Why does an apple fall but the Moon does not? The same invisible force — gravity — pulls the apple down AND keeps the Moon in orbit. Newton realised the heavens and the Earth obey one single law!
Gravitation
Every object in the universe attracts every other object with a force.
Universal law
F = G m1m2 ÷ d² — works everywhere.
g (free fall)
Acceleration given to objects by Earth’s gravity, about 9.8 m/s².
Mass vs Weight
Mass stays fixed; weight (W = mg) changes with place.
1. The Universal Law of Gravitation
Sir Isaac Newton proposed that every object in the universe attracts every other object along the line joining their centres. The force of attraction between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. If two objects of masses m1 and m2 are separated by a distance d, the gravitational force is F = G m1m2 ÷ d². Because the distance is squared, this is called an inverse-square law — if you double the distance, the force becomes one-fourth; if you triple it, the force becomes one-ninth. The law is called universal because it applies to every pair of bodies, whether they are tiny dust grains or giant planets and stars.
2. The Gravitational Constant (G)
The symbol G is the universal gravitational constant. Its value, measured by experiment, is G = 6.67 × 10-11 N m² kg-2. The unit comes from rearranging the formula: G = F d² ÷ (m1m2). The value of G is the same everywhere in the universe and at all times — it does not depend on the medium between the objects or on the nature of the bodies. Because G is such a tiny number, the gravitational force between everyday objects (like two students sitting together) is far too small to feel. We notice gravity only when at least one of the bodies is huge, like the Earth.
3. Importance of the Universal Law
This single law beautifully explains many things at once: (a) the force that binds us to the Earth; (b) the motion of the Moon around the Earth; (c) the motion of planets around the Sun; and (d) the tides in oceans caused by the Moon and the Sun. The Moon does not fall onto the Earth because it has a sideways (tangential) velocity; gravity continuously bends its straight-line path into a curved orbit. This required centripetal force is supplied by gravity.
4. Free Fall and the Acceleration g
When an object falls towards the Earth with only the force of gravity acting on it (no other push or support), we say it is in free fall. During free fall the object accelerates. This acceleration produced by Earth’s gravitational pull is called the acceleration due to gravity, written as g. Its average value near the Earth’s surface is about 9.8 m/s². We can find g from the universal law: the Earth (mass M, radius R) pulls an object of mass m with force F = G M m ÷ R². But this force also equals m g (from F = ma). Cancelling m gives g = G M ÷ R². Notice that g does NOT depend on the mass of the falling object — this is why a heavy stone and a light stone, dropped together (ignoring air resistance), reach the ground at the same time.
5. Why g Varies
Since g = G M ÷ R², the value of g changes if R changes. The Earth is slightly flattened, so R is smaller at the poles and larger at the equator. This makes g greatest at the poles and least at the equator. The value of g also decreases as we go higher above the surface (greater distance from the centre) and as we go deep inside the Earth. On the Moon, g is only about one-sixth of its value on Earth, because the Moon has much less mass and a smaller radius.
6. Mass and Weight
Mass is the amount of matter contained in a body. It is a scalar, measured in kilograms (kg), and is constant everywhere — on Earth, Moon, or in space. Weight is the force with which the Earth attracts a body: W = m g. Weight is a force, so it is a vector measured in newtons (N), and it changes from place to place because g changes. On the Moon, since g is one-sixth, your weight becomes one-sixth, even though your mass is unchanged. This is why astronauts can take giant leaps on the Moon.
7. Thrust and Pressure
The force acting perpendicular to a surface is called thrust. The effect of thrust depends on the area over which it acts. Pressure is the thrust per unit area: P = Thrust ÷ Area. Its SI unit is the pascal (Pa), where 1 Pa = 1 N/m². A smaller area gives larger pressure for the same force — this is why a sharp knife cuts well and why a camel’s broad feet stop it sinking in sand.
8. Buoyancy and Archimedes’ Principle
When an object is placed in a fluid (liquid or gas), the fluid exerts an upward force on it called the buoyant force or upthrust. This is why a bucket of water feels lighter under water and why objects float. Archimedes’ Principle states: when a body is wholly or partially immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by it. Whether an object floats or sinks depends on density. If the object’s density is less than the fluid’s, it floats; if greater, it sinks. Relative density = density of substance ÷ density of water; it has no unit because it is a ratio.
