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Gravitation: Class 9 Physics
Newton's law of gravitation, the gravitational constant G, mass of the Earth, variation of g with height, weight in a satellite, orbital motion, artificial satellites and the speed of a satellite.
Key points
Every body in the universe attracts every other body. This force is called gravitation. It is a force of attraction only, it acts at a distance and it is always present. We do not notice it between small objects because it is very weak. It becomes large when one of the bodies is as massive as the Earth, the Moon or the Sun.
Newton's 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. In symbols F = G m₁m₂/r². The force acts along the line joining the centres. The two bodies pull each other with equal and opposite forces, as the third law requires.
G is the universal gravitational constant. Its value is 6.67 × 10⁻¹¹ N m²/kg². It has the same value everywhere in the universe, which is why it is called universal. Its small size shows why gravitation between everyday objects cannot be seen. G was first measured by Henry Cavendish using a sensitive torsion balance. Doubling the distance between two bodies reduces the force to one quarter. Doubling one mass doubles the force. The law is an inverse square law. It holds exactly for point masses and for spheres, whose mass can be taken at the centre.
The acceleration due to gravity g is the acceleration given to a freely falling body by the Earth. The weight of a body of mass m is mg. The Earth's pull on this body is also GMm/R², where M and R are the mass and radius of the Earth. Equating the two gives g = GM/R². This does not depend on the mass of the falling body, so all bodies fall with the same acceleration.
The mass of the Earth can be found from g = GM/R², so M = gR²/G. With g = 9.8 m/s², R = 6.4 × 10⁶ m and G = 6.67 × 10⁻¹¹ N m²/kg², M is about 6 × 10²⁴ kg. Henry Cavendish's measurement of G is what made it possible to find the mass of the Earth.
The value of g changes with position:
- With height: g decreases as we go higher, because r in GM/r² becomes larger.
- With depth: g decreases towards the centre of the Earth.
- With latitude: g is slightly larger at the poles than at the equator, because the Earth is slightly flattened at the poles and its radius there is smaller.
- On the Moon g is about 1.6 m/s², roughly one sixth of its value on Earth.
At height h above the surface, the distance from the centre of the Earth is R + h. So g at that height is GM/(R + h)². The value is smaller than at the surface, and it becomes zero only at an infinite distance.
Weight depends on g. If a body is taken to a place where g is smaller, its weight is smaller but its mass is the same. An astronaut inside an orbiting satellite feels weightless. This is not because gravity is zero. The astronaut and the satellite are both falling freely towards the Earth with the same acceleration, so there is no contact force between them. This is apparent weightlessness.
An artificial satellite is a man-made body that orbits the Earth or another planet. The Moon is a natural satellite. A satellite keeps moving in a circle because the Earth's gravity supplies the centripetal force: mv²/r = GMm/r². Its orbital speed is v = √(GM/r), where r is the distance from the Earth's centre. A higher orbit means a smaller speed and a longer time period.
The Earth's gravitational pull on a satellite is the centripetal force, and the satellite is always falling towards the Earth, but it also moves sideways so fast that the surface curves away beneath it. A satellite near the surface needs a speed of about 8 km/s.
Artificial satellites are used for:
- Communication: television, telephone and internet signals.
- Weather forecasting and observing storms and floods.
- Navigation systems such as GPS.
- Remote sensing and studying the Earth and space.
A geostationary satellite orbits above the equator at about 36,000 km above the surface. It has a time period of 24 hours, the same as the Earth's rotation, and moves in the same direction. So it always stays above the same place and is used mostly for communication. Pakistan's PakSat-1R is an example of a communication satellite.
Study tip: Write the law, its formula, and what each symbol means before substituting. Convert every distance to metres and keep powers of ten in order. For explanation questions on weightlessness or satellites, say clearly that gravity is still acting and that free fall is the reason.
Definitions and flashcards
Say the meaning first, then open the card to check.
Gravitation
The force of attraction between any two bodies in the universe.
Gravitational constant G
The constant in Newton's law of gravitation, 6.67 × 10⁻¹¹ N m²/kg².
Inverse square law
A law in which the force falls to one quarter when the distance is doubled.
Acceleration due to gravity
The acceleration of a freely falling body, g = GM/R².
Weightlessness
The feeling of no weight when a body and its surroundings fall freely together.
Artificial satellite
A man-made body that orbits a planet.
Orbit
The path followed by a satellite around a planet.
