Gravitation

Isaac Newton showed that the same force that pulls an apple to the ground also keeps the Moon revolving around the Earth. This force of attraction between any two objects in the universe is called gravitation.

Newton's 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 their centres.

If two objects of masses m₁ and m₂ are separated by a distance r, the force of attraction F between them is given by:

F = G m₁m₂/r²

Here G is the universal gravitational constant. Its value is the same everywhere in the universe: G = 6.7 × 10^-11 N m² kg^-2. This force always pulls objects towards each other (it is never repulsive), and it acts even when the objects are far apart and not in contact.

Worked example: Two spheres of mass 40 kg and 60 kg have their centres 3 m apart. Find the gravitational force of attraction between them.

Given: m₁ = 40 kg, m₂ = 60 kg, r = 3 m, G = 6.7 × 10^-11 N m² kg^-2

F = G m₁m₂/r² = (6.7 × 10^-11 × 40 × 60) / 3² = (6.7 × 10^-11 × 2400) / 9

F ≈ 1.79 × 10^-8 N

This force is extremely small for objects of everyday mass, which is why we never notice ourselves being pulled towards nearby objects — the effect becomes significant only when at least one mass is very large, like a planet.

The universal law of gravitation explains many everyday and astronomical events: it is why objects fall towards the Earth, why the Moon revolves around the Earth, why planets revolve around the Sun, and why ocean tides occur due to the pull of the Moon and the Sun.

When an object falls towards the Earth under the effect of gravitational force alone (with no other force such as air resistance acting on it), it is said to be in free fall. During free fall, the velocity of the object changes at a constant rate — this constant acceleration is called the acceleration due to gravity, denoted g.

Near the Earth's surface, g ≈ 9.8 m/s². It is obtained from Newton's law of gravitation by taking one of the masses to be the Earth's mass M and the distance to be the Earth's radius R:

g = GM/R²

An important result, first demonstrated by Galileo, is that g does not depend on the mass of the falling object — a heavy stone and a light stone dropped together from the same height (in the absence of air resistance) hit the ground at the same time.

Since g is constant during free fall, the equations of motion can be used with g in place of acceleration a (taking the downward direction as positive for a falling object):

v = u + gt s = ut + (1/2)gt² v² = u² + 2gs

When an object is thrown upward, gravity acts against its motion, so g is taken as negative (retardation) until the object reaches its highest point, where its velocity becomes zero.

Worked example: A ball is dropped from a height of 20 m. Find (i) the velocity with which it strikes the ground and (ii) the time it takes to fall. (Take g = 9.8 m/s²)

Given: u = 0, s = 20 m, g = 9.8 m/s²

(i) v² = u² + 2gs = 0 + 2 × 9.8 × 20 = 392

v = √392 ≈ 19.8 m/s

(ii) v = u + gt, so t = v/g = 19.8/9.8 ≈ 2.02 s

Mass is the quantity of matter contained in an object. It is a scalar quantity, measured in kilograms (kg), and it does not change no matter where the object is placed — on the Earth, on the Moon, or in outer space.

Weight is the force with which a planet or moon attracts an object towards its centre. Since weight is a force, it is a vector quantity, its SI unit is the newton (N), and it is given by:

W = mg

Because the value of g is different on different celestial bodies, the weight of the same object changes from place to place, even though its mass stays exactly the same. On the surface of the Moon, the acceleration due to gravity is only about one-sixth of that on the Earth (g_moon ≈ 1.63 m/s², compared with g_earth ≈ 9.8 m/s²), so an object weighs about one-sixth as much on the Moon as it does on the Earth, though its mass is unchanged.

Worked example: A boy has a mass of 60 kg. Find his weight (i) on the Earth and (ii) on the Moon. (Take g_earth = 9.8 m/s², g_moon = 1.63 m/s²)

Given: m = 60 kg

(i) W_earth = mg_earth = 60 × 9.8 = 588 N

(ii) W_moon = mg_moon = 60 × 1.63 = 97.8 N (close to 588/6 ≈ 98 N)

His mass, however, remains 60 kg on both the Earth and the Moon.

A force acting on an object perpendicular to its surface is called thrust. Thrust is measured in newtons (N), the same unit as any other force.

The effect of a thrust depends not just on its size but also on the area over which it acts. Pressure is defined as the thrust acting per unit area of the surface:

Pressure = Thrust/Area, or P = F/A

The SI unit of pressure is the pascal (Pa), where 1 Pa = 1 N/m². For the same thrust, a smaller area of contact produces a larger pressure, and a larger area of contact produces a smaller pressure. This is why a sharp knife (small contact area) cuts more easily than a blunt one, why a camel's broad feet keep it from sinking into desert sand, and why a school bag with a wide strap is more comfortable to carry than one with a thin strap.

Worked example: A wooden block of mass 5 kg and dimensions 40 cm × 20 cm × 10 cm rests on a table. Find the pressure it exerts when it lies on its smallest face, of area 20 cm × 10 cm. (Take g = 9.8 m/s²)

Given: m = 5 kg, area A = 20 cm × 10 cm = 0.20 m × 0.10 m = 0.02 m²

Thrust, F = weight of block = mg = 5 × 9.8 = 49 N

Pressure, P = F/A = 49/0.02 = 2450 Pa

Pressure in fluids: Liquids and gases (fluids) also exert pressure — on the base and walls of their container, and on any object placed in them. Fluid pressure acts equally in all directions at a given point, and it increases with depth. The pressure at a depth h in a fluid of density ρ is given by:

P = hρg

This is why a dam is built thicker at the bottom than at the top, and why a diver feels greater pressure on the ears at greater depths in water.

When an object is placed in a fluid (liquid or gas), the fluid pushes up on it with an upward force called the buoyant force or upthrust, and this phenomenon is called buoyancy. It occurs because the pressure of the fluid at greater depth (on the lower surface of the object) is more than the pressure at shallower depth (on its upper surface), so the net pressure force is upward.

Archimedes' principle states that when an object is fully or partially immersed in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces.

Whether an object floats or sinks in a fluid depends on comparing its density with the density of the fluid:

• If the density of the object is less than the density of the fluid, the object floats (with part of it above the surface).
• If the density of the object is greater than the density of the fluid, the object sinks.
• If the density of the object equals the density of the fluid, the object remains suspended, fully submerged, in equilibrium.

The relative density of a substance is the ratio of its density to the density of water:

Relative density = Density of substance / Density of water

Relative density has no unit, since it is a ratio of two densities. It can also be found experimentally using the loss of weight of an object in water:

Relative density = Weight in air / (Weight in air − Weight in water)

Worked example: A piece of metal weighs 300 N in air. When fully immersed in water, it weighs 260 N. Find (i) the loss in weight and (ii) the relative density of the metal.

Given: Weight in air = 300 N, Weight in water = 260 N

(i) Loss in weight = 300 − 260 = 40 N (this equals the weight of water displaced, by Archimedes' principle)

(ii) Relative density = Weight in air / Loss in weight = 300/40 = 7.5

This principle is used in hydrometers and lactometers, which measure the relative density of liquids, and it explains why a solid iron ball sinks in water while a hollow iron ship of the same material floats — the ship's shape displaces a much larger volume of water, so its average density (mass divided by total volume, including the enclosed air) becomes less than the density of water.