Gravitation

Every object in the universe pulls every other object towards itself; this attractive force is called gravitation. The Earth's pull on nearby objects is specifically called gravity.

The idea builds historically from planetary motion. Nicolaus Copernicus proposed that planets, including Earth, revolve around the Sun. Johannes Kepler, using Tycho Brahe's observations, worked out three laws of planetary motion:

• Law of orbits: every planet moves in an elliptical orbit with the Sun at one focus.
• Law of areas: the line joining the planet to the Sun sweeps out equal areas in equal intervals of time (so a planet moves faster when nearer the Sun).
• Law of periods: the square of the time period of revolution of a planet is proportional to the cube of the semi-major axis of its orbit, T² ∝ r³.

Isaac Newton used Kepler's laws to reason that a force must be pulling the planets towards the Sun, and that the same force makes an apple fall to the ground — the starting point of the universal law of gravitation.

Newton's universal law of gravitation states: 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. The direction of this force is along the line joining the two objects.

For masses m₁, m₂ separated by distance d: F ∝ m₁m₂ and F ∝ 1/d², so F ∝ m₁m₂/d², giving F = G·m₁m₂/d², where G is the universal gravitational constant.

G = 6.673 × 10⁻¹¹ N m² kg⁻², the same everywhere — independent of the medium or the nature of the objects, which is why the law is universal. G was first measured experimentally by Henry Cavendish.

This one law explains many facts together: why a dropped object falls to Earth, why the Moon orbits the Earth, why planets orbit the Sun, why tides occur, and why our atmosphere stays close to the Earth instead of drifting into space.

NCERT lists everyday phenomena this single law explains — a favourite source of short-answer questions:

• The force that binds us to the Earth.
• The motion of the Moon around the Earth.
• The motion of planets around the Sun.
• The tides caused by the Moon and the Sun.

Before Newton, these were thought to be separate, unrelated phenomena; his insight was that one universal force of attraction between any two masses explains all of them.

These four points are worth memorising as a list, since a 2-mark question often asks you to name any two phenomena explained by the universal law of gravitation.

Whenever an object falls towards the Earth under the effect of gravitational force alone (no other force acting on it), it is said to be in free fall. The Earth's gravitational pull produces an acceleration in the falling object, called the acceleration due to gravity, denoted g. It has the same units as acceleration, m s⁻².

Since F = GMm/R² (using Earth's mass M, an object's mass m, and R as the distance from the Earth's centre to the object) and, by Newton's second law, F = mg, equating the two gives the relation between g and G:

mg = GMm/R² ⇒ g = GM/R²

Near Earth's surface R is essentially constant, so g works out to about 9.8 m s⁻². Since m cancels out of the derivation, g does not depend on the falling object's mass — all objects, light or heavy, fall with the same acceleration in the absence of air resistance (famously demonstrated on the airless Moon).

g varies slightly over the Earth: it is greater at the poles and smaller at the equator, because the Earth is not a perfect sphere (it is flattened at the poles, so the poles are closer to the centre) and because R increases as one moves from the poles to the equator. g also decreases with altitude and with depth below the surface.

Substituting M = 6 × 10²⁴ kg, R = 6.4 × 10⁶ m, and G = 6.7 × 10⁻¹¹ N m² kg⁻² into g = GM/R² gives g ≈ 9.8 m s⁻², the standard value used throughout this chapter (numericals sometimes round it to 10 m s⁻² — always use the value the question specifies).

Note that G is extremely small while M is extremely large, and R is squared in the denominator — it is the combination of a very large mass with a comparatively small radius that gives Earth's surface a measurable, everyday value of g.

Because free fall is uniformly accelerated motion (acceleration = g, constant), the three equations of motion apply directly, with a replaced by g (taken positive for falling motion, negative for upward motion since gravity then decelerates the object):

v = u + gt; s = ut + ½gt²; v² = u² + 2gs

For an object dropped from rest (u = 0): v = gt, h = ½gt², v² = 2gh. For an object thrown upward with speed u, g opposes the motion, so the object decelerates, momentarily stops at the top (v = 0), then falls back with the same launch speed; time up = time down, total flight time = 2u/g. These equations are the basis of nearly every numerical in this chapter.

