Gravitation

Statement: Every object in the universe attracts every other object with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. The force acts along the line joining the centres of the two objects.

Mathematically, if two objects of masses m₁ and m₂ are separated by a distance r, the gravitational force F between them is:

Where G is the Universal Gravitational Constant:

This law was given by Sir Isaac Newton in 1687. It applies universally — between planets, stars, and everyday objects. The force is always attractive (pulls the two masses toward each other) and acts along the line joining their centres. The value of G was first determined experimentally by Henry Cavendish in 1798.

The gravitational force between the Earth and an object on its surface is given by Newton's Law of Gravitation applied with the Earth's parameters:

Where:

• F = gravitational force between Earth and the object (in N)
• G = Universal Gravitational Constant = 6.674 × 10⁻¹¹ N m² kg⁻²
• M = mass of the Earth = 6 × 10²⁴ kg
• m = mass of the object (in kg)
• R = radius of the Earth = 6.4 × 10⁶ m (the distance from the object to the Earth's centre, approximately equal to Earth's radius for objects on the surface)

This gravitational force is what we experience as the weight of the object (W = mg), where g = GM/R² ≈ 9.8 m/s².

Free fall is the motion of an object when it falls solely under the influence of the Earth's gravitational force, with no other force (such as air resistance, upthrust, or any contact force) acting on it.

Key characteristics of free fall:

(i) Uniform acceleration: A freely falling object accelerates downward with the acceleration due to gravity, g ≈ 9.8 m/s² (near Earth's surface). The equations of motion apply with a = g.

(ii) Mass-independent: All objects in free fall have the same acceleration (g) regardless of their mass. A feather and a hammer dropped in a vacuum (no air) will fall at the same rate — this was famously demonstrated on the Moon by Apollo astronauts.

(iii) Weightlessness: An object in free fall appears weightless because there is no normal reaction force from any surface. This is why astronauts in orbit (which is essentially continuous free fall) experience weightlessness.

Example: A stone released from a height falls freely to the ground (neglecting air resistance).

Acceleration due to gravity (g) is the acceleration experienced by a freely falling object due to the gravitational force of the Earth. Near the Earth's surface, g = 9.8 m/s², directed downward (toward the centre of the Earth).

It is derived from Newton's Law of Gravitation and Newton's Second Law:

Where M = mass of Earth, R = radius of Earth, G = gravitational constant.

Key facts about g:

(i) g decreases as we go higher above the Earth's surface (R increases in the formula).
(ii) g decreases slightly as we go inside the Earth (towards the centre, g = 0 at centre).
(iii) g is slightly greater at the poles (Earth is slightly flatter, so R is smaller at poles) than at the equator.
(iv) g on the Moon ≈ 1.6 m/s² (about 1/6th of Earth's g) due to the Moon's smaller mass and radius.
(v) g is a vector quantity directed toward the Earth's centre (downward).

Mass:

(i) The amount of matter contained in an object.
(ii) A scalar quantity (no direction).
(iii) Measured in kilograms (kg).
(iv) Constant everywhere in the universe — mass does not change with location, altitude, or gravitational field.
(v) Measured using a balance (equal-arm balance) by comparing with standard masses.
(vi) Mass can never be zero for a physical object.

Weight:

(i) The gravitational force exerted on an object by a planet or celestial body.
(ii) A vector quantity (directed toward the centre of the Earth / downward).
(iii) Measured in Newtons (N).
(iv) Varies with location — weight is more at the poles than at the equator, less on the Moon (1/6 of Earth), and zero in deep space far from any massive body.
(v) Measured using a spring balance (weighing scale).
(vi) Calculated as: W = mg

Example: A person of mass 60 kg has a weight of 60 × 9.8 = 588 N on Earth, and 60 × 1.6 = 96 N on the Moon — but their mass remains 60 kg on both.

Weight = mg, so weight depends on the acceleration due to gravity (g). The value of g depends on the mass (M) and radius (R) of the body: g = GM/R².

The Moon has:

• Much smaller mass than Earth (M_moon ≈ 7.35 × 10²² kg vs. M_earth ≈ 6 × 10²⁴ kg)
• Smaller radius than Earth (R_moon ≈ 1.74 × 10⁶ m vs. R_earth ≈ 6.4 × 10⁶ m)

Calculating g on Moon:

Since g_moon ≈ 1.6 m/s² and g_earth ≈ 9.8 m/s²:

Since W = mg, and mass (m) is the same on Earth and Moon:

Therefore, the weight of an object on the Moon is approximately one-sixth of its weight on Earth. A person who weighs 600 N on Earth would weigh only 100 N on the Moon.

Given: Initial velocity u = 49 m/s (upward). g = 9.8 m/s² (acting downward). At maximum height, final velocity v = 0.

