10 min readConcept sheetSection A MCQ (1 mark) and fill in the blanks (2 marks, two parts); Section B very short answer (2 marks) or short answer (3 marks), often with a choice

Acceleration due to gravity

Near the earth every freely falling body speeds up at the same rate, g = GM/R², about 9.8 m s⁻². It does not depend on the body's mass.

gThe acceleration of a body falling under the earth's pull alone; average 9.8 m s⁻² near the surface

Formulag = GM/R² (M = mass of the earth, R = radius of the earth, G = 6.67 × 10⁻¹¹ N m² kg⁻²)

Unitm s⁻², the same as any acceleration

Variationg is more at the poles than at the equator, because the earth's radius is smaller at the poles

What g is

Drop a stone from 1 m and then from 5 m. It hits the ground faster the second time. Its speed keeps increasing as it falls, so it is accelerating. This acceleration, caused by the earth's pull, is the acceleration due to gravity, written g. Its average value near the earth's surface is 9.8 m s⁻². So a falling body gains about 9.8 m s⁻¹ of speed every second.

Derivation: g = GM/R²

1
Let a body of mass m fall towards the earth with acceleration g.
2
By the second law, the force on it is F = mg.
3
By the law of gravitation, the same force is F = G M m/r². Here M is the earth's mass and r is the body's distance from the earth's centre.
4
So mg = G M m/r². The mass m cancels on both sides, giving g = GM/r².
5
On or near the surface, r is the earth's radius R.
6★
So, g = GM/R². The value of g does not depend on the mass of the falling body.

★ key step

Worked example: g on the moon

Problem

The moon's mass is 7.4 × 10²² kg and its radius is 1.74 × 10⁶ m. Find the acceleration due to gravity on its surface. (G = 6.67 × 10⁻¹¹ N m² kg⁻²)

Working

g = GM/R².
GM = 6.67 × 10⁻¹¹ × 7.4 × 10²² = 49.36 × 10¹¹ = 4.936 × 10¹².
R² = (1.74 × 10⁶)² = 3.028 × 10¹².
g = 4.936 × 10¹² ÷ 3.028 × 10¹² ≈ 1.63 m s⁻².

Answer

About 1.63 m s⁻², roughly one sixth of the 9.8 m s⁻² on the earth (9.8 ÷ 6 ≈ 1.63)

Why g changes from place to place

The earth is not a perfect sphere. It is slightly flattened at the poles and bulges at the equator.
So the radius R is smaller at the poles and larger at the equator.
Since g = GM/R², a smaller R gives a larger g. So g is greatest at the poles and least at the equator.
g also becomes smaller as you go higher above the ground, because the distance r from the earth's centre grows.
For problems, use the average value g = 9.8 m s⁻², or 10 m s⁻² when a question says so.

Activity: drop a coin and a paper together

You need a coin and a sheet of paper about 15 cm × 15 cm.

1
Hold the coin and the flat sheet at the same height. Drop them together.
2
Observe: the coin lands first, while the flat paper floats down slowly.
3
Crumple the paper into a tight ball. Drop the coin and the paper ball together from the same height.
4★
Observe: both land at almost the same time.
5
Conclude: the flat paper was slowed by air resistance, not by its smaller mass. Without air, all bodies fall with the same acceleration g.
In a glass tube with the air pumped out, a coin and a feather fall together.

★ key step

In real life

A ripe mango and a dry leaf from the same branch fall at different speeds only because air slows the leaf. On the airless moon, a hammer and a feather dropped together land together.

Exam angle

2 marks: “What is acceleration due to gravity? Give its SI unit and its average value near the earth's surface.” It is the acceleration of a body falling freely under the earth's pull. Its SI unit is m s⁻², and its average value near the surface is 9.8 m s⁻².

3 marks: “Show that the acceleration of a freely falling body does not depend on its mass, and obtain the formula for g at the earth's surface.” Force on a mass m: F = mg (second law) and F = GMm/R² (law of gravitation). So mg = GMm/R². The mass m cancels, giving g = GM/R². Since m cancels, g is the same for all bodies at a place.

2 marks: “Why is g greater at the poles than at the equator?” g = GM/R². The earth is flattened at the poles, so R is smaller there. A smaller R gives a larger g.

Remember it

Mass cancels, so all bodies fall alike: g = GM/R², about 9.8 m s⁻².