R is smaller there, so g is greater at the poles than at the equator. g also decreases as you go higher above the surface.
The equations of motion work for free fall with a replaced by g (taking g as positive downward):
v = u + gt
s = ut + ½gt²
v² = u² + 2gs
Example. A stone is dropped from a cliff and hits the ground after 3 s. Taking g = 10 m/s², its speed on impact is 0 + 10 × 3 = 30 m/s, and the cliff's height is ½ × 10 × 3² = 45 m.
Mass and weight
Mass | Weight |
|---|---|
amount of matter in an object | force with which the Earth (or another body) pulls the object |
same everywhere | changes with g |
measured in kg | measured in newtons (N) |
scalar | vector, directed towards the centre of the Earth |
W = m × g
Weight on the Moon. The Moon's g is about 1/6 of the Earth's, so an object weighs about one-sixth as much there. A person of mass 60 kg weighs 60 × 9.8 = 588 N on Earth and about 98 N on the Moon, while their mass is still 60 kg.
Thrust and pressure
Thrust is the force acting on an object perpendicular to its surface. Pressure is the thrust per unit area:
P = thrust ⁄ area
The SI unit is the pascal (Pa), which is 1 N/m².
The same force produces more pressure on a smaller area:
Knives are sharpened so that a small edge produces a large pressure.
Buildings have wide foundations, and trucks and tractors have wide tyres, to reduce pressure on the ground.
Standing on sand, you sink less when lying down than standing.
Example. A block of weight 50 N rests on a face of area 0.25 m². Pressure = 50 ÷ 0.25 = 200 Pa. On a face of 0.05 m² it would be 1000 Pa.
Fluids exert pressure in all directions.
Buoyancy and Archimedes' principle
Push an empty plastic bottle into water and you feel it pushing back up. This upward force exerted by a fluid on an object immersed in it is upthrust or buoyant force.
An object sinks if its density is greater than the liquid's; it floats if its density is less.
A cork floats on water, while an iron nail of the same mass sinks, because iron is much denser.
Archimedes' principle: when an object is fully or partially immersed in a fluid, it experiences an upward force equal to the weight of the fluid it displaces.
Uses: designing ships and submarines; lactometers to check the purity of milk; hydrometers to measure the density of liquids.
Relative density = density of a substance ÷ density of water. It has no unit. The relative density of silver is 10.8, meaning silver is 10.8 times as dense as water.
Summary
F = GMm/d²; G = 6.67 × 10⁻¹¹ N m² kg⁻²; force falls with the square of distance.
g = GM/R² ≈ 9.8 m/s² on Earth and is independent of the falling mass.
Weight = mg; mass is constant, but weight on the Moon is about 1/6 of that on Earth.
Pressure = thrust/area; upthrust equals the weight of fluid displaced.