Physics (9630) | Newton’s gravitational law

A-Level Physics (9630): Newton’s Gravitational Law (3.7.1)

Welcome to one of the most fundamental and awe-inspiring laws of the universe! This chapter marks your journey into the vastness of space and the forces that govern it, building upon your prior work in mechanics. Don't worry if the formulae look intimidating; we will break them down piece by piece.

This topic (Section 3.7) is part of the International A-level content. It connects forces and fields, helping you understand everything from how an apple falls to how satellites stay in orbit!

1. Gravity: The Universal Attractive Force

Before Newton, people knew objects fell down. Newton's genius was realizing that the force pulling an apple to Earth is the same force that keeps the Moon orbiting the Earth.

• Definition: Gravity is a universal attractive force that acts between all matter.
• Every object with mass attracts every other object with mass. This attraction is mutual, meaning if the Earth pulls on you, you pull on the Earth with an equal and opposite force (Newton's Third Law!).
• This force is always attractive (it only pulls things together, never pushes them apart).

Did You Know?

The gravitational force between you and your Physics textbook is absolutely real! It's just incredibly small compared to the force between you and the massive Earth.

2. Newton’s Law of Universal Gravitation

Newton quantified this universal attraction. The law states that the gravitational force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

2.1 The Mathematical Definition

The force \(F\) between two masses, \(m_1\) and \(m_2\), separated by a distance \(r\), is given by the formula:

$$F = \frac{G m_1 m_2}{r^2}$$

2.2 Understanding the Variables

• \(F\) (Force): The gravitational force of attraction, measured in Newtons (N).
• \(m_1\) and \(m_2\) (Masses): The masses of the two objects, measured in kilograms (kg).
• \(r\) (Distance): The distance between the centres of the two masses, measured in metres (m). Crucially, always measure from the centre, not the surface!
• \(G\) (Universal Gravitational Constant): This is the constant of proportionality that makes the equation work.
  \(G\) value: \(6.67 \times 10^{-11} \, \text{N m}^2 \text{kg}^{-2}\).

2.3 The Concept of Point Masses

The formula strictly applies to point masses (objects considered to have all their mass concentrated at a single point).

Hold on, planets aren't points! That's okay. For uniform spherical objects (like planets or stars), the entire mass can be considered concentrated at their geometrical centre, as long as we measure the distance \(r\) from that centre.

Key Takeaway: When dealing with planets or spheres, \(r\) must be measured from centre to centre.

3. The Relationships: Proportionality and the Inverse Square Law

3.1 Dependence on Mass (\(m_1 m_2\))

The force \(F\) is directly proportional to the product of the masses (\(F \propto m_1 m_2\)).

• If you double the mass of one object, the force doubles.
• If you double both masses, the force increases by a factor of four (\(2 \times 2 = 4\)).

3.2 Dependence on Distance (\(\frac{1}{r^2}\)) – The Inverse Square Law

This is often the trickiest part, but it's vital! The force \(F\) is inversely proportional to the square of the distance (\(F \propto \frac{1}{r^2}\)).

What this means: As the objects get farther apart, the force drops off very quickly.

• If you double the distance \(r\) (e.g., move a satellite twice as far from Earth), the force decreases by a factor of \(2^2 = 4\).
• If you triple the distance \(r\), the force decreases by a factor of \(3^2 = 9\).

Analogy: Light and Gravity

Think of how the light from a bulb spreads out. If you double your distance from the bulb, you only receive a quarter (\(1/4\)) of the intensity of light. Gravity behaves the exact same way—it dilutes over space according to the square of the distance.

4. Common Mistakes and Calculations

Mistake 1: Misusing the Distance \(r\)

The Scenario: A satellite orbits Earth 500 km above the surface. The Earth's radius is 6370 km.

The Mistake: Using \(r = 500 \, \text{km}\) in the formula.

The Correction: The distance \(r\) must be measured from centre to centre.
\(r = (\text{Earth's Radius}) + (\text{Altitude})\)
\(r = 6370 \, \text{km} + 500 \, \text{km} = 6870 \, \text{km}\) (Remember to convert to metres before calculating \(F\)).

Mistake 2: Confusing \(G\) and \(g\)

\(G\) (Universal Gravitational Constant): The constant in Newton's Law. It is the same everywhere in the universe.

\(g\) (Gravitational Field Strength): This is the acceleration due to gravity (or force per unit mass) at a specific point near a mass. It changes depending on location (it's different on the Moon than on Earth).

While we look at \(g\) in detail in the next section (3.7.2), remember that \(G\) is the anchor of the fundamental gravitational formula (3.7.1).

Example Calculation Step-by-Step

Calculate the gravitational force between two asteroids, \(m_1 = 1.0 \times 10^{12} \, \text{kg}\) and \(m_2 = 5.0 \times 10^{12} \, \text{kg}\), separated by \(200 \, \text{m}\).

1. Identify knowns: \(G = 6.67 \times 10^{-11}\), \(m_1 = 10^{12}\), \(m_2 = 5 \times 10^{12}\), \(r = 200\).
2. Write the formula: \(F = \frac{G m_1 m_2}{r^2}\).
3. Substitute values: $$F = \frac{(6.67 \times 10^{-11}) \times (1.0 \times 10^{12}) \times (5.0 \times 10^{12})}{ (200)^2}$$
4. Calculate the numerator: \(6.67 \times 10^{-11} \times 5.0 \times 10^{24} = 3.335 \times 10^{14}\)
5. Calculate the denominator: \(200^2 = 40,000\)
6. Final Calculation: \(F = \frac{3.335 \times 10^{14}}{40,000} \approx 8.3 \times 10^{9} \, \text{N}\)

Key Takeaway (Section 3.7.1)

Newton's Law of Universal Gravitation, \(F = \frac{G m_1 m_2}{r^2}\), defines gravity as a force that depends only on the masses of the two objects and the square of the distance between their centres. It is the foundational equation for all orbital mechanics and field theory.

You’ve mastered the core force calculation! Next, we will use this concept to explore Gravitational Fields.