Comprehensive theory, key formulas, diagrams, and memory aids for Gravitation.
Why does an apple fall from a tree to the ground? Why does the moon revolve around the Earth? Why do all planets revolve around the Sun? Isaac Newton pondered these questions and realized that the same force is responsible for all these phenomena. This force is called the gravitational force.
In this chapter, we will learn about gravitation and the Universal Law of Gravitation. We will discuss the motion of objects under the influence of gravitational force on Earth, and understand how weight varies from place to place. We will also learn about thrust and pressure, and the principle of flotation.
Every object in the universe attracts every other object with a force. The Universal Law of Gravitation states that: Every object in the universe attracts every other object with a force which is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is along the line joining the centres of two objects.
Mathematically, if two objects of masses $M$ and $m$ lie at a distance $d$ from each other, the force of attraction $F$ is given by: $$F = G \frac{M \times m}{d^2}$$ Where $G$ is the constant of proportionality and is called the Universal Gravitation Constant. * The accepted value of $G$ is $6.673 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2$.
This law successfully explained several phenomena which were believed to be unconnected: 1. The force that binds us to the Earth. 2. The motion of the moon around the Earth. 3. The motion of planets around the Sun. 4. The tides due to the moon and the Sun.
When objects fall towards the Earth under the influence of gravitational force alone, we say that the objects are in free fall. * Does the velocity of a falling object change? Yes. While the direction of motion remains the same (downwards), the magnitude of velocity increases due to Earth's attraction. * Any change in velocity involves acceleration. The acceleration experienced by an object during free fall is called the acceleration due to gravity, denoted by $g$. * The unit of $g$ is the same as that of acceleration, i.e., $\text{m/s}^2$.
From the Universal Law of Gravitation, the force $F$ exerted by the Earth (mass $M$) on an object (mass $m$) at its surface (distance $R$, radius of Earth) is: $F = G \frac{M \times m}{R^2}$
From Newton's second law, $F = mg$. Equating the two: $$mg = G \frac{M \times m}{R^2}$$ $$g = G \frac{M}{R^2}$$
Plugging in the values ($G = 6.67 \times 10^{-11}$, $M = 6 \times 10^{24} \text{ kg}$, $R = 6.4 \times 10^6 \text{ m}$): $g = 9.8 \text{ m/s}^2$ (approx).
Note: Because the Earth is not a perfect sphere (bulges at the equator), the radius $R$ increases from the poles to the equator. Hence, the value of $g$ is slightly greater at the poles than at the equator.
Since $g$ is constant near the Earth, all equations for uniformly accelerated motion are valid, just replace $a$ with $g$. 1. $v = u + gt$ 2. $h = ut + \frac{1}{2}gt^2$ 3. $v^2 = u^2 + 2gh$ * In applying these equations, take $g$ as positive if the object is falling (direction of motion), and negative if the object is thrown upwards (opposite to motion).
We often use mass and weight interchangeably in daily life, but they are fundamentally different in physics.
The mass of the moon is less than that of the Earth. Due to this, the moon exerts lesser gravitational force on objects. Calculations show that the acceleration due to gravity on the moon is about one-sixth (1/6) of that on Earth. * $\text{Weight of object on moon} = \frac{1}{6} \times \text{Its weight on Earth}$.
Have you ever wondered why a camel can run easily in a desert? Why an army tank weighing thousands of tonnes rests on continuous chains? Why a truck has much wider tyres?
The same force acting on a smaller area exerts a larger pressure, and a smaller pressure on a larger area. This is why a nail has a pointed tip (small area -> high pressure to pierce wood), and why wide straps are provided on school bags (large area -> low pressure on shoulders).
All liquids and gases are fluids. A solid exerts pressure on a surface due to its weight. Similarly, fluids have weight, and they also exert pressure on the base and walls of the container in which they are enclosed.
Have you ever had a swim in a pool and felt lighter? Or drawn water from a well and felt that the bucket of water is heavier when it is pulled out of the water? When an object is immersed in a fluid, it experiences an upward force exerted by the fluid. This upward force is called the buoyant force or upthrust, and the phenomenon is called buoyancy.
Density is mass per unit volume ($\text{Density} = \frac{\text{Mass}}{\text{Volume}}$). * Objects of density less than that of a liquid float on the liquid. (e.g., cork on water). * Objects of density greater than that of a liquid sink in the liquid. (e.g., iron nail in water).
Archimedes, a Greek scientist, formulated a principle which states: When a body is immersed fully or partially in a fluid, it experiences an upward force that is equal to the weight of the fluid displaced by it.
Applications of Archimedes' Principle: 1. In designing ships and submarines. 2. In lactometers, used to determine the purity of a sample of milk. 3. In hydrometers used for determining the density of liquids.
Gravitation is a universal phenomenon. Newton’s law provides the mathematical basis for calculating this attractive force between any two masses. On Earth, this force causes all objects to fall with a uniform acceleration ($g \approx 9.8 \text{ m/s}^2$). Understanding the distinction between constant mass and variable weight helps us comprehend physical behavior in different environments like the moon. The concepts of thrust, pressure, and buoyancy explain everyday occurrences, from why sharp knives cut better to why massive steel ships float on water according to Archimedes' Principle.