CDS Physics · Gravitation

Gravitation and Weight

Newton's inverse-square law of gravitation, how the force changes with mass and distance, and weight as the pull of a planet that depends on its g.

All Gravitation notes

Why this matters

Eight CDS questions from 2020 (II) to 2025 (I), six of them EASY. Three ask how the force changes when a mass or the distance changes, and four are about weight and g.

Concept 1 of 2: Newton's law of gravitation

Every body in the universe pulls every other body. The pull grows with each of the two masses and weakens quickly with distance: double the distance and the pull drops to a quarter, because the distance is squared. So to see how the force changes, multiply by each mass factor and divide by the square of the distance factor.

Definition

The law and how to use it:

$F = \dfrac{Gm_1m_2}{r^{2}}$ : proportional to each mass, inversely proportional to the square of the distance between their centres.
$G = 6.67 \times 10^{-11}\ \text{N m}^{2}\,\text{kg}^{-2}$ , the universal gravitational constant, the same everywhere.
The force is always attractive, and the two bodies pull each other with equal force (third law).
Scaling: new force = old force × (mass factors) ÷ (distance factor)².
The law of gravitation is a separate law found from observation (Newton used the Moon and the planets). It does not follow from the three laws of motion, which say how a body responds to a force but not what the force is.

Universal law of gravitation

F = \dfrac{Gm_1m_2}{r^{2}} \qquad F' = F \times \dfrac{(\text{mass factors})}{(\text{distance factor})^{2}}

m₁, m₂the two masses
rdistance between their centres
Guniversal gravitational constant

Worked example

Both masses of a pair are doubled and the distance between them is halved. How many times larger is the gravitational force?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — General Knowledge · Q6Easy

Example 1 · Gravitation · Gravitation and Weight

What happens to the gravitational force between two objects if the mass of one object is doubled and the distance between them is also doubled ?

The distance is squared

Doubling the distance makes the force one quarter, not one half. Halving it makes the force four times. Always square the distance factor before dividing.

Gravitation is its own law

Newton's law of gravitation does not follow from the laws of motion. A statement claiming it does is the false one.

G is not g

G is the universal constant, the same everywhere. g is the acceleration due to gravity at a place, about 9.8 m/s² on the earth's surface, and it changes from place to place and planet to planet.

Drill 3 more on newton's law of gravitation

Concept 2 of 2: Weight, mass and g

Mass is how much matter a body has, and it is the same anywhere. Weight is the force with which a planet pulls that body, so it depends on where you are. On the Moon your mass is unchanged, but your weight is about one sixth of your earth weight. A planet's surface gravity g depends on both its mass and its radius.

Definition

Weight and g:

Weight is the force with which the earth (or a planet) attracts a body: $W = mg$ , measured in newtons.
Mass is the amount of matter, measured in kilograms, and does not change from place to place.
At the surface, $g = \dfrac{GM}{R^{2}}$ : it depends on both the planet's mass $M$ and its radius $R$ .
So if g is k times larger, the weight is k times larger, with the same mass.
g is about 9.8 m/s² on the earth, about one sixth of that on the Moon, and zero at the centre of the earth.

Weight and surface gravity

W = mg \qquad g = \dfrac{GM}{R^{2}}

Wweight, in newtons
mmass of the body, in kilograms
M, Rmass and radius of the planet

Worked example

An astronaut has a mass of 60 kg. Find her mass and her weight on the earth (

g = 10

m/s²) and on the Moon, where g is one sixth as large.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (I) 2022 — General Knowledge · Q21Easy

Example 2 · Gravitation · Gravitation and Weight

An object weighs 9 N on the surface of the Earth. What would be its weight, when measured on the surface of a planet where the acceleration due to gravity is 9 times that on the surface of the Earth ?

Weight is a force, mass is not

Weight is the earth's pull on a body, in newtons, and it changes with g. Mass is the amount of matter, in kilograms, and stays fixed. Options like 'mass' or 'density' for the earth's pull are wrong.

g needs both mass and radius

g = GM/R^{2}

contains the planet's mass and its radius. Neither alone fixes g.

Weight scales with g, not with its square

With the mass fixed,

W = mg

: if g becomes 5 times, the weight becomes 5 times. No squaring is needed here; the square belongs to the radius inside g.

Drill 3 more on weight, mass and g

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Newton's law of gravitation
Universal law of gravitation
$F = \dfrac{Gm_1m_2}{r^{2}} \qquad F' = F \times \dfrac{(\text{mass factors})}{(\text{distance factor})^{2}}$
Weight, mass and g
Weight and surface gravity
$W = mg \qquad g = \dfrac{GM}{R^{2}}$

Watch out for (6)

The distance is squared
→ Newton's law of gravitation
Gravitation is its own law
→ Newton's law of gravitation
G is not g
→ Newton's law of gravitation
Weight is a force, mass is not
→ Weight, mass and g
g needs both mass and radius
→ Weight, mass and g
Weight scales with g, not with its square
→ Weight, mass and g

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