Gravitation

• Newton's Universal Law of Gravitation and what each symbol means
• Why g = 9.8 m/s² on Earth and how it is derived
• The difference between mass and weight
• Why the Moon orbits rather than crashes into Earth

Newton's Universal Law of Gravitation Every object in the universe attracts every other object with a gravitational force. The magnitude of this force is:

F = G × m₁ × m₂ / r²

where:

• F = gravitational force (N)
• G = Universal Gravitational Constant = 6.674 × 10⁻¹¹ N·m²/kg²
• m₁, m₂ = masses of the two objects (kg)
• r = distance between their centres (m)

Two consequences: (1) Force increases as mass increases. (2) Force decreases as the square of the distance — doubling the distance reduces force to one-quarter (inverse square law).

Gravitational acceleration on Earth's surface Setting m₁ = mass of Earth (M_E = 5.97 × 10²⁴ kg) and r = radius of Earth (R_E = 6.37 × 10⁶ m):

g = G × M_E / R_E² ≈ 9.8 m/s²

This is the acceleration any freely falling object experiences near Earth's surface (ignoring air resistance). All objects — feather or stone — fall with the same acceleration g because g doesn't depend on the falling object's mass.

Free fall An object is in free fall when gravity is the only force acting on it. In free fall, every object accelerates downward at g regardless of mass. Galileo demonstrated this by dropping two spheres of different masses from the Leaning Tower of Pisa — they landed together. Air resistance is what makes a feather fall slower than a stone in normal conditions; in a vacuum, they fall identically.

Mass vs Weight

• Mass: the amount of matter in an object. Measured in kilograms (kg). Scalar. Constant everywhere — your mass on the Moon is the same as on Earth.
• Weight: the gravitational force on an object. Weight = mg. Measured in Newtons (N). Vector (acts downward). Varies with location — where g is different, weight differs.

On Earth: g = 9.8 m/s² On Moon: g_moon = 1.62 m/s² (about 1/6 of Earth's) On Mars: g_mars = 3.72 m/s²

Why the Moon doesn't crash into Earth Gravity IS pulling the Moon toward Earth. But the Moon also has a sideways velocity of about 1 km/s. As the Moon falls toward Earth due to gravity, it is simultaneously moving sideways fast enough that the Earth's surface curves away beneath it at the same rate. The result: the Moon is perpetually "falling around" Earth — this is an orbit. If the Moon stopped moving sideways, it would indeed fall straight into Earth. This is how all satellites — natural and artificial — stay in orbit.

Problem: A person has a mass of 60 kg. Calculate their weight on (a) Earth (g = 9.8 m/s²) and (b) the Moon (g = 1.62 m/s²).

Solution:

(a) Weight on Earth: W_Earth = mg = 60 × 9.8 = 588 N

(b) Weight on Moon: W_Moon = mg_moon = 60 × 1.62 = 97.2 N

The person's mass remains 60 kg on both bodies — only the weight changes. On the Moon, they feel about 1/6 as heavy as on Earth. Note: the "16 kg on Moon" phrasing you sometimes see is loose everyday language meaning 60/6 ≈ 10 kg-force equivalent; the actual force is about 97 N.

• Saying mass and weight are the same thing. Mass is kg (amount of matter), weight is N (force due to gravity). They are different quantities with different units.
• Thinking heavier objects fall faster. Ignoring air resistance, all objects fall at the same rate g regardless of mass.
• Forgetting r is distance between centres, not surfaces. When using F = Gm₁m₂/r², r is measured from centre to centre.
• Thinking weightlessness means no gravity. Astronauts in the ISS experience about 90% of Earth's gravity. They appear weightless because they and the station are in free fall together — there is no normal force beneath them.

1. If the distance between two masses is tripled, by what factor does the gravitational force change?
2. A 70 kg astronaut travels to Mars (g = 3.72 m/s²). What is their weight there?
3. Why do a cricket ball and a tennis ball hit the ground at the same time when dropped from the same height (ignoring air resistance)?