Geostationary satellite
A satellite that stays above the same point on the equator with a period of 24 hours.
Natural satellite
A body such as the Moon that orbits a planet by itself.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Law of gravitation | F = G m₁m₂/r² | Force between two masses a distance r apart |
| Value of G | G = 6.67 × 10⁻¹¹ N m²/kg² | Universal gravitational constant |
| Acceleration due to gravity | g = GM/R² | At the surface of a planet of mass M and radius R |
| Mass of the Earth | M = gR²/G | From g, R and G |
| g at height h | g' = GM/(R + h)² | Smaller than g at the surface |
| Orbital speed | v = √(GM/r) | Speed of a satellite in a circular orbit of radius r |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
State Newton's law of gravitation. (2 marks)
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 their centres. In symbols F = G m₁m₂/r².
Why do we not feel the gravitational force between two students sitting next to each other? (2 marks)
The force is proportional to the product of the masses and G is very small, 6.67 × 10⁻¹¹ N m²/kg². The masses of students are small compared with the Earth, so the force is tiny and cannot be noticed.
How does the force between two bodies change if the distance between them is doubled? (2 marks)
The force is inversely proportional to the square of the distance. If the distance becomes 2r, the force becomes F/2², which is one quarter of the original force.
Derive the relation g = GM/R². (3 marks)
The weight of a body of mass m on the Earth's surface is mg. By the law of gravitation the pull of the Earth, of mass M and radius R, on this body is GMm/R². These are the same force, so mg = GMm/R². Cancelling m gives g = GM/R². The value of g does not depend on m.
Why does an astronaut in an orbiting satellite feel weightless? (3 marks)
Gravity still acts on the astronaut. But the astronaut and the satellite fall towards the Earth with the same acceleration. There is no force of contact between the astronaut and the floor of the satellite, so no weight is felt. This is apparent weightlessness.
What is a geostationary satellite? Describe its properties and uses. (5 marks)
A geostationary satellite is a satellite that appears fixed above one point of the Earth. It orbits above the equator at a height of about 36,000 km, in the same direction as the Earth's rotation. Its time period is 24 hours, equal to that of the Earth, so it is always above the same place. Its speed is constant in the circular orbit, and gravity provides the centripetal force. It is used for telecommunication, television broadcasting and weather observation of a fixed region.
Numericals with worked steps
Two bodies of masses 50 kg and 100 kg are 2 m apart. Find the gravitational force between them. Take G = 6.67 × 10⁻¹¹ N m²/kg².
- F = G m₁m₂/r²
- m₁m₂ = 50 × 100 = 5000 kg², r² = 4 m²
- F = 6.67 × 10⁻¹¹ × 5000/4 = 6.67 × 10⁻¹¹ × 1250
- F = 8.34 × 10⁻⁸ N (about)
Answer: About 8.3 × 10⁻⁸ N
The distance between two bodies is doubled and one of the masses is also doubled. By what factor does the force change?
- Original F = G m₁m₂/r²
- New F = G (2m₁) m₂/(2r)² = 2 G m₁m₂/(4r²)
- New F = ½ × (G m₁m₂/r²)
Answer: The force becomes half
Find the mass of the Earth. Take g = 10 m/s², R = 6.4 × 10⁶ m and G = 6.67 × 10⁻¹¹ N m²/kg².
- M = gR²/G
- R² = (6.4 × 10⁶)² = 40.96 × 10¹² = 4.096 × 10¹³ m²
- gR² = 10 × 4.096 × 10¹³ = 4.096 × 10¹⁴
- M = 4.096 × 10¹⁴/(6.67 × 10⁻¹¹) = 0.614 × 10²⁵ = 6.1 × 10²⁴ kg
Answer: About 6 × 10²⁴ kg
On the Moon g = 1.6 m/s². Find the weight of a 60 kg astronaut on the Moon and on Earth (g = 10 m/s²).
- Mass is the same on both: 60 kg
- Weight on Earth = mg = 60 × 10 = 600 N
- Weight on Moon = 60 × 1.6 = 96 N
Answer: 600 N on Earth, 96 N on the Moon
Common mistakes
- Measuring r from the surface instead of from the centre of the Earth. At height h, use R + h.
- Saying there is no gravity in a satellite. Gravity is there and supplies the centripetal force.
- Confusing G with g. G is a universal constant. g depends on the planet and on the place.
- Forgetting to square the distance in the inverse square law.
- Giving the weight in kg. Mass is in kg, and weight is in N.