Gravitation, mass and weight

An apple falls to the ground. The moon circles the earth. The earth circles the sun. Newton saw that all three have the same cause: every object in the universe attracts every other object. This attraction is called gravitation.

The universal law of gravitation

Every body attracts every other body with a force that is

• directly proportional to the product of their masses, and
• inversely proportional to the square of the distance between their centres.

F = G × (m₁ × m₂) ÷ d²

Here G is the universal gravitational constant. Its value, 6.67 × 10⁻¹¹ N m²/kg², was first measured by Henry Cavendish. Because G is so small, the pull between two ordinary objects, such as two people, is far too weak to notice. It becomes large only when at least one of the bodies is huge, like the earth.

The law explains the force that binds us to the earth, the motion of the moon and the planets, and the tides, which are caused mainly by the pull of the moon on the sea.

Free fall and g

When an object falls under the earth's gravity alone, it is in free fall. Its velocity increases as it falls. This acceleration is called the acceleration due to gravity, g. Near the surface of the earth its value is about 9.8 m/s²: the velocity of a falling body rises by 9.8 m/s every second.

The value of g does not depend on the mass of the falling body. Galileo argued that a heavy stone and a light stone dropped together would land together. A feather falls slowly only because of air resistance. In a tube with the air pumped out, a coin and a feather fall side by side.

g is slightly greater at the poles than at the equator, because the earth is a little flattened at the poles, and it decreases as we go high above the surface.

Mass and weight

Mass is the amount of matter in a body. It is measured in kilograms and is the same everywhere: on the earth, on the moon or in deep space.

Weight is the force with which a planet pulls the body towards its centre.

Weight = mass × g, or W = mg

Weight is a force, so its SI unit is the newton. A body of mass 10 kg weighs 10 × 9.8 = 98 N on the earth.

The moon is smaller and lighter than the earth, so its pull is weaker: about one-sixth of the earth's. A person who weighs 600 N on the earth weighs only 100 N on the moon, though the mass is unchanged. That is why astronauts could leap so easily there.