Gravitation

Every object in the universe attracts every other object with a gravitational force. The force is attractive and acts along the line joining the two objects.

Newton's universal law of gravitation states that the gravitational force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

This means if you double the mass of one body, the force doubles. But if you double the distance between them, the force becomes one-fourth. This inverse-square relationship is important and comes up in many exam questions.

The gravitational constant G has a fixed value everywhere in the universe:

G is a universal constant, meaning its value is the same on Earth, on the Moon, and anywhere else in space. It was first measured experimentally by Henry Cavendish.

Because G is extremely small, gravitational force between ordinary objects of everyday mass is negligibly tiny. Gravitational effects are noticeable only when at least one of the masses is very large, such as a planet or a star.

When an object moves under the influence of gravity alone, with no other force acting on it, it is said to be in free fall. A freely falling body accelerates toward the Earth at a rate called the acceleration due to gravity, denoted g.

Near the Earth's surface, g is approximately 9.8 m/s². This means a freely falling body gains 9.8 m/s of speed every second.

A remarkable fact shown by Galileo and confirmed in vacuum experiments is that all freely falling bodies experience the same acceleration regardless of their mass. A heavy iron ball and a light feather dropped together in vacuum reach the ground at exactly the same time.

In normal conditions, air resistance affects light objects more than heavy ones. That is why a feather falls slower than a coin when dropped from the rooftop of a building in a Delhi neighbourhood.

The same three equations of motion from Chapter 1 apply to free fall, with acceleration replaced by g for downward motion and −g for upward motion.

When a ball is thrown upward from a terrace in Jaipur, it decelerates at 9.8 m/s² going up and accelerates at 9.8 m/s² coming down. At the highest point, velocity is zero but acceleration is still g downward.

Mass is the amount of matter in a body. It is a scalar quantity measured in kilograms. Mass stays the same everywhere — on the Earth, on the Moon, or in deep space.

Weight is the gravitational force exerted by the Earth (or any other body) on an object. Since it is a force, weight is measured in newtons. Weight depends on the value of g at the location.

A student who has a mass of 60 kg has the same mass everywhere. But the student's weight on Earth is about 588 N, while on the Moon it would be only about 98 N because the Moon's gravity is weaker.

The Moon has much less mass than the Earth, so its gravitational pull is weaker. The acceleration due to gravity on the Moon is about which is roughly one-sixth of g on Earth.

ISRO scientists working on Chandrayaan missions carefully account for this difference when designing landers and rovers. Instruments, equipment, and mobility systems must work correctly under the Moon's weaker gravity.

An astronaut who weighs 600 N on Earth weighs only about 100 N on the Moon. This makes movement feel much lighter and jumping much easier. Mass, however, remains unchanged at 60 kg.

Thrust is the force acting perpendicular to a surface. Pressure is the thrust per unit area. The same thrust creates higher pressure on a smaller area.

A sharp knife cuts vegetables more easily than a blunt one because the same force from your hand is concentrated on a much smaller area at the sharp edge, creating greater pressure on the vegetable.

Tractors used in farms in Punjab and Haryana have wide tyres so that the weight of the tractor spreads over a large area, reducing pressure on soft soil and preventing the tractor from sinking.

When an object is immersed in a fluid (liquid or gas), the fluid exerts a pressure on all surfaces of the object. Because pressure increases with depth, the upward pressure on the bottom of the object is greater than the downward pressure on the top. This net upward force is called buoyant force or upthrust.

You can easily feel this force when you try to push a rubber ball or an inflated balloon into a bucket of water — the water pushes back upward. Children playing in a swimming pool or a river experience buoyancy as a lifting sensation in water.

Buoyant force depends on the volume of fluid displaced and the density of the fluid. It does not depend on the mass or density of the object itself.

Archimedes' principle states that a body wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by it.

This principle allows us to measure buoyant force precisely. If an object displaces 500 mL of water, the buoyant force on it equals the weight of 500 mL of water, which is about 4.9 N.

Archimedes' principle has many practical applications in India. Ships carrying cargo from Mumbai and Chennai ports are designed so that the weight of water they displace equals the total weight of the ship and cargo. Hydrometer instruments used in dairy farms to check the purity of milk also use this principle.

An object floats when the buoyant force acting on it equals its weight. This happens when the average density of the object is less than or equal to the density of the fluid.

An iron nail sinks in water because the volume of water it displaces weighs less than the nail itself. But a large steel ship shaped like a hollow bowl displaces a huge volume of water whose weight can equal the entire weight of the ship and cargo, so it floats. This is why ships made of steel float even though steel is denser than water.

Relative density (also called specific gravity) is the ratio of the density of a substance to the density of water at 4 °C. If relative density is less than 1, the substance floats in water. Wood, cork, and ice have relative density less than 1. Iron, copper, and stone have relative density greater than 1.

Universal gravitation: . G is universal; g varies by location.

Free fall: all bodies fall with same acceleration g ≈ 9.8 m/s² in vacuum.

Weight: . Weight on Moon ≈ Weight on Earth ÷ 6.

Pressure: . More area → less pressure; less area → more pressure.

Archimedes' principle: buoyant force = weight of fluid displaced.

Flotation: object floats when density ≤ density of fluid.