(i) Maximum height — using v² = u² − 2gh:

(ii) Total time — time to reach maximum height using v = u − gt:

By symmetry of projectile motion, the time to fall back = time to go up = 5 s.

The ball reaches a maximum height of 122.5 m and takes a total time of 10 s to return to the ground.

Given: Initial velocity u = 0 (released from rest). Height h = 19.6 m. g = 9.8 m/s².

Using v² = u² + 2gh:

The final velocity of the stone just before it touches the ground is 19.6 m/s (directed downward).

Given: u = 40 m/s (upward). v = 0 at maximum height. g = 10 m/s² (taking deceleration as −10 m/s² during ascent).

Maximum height — using v² = u² − 2gh:

Net displacement:
The stone goes up 80 m and then comes back down 80 m to its starting point (same initial and final position).
Net displacement = 0 m (initial position = final position).

Total distance covered:

Maximum height = 80 m. Net displacement = 0. Total distance = 160 m.

Given: M_Earth = 6 × 10²⁴ kg. M_Sun = 2 × 10³⁰ kg. r = 1.5 × 10¹¹ m. G = 6.674 × 10⁻¹¹ N m² kg⁻².

Using F = G × M_Earth × M_Sun / r²:

The gravitational force between the Earth and the Sun is approximately 3.56 × 10²² N. This enormous attractive force keeps the Earth in its orbit around the Sun.

Mass on Earth:

Mass on Moon:
Mass is a fundamental property of matter and does not change with location. Therefore, his mass on the Moon is also 60 kg.

Acceleration due to gravity on the Moon:

Mass on Earth = Mass on Moon = 60 kg. Acceleration due to gravity on Moon = 1.67 m/s² (which is approximately 1/6 of Earth's g = 10 m/s²).

Thrust:
The force acting on an object perpendicular (normal) to its surface is called thrust. It is the total force applied on a surface. Thrust is a vector quantity measured in Newtons (N).

Example: The weight of a book lying on a table is the thrust it exerts on the table surface.

Pressure:
Pressure is defined as the thrust (force) acting per unit area of the surface on which it acts.

SI unit of pressure is Pascal (Pa), where 1 Pa = 1 N/m².

Key concept: The same thrust (force) produces greater pressure on a smaller area and lesser pressure on a larger area. This is why:

• A sharp knife (small area) cuts better than a blunt knife (larger area) — same force, higher pressure.
• A bed of nails is safe to lie on because the total weight is distributed over many nail points, reducing pressure per nail.
• Wide tyres on heavy vehicles reduce pressure on the road surface.

Given: Mass m = 5 kg. g = 9.8 m/s². Thrust (weight) = mg = 5 × 9.8 = 49 N.

(a) Face with dimensions 20 cm × 10 cm:

(b) Face with dimensions 40 cm × 20 cm:

Conclusion: When the block lies on its smallest face (20 × 10 cm), it exerts a greater pressure (2450 Pa) compared to when it lies on its largest face (40 × 20 cm), which gives lesser pressure (612.5 Pa). The thrust (weight = 49 N) is the same in both cases.

Archimedes' Principle: When a body is immersed wholly or partially in a fluid (liquid or gas), it experiences an upward buoyant force (upthrust) equal to the weight of the fluid displaced by the body.

Where ρ_fluid is the density of the fluid, V_displaced is the volume of fluid displaced, and g is the acceleration due to gravity.

Consequences and applications:

(i) Apparent loss of weight: A body immersed in a fluid appears lighter (loses weight = buoyant force). This is why heavy objects feel lighter when lifted in water.
(ii) Floating: An object floats when buoyant force = weight of object (i.e., density of object ≤ density of fluid).
(iii) Sinking: An object sinks when its weight > buoyant force (density of object > density of fluid).
(iv) Applications: Design of ships and submarines (by changing their density using ballast tanks), hydrometers (to measure density of liquids), lactometers (to test purity of milk).

This principle was discovered by the ancient Greek scientist Archimedes (~250 BCE), according to legend while taking a bath.

Relative density (also called specific gravity) of a substance is the ratio of the density of the substance to the density of water at 4°C (where water has its maximum density of 1000 kg/m³).

Key properties of relative density:

(i) Dimensionless (no unit): Since both densities are in the same unit (kg/m³), their ratio has no unit. This makes relative density very convenient for comparing materials.

(ii) Relation to floating/sinking:
• If relative density > 1: the substance is denser than water and will sink in water. (e.g., iron with RD = 7.8)
• If relative density < 1: the substance is less dense than water and will float in water. (e.g., wood with RD ≈ 0.5–0.8)
• If relative density = 1: the substance has the same density as water (e.g., pure water itself).

(iii) Practical application: Relative density can be measured using Archimedes' principle:

It is also used in hydrometers (instruments that directly measure the relative density of liquids) for checking the density of battery acid, milk (lactometer), or soil solutions.