- Universal law: F = G m1m2 ÷ d²
- G = 6.67 × 10-11 N m² kg-2 (same everywhere)
- g = G M ÷ R², value ≈ 9.8 m/s² on Earth
- Weight: W = m g (unit: newton, N)
- g on Moon = (1/6) of g on Earth
- Pressure: P = Thrust ÷ Area (unit: pascal, Pa)
- Relative density = density of substance ÷ density of water (no unit)
- Equations of motion for free fall (take a = g): v = u + g t, h = u t + ½ g t², v² = u² + 2 g h
Two bodies of masses 80 kg and 1200 kg are kept at a distance of 6 m. Calculate the gravitational force of attraction between them. (G = 6.67 × 10-11 N m² kg-2)
- Write the formula: F = G m1m2 ÷ d²
- Put values: F = (6.67 × 10-11 × 80 × 1200) ÷ (6)²
- Numerator = 6.67 × 10-11 × 96000 = 6.4032 × 10-6
- Denominator = 6² = 36
- F = 6.4032 × 10-6 ÷ 36 = 1.78 × 10-7 N
Calculate the acceleration due to gravity g on the surface of the Earth, given mass of Earth M = 6 × 1024 kg, radius R = 6.4 × 106 m, and G = 6.67 × 10-11 N m² kg-2.
- Write the formula: g = G M ÷ R²
- Find R² = (6.4 × 106)² = 4.096 × 1013 m²
- Numerator = G M = 6.67 × 10-11 × 6 × 1024 = 4.002 × 1014
- g = 4.002 × 1014 ÷ 4.096 × 1013
- g = 9.77 m/s² ≈ 9.8 m/s²
For weight on the Moon, remember “Moon = 1/6”. Mass never changes — only Weight Wobbles (because W = mg, and g wobbles). Mass is the “matter you carry”; weight is “how hard gravity tugs”.
The most common mistake is forgetting to square the distance (using d instead of d²) in the universal law, and confusing mass (kg) with weight (N). Always check units: G has units N m² kg-2, and g is in m/s². Never write that the value of g is constant everywhere — it varies with height, depth, and latitude.
Q1. State the universal law of gravitation. Why is it called “universal”?
Answer: The universal law of gravitation states that every object in the universe attracts every other object with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them, i.e. F = G m1m2 ÷ d². The force acts along the line joining the centres of the two objects. It is called “universal” because it applies to all objects everywhere in the universe — from a falling apple to planets and stars — regardless of the medium between them.
Q2. What is the difference between mass and weight? Why does the weight of an object change on the Moon?
Answer: Mass is the amount of matter in a body; it is a scalar, measured in kilograms, and remains constant everywhere. Weight is the gravitational force with which a body is attracted, W = m g; it is a vector, measured in newtons, and varies with location because g varies. On the Moon the acceleration due to gravity is about one-sixth of that on Earth (since the Moon has less mass and smaller radius), so an object’s weight on the Moon is one-sixth of its weight on Earth, even though its mass stays the same.
Q3. Define acceleration due to gravity g and derive an expression for it. State two factors on which it depends.
Answer: The acceleration due to gravity g is the acceleration produced in a freely falling body due to the Earth’s gravitational pull. For a body of mass m near the Earth (mass M, radius R), the gravitational force is F = G M m ÷ R². This force also equals m g. Equating and cancelling m gives g = G M ÷ R². It depends on (i) the mass of the Earth M and (ii) the distance from the Earth’s centre R (so it changes with height, depth, and latitude). It does not depend on the mass of the falling object.
Q4. State Archimedes’ principle. Why does an iron nail sink in water but an iron ship float?
Answer: Archimedes’ principle states that when a body is wholly or partially immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by it. An iron nail sinks because the average density of solid iron is greater than that of water, so the weight of water it displaces is less than its own weight. A ship is hollow and shaped so that its average density (iron plus the large volume of air inside) is less than that of water; it displaces a large volume of water whose weight equals the ship’s weight, so the upthrust balances gravity and the ship floats.
- ✅ Every mass attracts every other mass: F = G m1m2 ÷ d² (inverse-square law).
- ✅ g = G M ÷ R² ≈ 9.8 m/s²; it is independent of the falling body’s mass.
- ✅ Mass is constant (kg); weight W = mg is a force (N) that changes with g.
- ✅ Pressure = thrust ÷ area; buoyant force = weight of fluid displaced (Archimedes).
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