Newton's universal law of gravitation
Every body attracts every other body. The force is F = Gm₁m₂/r²: it grows with the masses and falls with the square of the distance.
A force between all bodies
A stone thrown up comes back down. Ripe mangoes fall from the tree. The earth pulls them towards itself. Newton saw that the same pull keeps the moon moving round the earth, and the planets round the sun. He concluded that every object in the universe attracts every other object. This force is the gravitational force. It is always attractive.
Newton's law of gravitation
Every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining them. In symbols, F = G m₁m₂/r². G is the universal gravitational constant, 6.67 × 10⁻¹¹ N m² kg⁻². It is called universal because its value is the same everywhere.
How the force changes
| Change | Effect on F | Why |
|---|---|---|
| One mass doubled | F doubles | F ∝ m₁m₂ |
| Both masses doubled | F becomes 4 times | 2 × 2 = 4 |
| Distance doubled | F becomes one fourth | F ∝ 1/r², and 2² = 4 |
| Distance halved | F becomes 4 times | r² becomes one fourth |
| Distance made 4 times | F becomes 1/16 | 4² = 16 |
Worked example: two people 2 m apart
Two people of masses 60 kg and 45 kg stand 2 m apart. Find the gravitational force between them. (G = 6.67 × 10⁻¹¹ N m² kg⁻²)
- F = G m₁m₂/r².
- m₁m₂ = 60 × 45 = 2700 kg², and r² = 2² = 4 m².
- F = 6.67 × 10⁻¹¹ × 2700 ÷ 4 = 6.67 × 10⁻¹¹ × 675.
- F = 4502 × 10⁻¹¹ N = 4.5 × 10⁻⁸ N.
About 4.5 × 10⁻⁸ N, far too small to feel. That is why we do not notice the pull between ordinary objects
Worked example: the earth and a student
Find the force of the earth on a 50 kg student standing on the ground. Mass of the earth M = 6 × 10²⁴ kg, radius R = 6.4 × 10⁶ m, G = 6.67 × 10⁻¹¹ N m² kg⁻².
- The distance from the earth's centre is its radius, so r = R = 6.4 × 10⁶ m.
- F = G M m/R² = 6.67 × 10⁻¹¹ × 6 × 10²⁴ × 50 ÷ (6.4 × 10⁶)².
- Top: 6.67 × 6 × 50 = 2001, so the top is 2001 × 10¹³ = 2.001 × 10¹⁶.
- Bottom: (6.4 × 10⁶)² = 40.96 × 10¹² = 4.096 × 10¹³.
- F = 2.001 × 10¹⁶ ÷ 4.096 × 10¹³ ≈ 489 N.
About 489 N. This large pull is the student's weight. It is large because the earth's mass is huge
Why we feel the earth's pull but not each other's
- G is very small, so the force between ordinary objects is tiny.
- The earth's mass is huge (about 6 × 10²⁴ kg), so its pull on us is large.
- The same force holds the moon in its orbit round the earth, and the earth round the sun.
- The gravitational pull of the earth on a body is called gravity.
ISRO's weather and TV satellites stay in orbit because the earth's gravity keeps pulling them round. Without it they would fly off into space in a straight line.
2 marks: “State Newton's universal law of gravitation and write it as a formula.” Every body attracts every other body with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. F = G m₁m₂/r².
1 marks: “How does the gravitational force change if the distance between two bodies is made three times?” It becomes one ninth of its value, because F ∝ 1/r² and 3² = 9.
3 marks: “Find the gravitational pull of the sun on Jupiter. Take the mass of the sun as 2 × 10³⁰ kg, the mass of Jupiter as 1.9 × 10²⁷ kg and the distance between them as 7.8 × 10¹¹ m. (G = 6.67 × 10⁻¹¹ N m² kg⁻²)” Method: F = G m₁m₂/r². Top = 6.67 × 10⁻¹¹ × 2 × 10³⁰ × 1.9 × 10²⁷ = 2.53 × 10⁴⁷. Bottom = (7.8 × 10¹¹)² = 6.08 × 10²³. F = 2.53 × 10⁴⁷ ÷ 6.08 × 10²³ ≈ 4.2 × 10²³ N.
Masses multiply; distance is squared and divides: F = Gm₁m₂/r².