CBSE Class 10 Physics Gravitation Notes Set 07

Universal Law of Gravitation

Every object in the universe pulls on every other object. The pull is proportional to the product of the two masses and inversely proportional to the square of the distance between their centres, and it acts along the line joining the centres.

\( F = \dfrac{G m_1 m_2}{r^2} \)

1. G is the universal gravitational constant: the force between two 1 kg masses kept 1 m apart. Its value is \( 6.67\times10^{-11} \) N m\( ^2 \) kg\( ^{-2} \), first measured by Cavendish.
2. The two forces form an action-reaction pair: equal in size, opposite in direction, acting on different bodies.
3. Doubling the distance cuts the force to one quarter; doubling one mass doubles the force.
4. The force is attractive, acts at a distance, and does not depend on the medium in between.
5. Gravitation is very weak for everyday objects but dominates for planets, stars and moons because their masses are huge.

Why the Moon Does Not Fall Onto the Earth

The moon moves with a velocity along the tangent to its orbit, while gravity pulls it towards the earth. Gravity supplies the centripetal force that keeps bending its path, so the moon keeps falling around the earth without hitting it. Without gravity it would move off along a straight tangent; without the sideways velocity it would fall straight down.

Kepler's Laws of Planetary Motion

Free Fall and Acceleration Due to Gravity

A body moving under gravity alone is in free fall. Its acceleration, g, is found by equating weight with the gravitational pull of the earth:

\( g = \dfrac{GM}{R^2} \) with g = 9.8 m/s\( ^2 \) on the earth (980 cm/s\( ^2 \)).

1. g does not depend on the mass of the falling body.
2. For two planets, \( g \propto M/R^2 \).
3. Height h above the surface: \( g' = \dfrac{GM}{(R+h)^2} \), which is about \( g(1 - 2h/R) \) for small h.
4. Depth d below the surface: \( g' = g(1 - d/R) \); at the centre of the earth g = 0.
5. g is smallest at the equator and largest at the poles, because the earth is flattened and rotates.
6. On the moon g is about one-sixth of its value on the earth.

Mass and Weight

A freely falling body, or one at the centre of the earth, has zero weight (apparent weight), but its mass is unchanged.

Satellites and Escape Speed

1. Orbital speed at height h: \( v_o = \sqrt{\dfrac{GM}{R+h}} \); close to the surface \( v_o = \sqrt{gR} \), about 7.9 km/s.
2. Time period of a near-earth satellite: \( T = 2\pi\sqrt{R/g} \), about 85 minutes.
3. A geostationary satellite has a period of 24 hours, orbits in the equatorial plane at about 35800 km above the surface and seems fixed in the sky.
4. A polar satellite is at 500 to 800 km height, passes over the poles and has a period near 100 minutes.
5. Escape speed: \( v_e = \sqrt{2GM/R} = \sqrt{2gR} \), about 11.2 km/s for the earth; it does not depend on the mass or the direction of projection.
6. Astronauts in orbit feel weightless because they and the spacecraft are in free fall together.

Worked Examples

Example 1. Two masses of 60 kg and 70 kg are 1 m apart. \( F = 6.67\times10^{-11} \times 60 \times 70 = 2.8\times10^{-7} \) N.

Example 2. A planet at four times the earth's distance from the sun has \( T = 4^{3/2} = 8 \) years.

Example 3. A body weighs 60 N on the earth. On the moon it weighs 60/6 = 10 N, but its mass is unchanged.