Newton's Law of Gravitation

Overview

This lesson turns gravitational field ideas into an inverse-square model. You will use Newton’s law of gravitation for point masses, derive the expression for field strength around a point mass, and explain why Earth’s gravitational field is approximately constant close to the surface.

What You Need to Know

• For a point outside a uniform sphere, the sphere’s mass can be treated as concentrated at its centre.
• Newton’s law of gravitation gives the attractive force between two point masses.
• The force follows an inverse-square relationship with separation.
• Combining F = GMm / r^2 with g = F / m gives g = GM / r^2 for the field strength due to a point mass.
• Near Earth’s surface, small changes in height make only a very small fractional change to r, so g is approximately constant.

How to Work Through It

1. Start by identifying when the point-mass model is appropriate.
2. Practise Newton’s law of gravitation calculations with clear values of mass and separation.
3. Derive g = GM / r^2 from force per unit mass.
4. Compare changes in g at different distances from Earth’s centre.

Check Your Understanding

• Why does doubling the separation reduce the gravitational force by a factor of four?
• When can Earth’s mass be treated as if it acts at the centre of Earth?
• How does Newton’s law lead to g = GM / r^2?
• Why is g almost constant over a few metres near Earth’s surface?

Common Mistakes

• Measuring r from the surface instead of from the centre of the mass.
• Forgetting that the force is attractive and acts along the line joining the masses.
• Using g = GM / r^2 without recognising which mass creates the field.
• Treating g as exactly constant everywhere around Earth.

Next Steps

• Practise inverse-square calculations until the role of r is secure.
• Bring force and field strength ideas into gravitational potential next.