Gravity, weight and circular motion

Newton’s apple and the Moon

Newton realized that the same force that makes an apple fall also keeps the Moon in orbit. The Moon is constantly "falling" toward Earth, but it’s moving sideways so fast that it keeps missing.

The law of universal gravitation

Newton’s law of universal gravitation:

F_g = G m₁m₂ ÷ r²

• G = 6.67 × 10⁻¹¹ N·m²/kg² (the universal gravitational constant)
• m₁, m₂ = masses (kg)
• r = distance between their centres (m)

Inverse square law

Henry Cavendish first measured G in 1798 using a torsion balance, which also let him calculate Earth’s mass.

Example: The gravitational force between two 70 kg people 1 m apart:
F = 6.67 × 10⁻¹¹ × 70 × 70 ÷ 1² ≈ 3.3 × 10⁻⁷ N, far too small to notice.

Weight and gravitational field strength

Weight is the gravitational force on an object:

F_g = mg

Gravitational field strength (g) is the force per kilogram:

g = GM ÷ r²

Astronauts on the ISS still experience about 89% of Earth’s gravity. They float because they and the station are in free fall together, continuously falling around Earth.

Mass stays the same everywhere; weight changes with g.

Uniform circular motion

In uniform circular motion, an object moves in a circle at constant speed. Its velocity keeps changing direction, so it’s accelerating.

Centripetal acceleration

a_c = v² ÷ r

Centripetal acceleration always points toward the centre of the circle.

The speed for one full circle: v = 2πr ÷ T, where T is the period (time for one revolution).

Centripetal force

F_c = mv² ÷ r

Centripetal force isn’t a new kind of force. It’s the net inward force provided by something real:

If the inward force disappears, the object flies off in a straight line tangent to the circle (Newton’s first law). There is no outward "centrifugal force" pulling it; that’s just inertia.

Example: A 1200 kg car rounds a curve of radius 50 m at 15 m/s.
F_c = 1200 × 15² ÷ 50 = 5400 N, supplied by friction. On ice, friction may not be enough, and the car slides straight.

Orbits

For a satellite in a circular orbit, gravity provides the centripetal force:

GMm ÷ r² = mv² ÷ r → v = √(GM ÷ r)

• The satellite’s mass cancels out.
• Lower orbits require higher speeds.

Kepler’s laws

Johannes Kepler described planetary orbits in the early 1600s:

1. Planets move in ellipses, with the Sun at one focus.
2. A planet sweeps equal areas in equal times (it moves faster when closer to the Sun).
3. T² ∝ r³: planets farther away take much longer to orbit.

Newton’s gravitation explained why Kepler’s laws work.

Common mistakes

• Thinking astronauts are weightless because there’s no gravity.
• Believing in an outward centrifugal force in an inertial frame.
• Forgetting to square r in the gravitation equation.

Try it yourself

1. If you moved to twice Earth’s radius from its centre, what would happen to your weight?
2. A 0.2 kg ball on a 0.8 m string moves at 4 m/s in a horizontal circle. Find the tension.
3. Why doesn’t the Moon fall into Earth?

(Answers: it becomes 1/4; T = 0.2 × 16 ÷ 0.8 = 4 N; its sideways speed means it keeps falling around Earth instead of into it)

Quick recap

• F_g = Gm₁m₂ ÷ r²; gravity follows an inverse-square law.
• Weight = mg; g varies by location.
• In circular motion, acceleration is v²/r toward the centre.
• Centripetal force is a net inward force from gravity, tension, friction or normal force.
• Orbits happen when gravity supplies exactly the needed centripetal force.