The force that makes an apple fall also holds the Moon in its orbit: this was Newton’s great discovery. The law of universal gravitation united earthly and heavenly phenomena: falling bodies, ocean tides, the motion of planets, artificial satellites and space flight are all explained by one law. The topic brings together Newton’s laws and circular motion.
§1. The law of universal gravitation
By analysing the Moon’s motion, Newton (1687) concluded that the force pulling it towards the Earth is of the same nature as the force on an apple, and he stated the general law:
F = G·m₁m₂/r², G = 6.67·10⁻¹¹ N·m²/kg²
Any two bodies attract each other; the force is proportional to the product of their masses and inversely proportional to the square of the distance between them.
The formula is exact for point masses and uniform spheres; for a sphere r is the distance between centres.
Gravitational forces obey Newton’s third law: the Earth pulls the apple exactly as hard as the apple pulls the Earth.
G is very small, so the attraction between ordinary objects is unnoticeable. Cavendish (1798) first measured it in the laboratory with a torsion balance.
Doubling the distance makes the force 4 times smaller; tripling it, 9 times smaller.
Worked example
The Moon is 3.84·10⁸ m (about 60 Earth radii) from the Earth’s centre and orbits once every 27.3 days. Find its centripetal acceleration and compare it with g/60².
T = 27.3·86 400 ≈ 2.36·10⁶ s.
a = 4π²r/T² = 4·9.87·3.84·10⁸/(2.36·10⁶)² ≈ 0.0027 m/s².
g/3600 = 9.8/3600 ≈ 0.0027 m/s², which agrees; Newton checked his law with exactly this calculation.
§2. Acceleration due to gravity
The weight of a body at the Earth’s surface is the Earth’s gravitational pull: mg = GMm/R². Hence
g = GM/R²; at height h: g_h = GM/(R + h)² = g·R²/(R + h)²
g does not depend on the body’s mass, which is why all bodies fall alike without air.
At a height equal to the Earth’s radius (h = R), g is 4 times smaller. On top of the highest mountain (about 8.8 km) it is only about 0.3 % smaller, so near the surface g can be treated as constant.
g is slightly larger at the poles than at the equator: the Earth is flattened at the poles and it rotates.
On another planet g depends on its mass and radius: g′/g = (M′/M)·(R/R′)².
This formula let scientists “weigh” the Earth: M = gR²/G.
Worked example
What is g on a planet with twice the Earth’s mass and twice its radius?
g′ = G·2M/(2R)² = (2/4)·GM/R².
g′ = g/2 = 5 m/s².
Worked example
Using g = 10 m/s² and R = 6.4·10⁶ m, find the mass of the Earth.
M = gR²/G = 10·(6.4·10⁶)²/(6.67·10⁻¹¹).
M ≈ 6.1·10²⁴ kg (the accepted value is ≈ 6·10²⁴ kg).
§3. Satellites and cosmic speeds
A satellite in a circular orbit feels only gravity, which provides the centripetal acceleration: GMm/r² = mv²/r.
Orbital speed: v = √(GM/r); period: T = 2πr/v
The first cosmic speed is the speed of a circular orbit just above the Earth’s surface:
v₁ = √(gR) = √(9.8·6.4·10⁶) ≈ 7.9 km/s
The second cosmic speed (escape speed) is the smallest speed needed to overcome the Earth’s gravity and leave it for ever:
v₂ = √2·v₁ ≈ 11.2 km/s
The third cosmic speed (leaving the Solar System) is ≈ 16.7 km/s. The larger the orbit, the lower the orbital speed and the longer the period. A geostationary satellite orbits above the equator with the Earth’s rotation period (≈ 24 h) at a height of about 36 000 km and seems to “hang” in one place, which is how communication and TV satellites work. The first artificial satellite was launched on 4 October 1957.
Worked example
A planet has 4 times the Earth’s mass and twice its radius. What is its first cosmic speed?
v₁ = √(GM/R) → v₁′/v₁ = √(4/2) = √2.
v₁′ = √2·7.9 ≈ 11.2 km/s.
§4. Kepler’s laws
Kepler (early seventeenth century) found three laws of planetary motion from Tycho Brahe’s observations; Newton later derived them from the law of gravitation.
Planets move in ellipses with the Sun at one focus.
The line joining a planet to the Sun sweeps out equal areas in equal times, so a planet moves faster near the Sun.
The square of the orbital period divided by the cube of the semi-major axis is the same for all planets:
T₁²/T₂² = a₁³/a₂³
For a circular orbit the third law reads T² = 4π²r³/(GM); it depends only on the mass of the central body.
Worked example
The semi-major axis of Mars’s orbit is 1.52 times the Earth’s. How many Earth years is a Martian year?
T_M²/T_E² = (1.52)³.
T_M = 1.52^1.5 ≈ 1.87 Earth years.
§5. Weightlessness and gravitational energy
An astronaut in orbit is weightless: both the astronaut and the spacecraft are in free fall under gravity alone with the same acceleration, so the astronaut does not press on the floor. Gravity has not disappeared; it is what keeps the orbit. The gravitational potential energy of two bodies is
E_p = −GMm/r
At infinity E_p = 0; as the bodies come closer the energy becomes negative and smaller. The escape speed follows from it: mv₂²/2 − GMm/R = 0 → v₂ = √(2GM/R) = √2·v₁. For small heights near the Earth’s surface the formula reduces to the familiar mgh.
§6. Gravity at work
Tides: the Moon pulls the near side of the Earth more strongly than the far side. This difference raises two “bulges” in the ocean, and as the Earth turns, the water rises twice a day in many places. The Sun contributes too, but its effect is roughly half the Moon’s.
Round planets and stars: in large bodies their own gravity pulls matter towards the centre and makes them spherical.
Everyday objects: the attraction between them is so small that only very sensitive instruments can detect it.
Worked example
Two ships of 50 000 t each are 1 km apart. With what force do they attract each other?
m = 5·10⁷ kg, r = 10³ m.
F = Gm²/r² = 6.67·10⁻¹¹·(5·10⁷)²/10⁶ ≈ 0.17 N, billions of times less than the ships’ weight.
Check yourself
The distance between two bodies becomes 3 times smaller. How does the gravitational force change?
Acceleration due to gravity at height h = R?
g on a planet with the Earth’s mass and half its radius?
The first cosmic speed is 7.9 km/s. The second?
The orbit radius increases 4 times. How does the orbital speed change?
Masses of 1 kg and 1 kg are 1 m apart. Gravitational force?
A planet’s semi-major axis is 4 times larger. By what factor is its period longer?
Does gravity act on an astronaut in orbit?
Common mistakes
Measuring r from the surface: it is measured from the centre, r = R + h.
Forgetting that the force is inversely proportional to the square of the distance, not the distance.
Thinking there is no gravity in orbit: weightlessness is not the absence of gravity.
Swapping the formulas for v₁ and v₂.
Calculating with G without converting kilometres to metres.