Why does an apple fall from a tree towards the ground, and not upwards? Why does the Moon revolve around the Earth, and the Earth around the Sun? The answer to all these questions is gravitation - the force of attraction that exists between any two objects in the universe. It was Sir Isaac Newton who first formulated the universal law of gravitation in 1687.
Gravitation is a universal force. Every object in the universe attracts every other object with a force, whether they are as small as dust particles or as large as stars and planets. This force of attraction keeps the planets in their orbits, holds the atmosphere around the Earth, and pulls objects towards the Earth's surface.
In this chapter we study Newton's universal law of gravitation, the acceleration due to gravity, free fall, mass and weight, and the motion of objects under the influence of gravity.
Newton's universal law of gravitation states that 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 their centres.
If two objects of masses m1 and m2 are separated by a distance d, the force of gravitation F between them is:
F proportional to (m1 x m2) / d^2
Introducing the universal gravitational constant G, we write:
F = G (m1 m2) / d^2
The universal gravitational constant G has a fixed value: G = 6.7 x 10^-11 N m^2 / kg^2. G is the same everywhere in the universe, hence the name universal gravitational constant.
The gravitational force: 1. Depends directly on the product of the two masses. 2. Depends inversely on the square of the distance between them. 3. Acts along the line joining the centres of the two objects.
The force with which the Earth attracts objects towards its centre is called gravity. Since the Earth is very massive, it attracts objects near it with a significant force, which is why things fall downwards.
When an object is dropped, it falls towards the Earth because of the gravitational attraction of the Earth. According to the third law of motion, the object also attracts the Earth with an equal force, but the Earth's huge mass means its acceleration is negligible.
The gravitational force of the Earth keeps the atmosphere around it, causes tides in the oceans (along with the Moon's attraction), and holds the Moon in its orbit.
When an object falls freely towards the Earth, it accelerates because of the gravitational force of the Earth. This acceleration is called the acceleration due to gravity and is denoted by the symbol g.
The value of g on the surface of the Earth is about 9.8 m/s^2. The acceleration due to gravity does not depend on the mass of the falling object - all objects fall with the same acceleration (ignoring air resistance). A feather and a stone would fall together in a vacuum.
Using the universal law of gravitation, the acceleration due to gravity is given by:
g = G x M / R^2
where M is the mass of the Earth and R is its radius. Substituting the values of G, M (6 x 10^24 kg) and R (6.4 x 10^6 m) gives g = 9.8 m/s^2.
The value of g decreases as we go higher above the Earth's surface, and it is also slightly less at the equator than at the poles.
When an object falls only under the influence of gravity, without any other force acting on it, its motion is called free fall. During free fall, the only force acting on the object is the gravitational force of the Earth, so the object moves with a constant acceleration g.
The equations of motion can be applied to freely falling bodies, taking the acceleration as g (with proper sign) and the displacement as the height fallen:
v = u + gt s = ut + (1/2)gt^2 v^2 = u^2 + 2gs
If an object is dropped from rest, u = 0, so: v = gt s = (1/2)gt^2 v^2 = 2gs
If an object is thrown upwards, gravity acts against the motion, so the acceleration is -g. The time taken to reach the highest point is t = u/g, and the maximum height reached is h = u^2 / (2g).
Mass: The mass of an object is the measure of the quantity of matter it contains. The mass of an object is constant everywhere in the universe and does not change from place to place. The SI unit of mass is the kilogram (kg).
Weight: The weight of an object is the force with which the Earth attracts it. Weight is a vector quantity and is given by:
Weight (W) = mass (m) x acceleration due to gravity (g)
The SI unit of weight is the newton (N). Since g = 9.8 m/s^2, the weight of a body of mass 1 kg is 1 x 9.8 = 9.8 N.
Weight is not constant; it changes with the value of g. On the Moon, the value of g is about one sixth of its value on the Earth, so an object weighs about one sixth as much on the Moon as on the Earth, even though its mass remains the same. On a mountain top or high above the Earth, weight decreases because g decreases.
Thrust: The force acting perpendicularly on a surface is called thrust. Its SI unit is the newton.
Pressure: Pressure is the thrust (force) acting on a unit area of a surface.
Pressure = Force / Area
The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m^2.
Pressure depends on both the force and the area over which it acts. A sharp knife cuts better than a blunt one because the force is concentrated over a smaller area, producing a higher pressure. A camel can walk on sand easily because its broad feet spread its weight over a large area, reducing the pressure. Nails and pins have sharp pointed ends so that a small force creates a large pressure.
Liquids and gases are called fluids because they can flow. Fluids exert pressure on the walls of the container that holds them and on any object immersed in them. The pressure in a fluid increases with depth.
Buoyancy: When an object is immersed in a fluid, the fluid exerts an upward force on the object. This upward force is called buoyant force or buoyancy. The buoyant force acts in the opposite direction to the weight of the object.
Archimedes' principle: Archimedes' principle states that when a body is immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by the body.
Whether an object sinks or floats depends on its density compared to the density of the fluid. A body sinks if its density is greater than that of the fluid, and floats if its density is less than that of the fluid. Ships float because their overall density is less than that of water, even though they are made of heavy metal. A balloon rises in air because its density is lower than the surrounding air. Objects of high density like iron sinks in water.
Relative density: The relative density of a substance is the ratio of its density to the density of water. Since it is a ratio, it has no unit. The relative density of a substance is numerically equal to the density of the substance in g/cm^3, since the density of water is 1 g/cm^3.
Relative density = Density of substance / Density of water
| Quantity | Definition | SI Unit |
|---|---|---|
| Mass | Quantity of matter in a body | kg |
| Weight | Force of gravity on a body | newton (N) |
| Thrust | Force acting perpendicular to a surface | N |
| Pressure | Thrust per unit area | pascal (Pa) |
| Object | Value of g |
|---|---|
| Earth surface | 9.8 m/s^2 |
| Moon surface | About 1.63 m/s^2 (one sixth of Earth) |
| Higher altitude | Decreases |
| Poles | Slightly more |
In this chapter we learned that gravitation is the universal force of attraction between all objects, described by Newton's law F = G m1 m2 / d^2. We studied the acceleration due to gravity g = 9.8 m/s^2 on the Earth, which is independent of the mass of the falling object, and applied the equations of motion to free fall. We distinguished mass, which is constant, from weight, which is the gravitational force on a body and changes with g. We also studied thrust and pressure, pressure in fluids, buoyancy, Archimedes' principle and relative density. Gravitation not only explains why apples fall and planets orbit the Sun, but also governs the weight we feel, the tides of the ocean, and the flight of balloons and ships.