Gravitation

Why does a mango fall from the tree to the ground? Why does the Moon orbit the Earth instead of flying off into space? Newton answered these questions with one of the most elegant laws of physics -- the Universal Law of Gravitation.

Newton's Law of Universal Gravitation

Every object in the universe attracts every other object with a force that is:

• Directly proportional to the product of their masses
• Inversely proportional to the square of the distance between them

F = G m1 m2 / r²

where G = 6.67 x 10⁻¹¹ N m² kg⁻² is the universal gravitational constant.

Gravitational Acceleration: g vs G

G is the universal gravitational constant -- it is the same everywhere in the universe.

g is the acceleration due to gravity at a planet's surface:

g = GM / R²

where M is the planet's mass and R is its radius. On Earth, g = 9.8 m/s².

Variation of g

With altitude (h): g decreases as you go higher.

g' = g (1 - 2h/R) for small h, or g' = g R² / (R + h)² for large h.

With depth (d): g decreases as you go below the surface.

g' = g (1 - d/R)

At the centre of the Earth (d = R), g = 0.

With latitude: g is slightly less at the equator than at the poles due to Earth's rotation.

Worked Example

Calculate the gravitational force between two objects of masses 50 kg and 70 kg separated by 2 m.

Solution:

• F = G m1 m2 / r² = 6.67 x 10⁻¹¹ x 50 x 70 / (2)²
• F = 6.67 x 10⁻¹¹ x 3500 / 4
• F = 5.84 x 10⁻⁸ N

This force is extremely small, which is why we do not feel gravitational attraction between everyday objects.

Satellites

A satellite orbits a planet because gravity provides the centripetal force:

Orbital velocity: v = sqrt(GM/r) = sqrt(gR²/r)

Time period: T = 2 pi sqrt(r³/GM)

For a satellite close to Earth's surface, v is approximately 7.9 km/s and T is approximately 84 minutes.

Nepal Connection

Nepal is home to Mount Everest (8,849 m). At the summit, g is slightly less than at sea level. Using the altitude formula: g' is approximately 9.77 m/s².

Key Takeaways

• Gravitational force is universal and always attractive
• g varies with altitude, depth, and latitude
• G is a universal constant; g depends on the planet
• Satellites orbit due to the balance of gravity and inertia