Newton's law of gravitation and acceleration due to gravity
Every two masses attract each other with a force F = Gm₁m₂/r². On Earth this pull gives every falling body the same acceleration, g = 9.8 m s⁻².
Newton's law of gravitation
Every particle of matter in the universe attracts every other particle with a force. This force is directly proportional to the product of their masses. It is inversely proportional to the square of the distance between their centres. It acts along the line joining the two masses. F = G m₁m₂ / r². Here G is the universal gravitational constant: 6.674 × 10⁻¹¹ N m² kg⁻². G is numerically equal to the force, in newtons, between two 1 kg masses kept 1 m apart. If the distance is doubled, the force becomes one-fourth.
Deriving g = GM/R² and the mass of the Earth
- 1Take a body of mass m on the Earth's surface. The Earth has mass M and radius R.
- 2Force of gravity on the body (law of gravitation): F = GMm/R².
- 3The same force gives the body an acceleration g, so F = mg (second law).
- 4★Equate the two: mg = GMm/R², so g = GM/R². g does not depend on the mass of the falling body.
- 5Rearrange: M = gR²/G. Take g = 9.8 m s⁻², R = 6.378 × 10⁶ m and G = 6.674 × 10⁻¹¹ N m² kg⁻². Then M = 9.8 × (6.378 × 10⁶)² ÷ (6.674 × 10⁻¹¹) ≈ 5.97 × 10²⁴ kg. Your textbook gives 5.972 × 10²⁴ kg.
★ key step
How the value of g changes
- g depends on the distance from the centre of the Earth, so it is not the same everywhere.
- With height: g decreases as you go up, because you are farther from the centre.
- With depth: g decreases as you go down a deep mine, and is zero at the centre of the Earth.
- Poles and equator: the Earth is slightly flattened at the poles, so the polar radius is smaller. g is greatest at the poles and least at the equator.
- On the Moon, g is about 1.625 m s⁻², roughly one-sixth of g on the Earth.
Applications of the law of gravitation
- Measuring heavenly bodies: the mass and radius of the Earth, and the value of g, can be found accurately.
- Discovering new stars and planets.
- A star can wobble (move slightly to and fro) because of the pull of a planet near it. The law helps find the mass of such a star.
- Explaining why plant roots grow downward: they respond to gravity (geotropism).
- Predicting the paths of planets, comets and other astronomical bodies.
Force between two bodies
Two bodies of masses 50 kg and 60 kg are 1 m apart. Find the gravitational force between them. (G = 6.674 × 10⁻¹¹ N m² kg⁻²)
- Given: m₁ = 50 kg, m₂ = 60 kg, r = 1 m.
- Formula: F = G m₁m₂ / r².
- F = 6.674 × 10⁻¹¹ × 50 × 60 ÷ 1² N.
- 50 × 60 = 3000 = 3 × 10³, so F = 6.674 × 3 × 10⁻⁸ N = 20.022 × 10⁻⁸ N.
F ≈ 2.0 × 10⁻⁷ N (rounded). This force is tiny, which is why we do not feel the pull between two people.
A coconut falls from the tree and a ball thrown up comes back because the Earth pulls them. Seeds sprouting in a pot always send their roots downward: that is gravity at work in plants.
1 marks: “What is the value of the universal gravitational constant G?” 6.674 × 10⁻¹¹ N m² kg⁻².
2 marks: “State Newton's law of gravitation.” Every particle attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them; F = Gm₁m₂/r².
4 marks: “Derive the relation between g and G. Use it to find the mass of the Earth.” F = GMm/R² and F = mg; equate to get g = GM/R²; M = gR²/G = 9.8 × (6.378 × 10⁶)² ÷ (6.674 × 10⁻¹¹) ≈ 5.97 × 10²⁴ kg.
Double the distance, a quarter of the force: gravity follows the inverse square of the distance.