The mass of a body is the quantity of matter contained in it. Mass is a scalar quantity, its SI unit is the kilogram (kg), and it does not change with location — the mass of an object is the same on the Earth, the Moon, or in outer space, because the amount of matter in it does not change.

Mass is also a measure of a body's inertia — its resistance to a change in motion, met in the previous chapter. A larger mass is harder to accelerate for a given force. Mass can never be zero or negative for a physical object, and simply adds up when objects are combined.

The weight of an object is the force with which the Earth (or any other body) attracts it, i.e. W = mg. Weight is a vector quantity, always directed towards the centre of the attracting body, and its SI unit is the newton (N), the same unit as force. Since g varies from place to place, the weight of a given object also varies from place to place, even though its mass stays constant.

Everyday spring balances actually measure weight but are calibrated to show 'mass' in kg assuming g is constant — strictly, a person doesn't 'weigh 60 kg'; 60 kg is the mass, and the weight is 60 × 9.8 = 588 N.

9.5.1 Weight on the Moon. The Moon's mass and radius are both much smaller than Earth's, and g_moon = GM_moon/R_moon² works out to about 1/6th of g on Earth (≈1.63 m s⁻²). Since W = mg, an object's weight on the Moon is about 1/6th its weight on Earth even though its mass is unchanged — why astronauts jump higher and carry heavier loads there.

When a body is in free fall — e.g. inside a lift whose cable has snapped, or an astronaut inside an orbiting spacecraft — it feels no support/reaction force, so it appears to have no weight, even though gravity still acts on it, giving it acceleration g. This is weightlessness. An orbiting satellite is technically in continuous free fall around the Earth — gravity supplies exactly the centripetal force needed for the orbit, which is why astronauts float inside it.

Thrust is the force acting perpendicular to a surface; SI unit newton (N). Pressure is thrust per unit area: P = F/A; SI unit pascal (Pa), 1 Pa = 1 N/m².

For the same thrust, a smaller contact area gives more pressure, a larger area gives less — why a sharp knife cuts more easily than a blunt one, why camels have broad feet to avoid sinking in sand, and why a nail has a pointed tip and a wide flat head.

Fluids (liquids and gases) also exert pressure on their container walls and on any immersed object. Unlike a solid on a surface, fluid pressure at a point acts equally in all directions, not just downward. It increases with depth, since a deeper layer has more fluid above pressing down, and it depends on the fluid's density too — denser fluids exert more pressure at the same depth.

This is why a dam is built much thicker at its base than at the top: the water pressure is greatest at the bottom, so the wall there must be strong enough to withstand the largest force. It is also why deep-sea divers must wear pressure-resistant suits, since the pressure of water increases rapidly with depth and would otherwise crush an unprotected body.

When an object is immersed in a fluid, the fluid exerts an upward force on it that opposes the object's weight; this upward force is called buoyant force (or upthrust), and the phenomenon is called buoyancy. It happens because pressure in a fluid increases with depth, so the upward pressure pushing on the bottom face of the object is greater than the downward pressure pushing on its top face, giving a net upward push.

You can feel this yourself: it takes more effort to push an empty bottle down into water than to lift it in air, because the water pushes back up. The buoyant force's size depends on how much fluid is displaced and its density — the exact relationship Archimedes' principle states precisely.

Whether an object floats or sinks depends on comparing its density with the fluid's density:

• Density less than the fluid: the object floats, rising until just enough of it is submerged for the buoyant force to balance its weight (wood, ice, cork).
• Density equal to the fluid: it stays suspended anywhere within it (neutral buoyancy).
• Density greater than the fluid: it sinks (a stone, or solid iron).

This is why a solid iron ball sinks but a hollow iron ship floats: the ship's shape displaces a large volume of water, so its average density (mass ÷ total volume including its air-filled hollows) is less than water's, even though solid iron itself is denser than water.

Archimedes' principle: when a body is immersed fully or partially in a fluid, it experiences an upward buoyant force equal to the weight of the fluid displaced by it.

Applications examiners test directly: design of ships and submarines (a submarine changes its average density using ballast tanks to sink, float, or hover); hydrometers/lactometers, which measure a liquid's density from how deep they sink; and why we feel lighter standing in water, since part of our weight is supported by the buoyant force.

Relative density = density of substance / density of water — a ratio, so it has no unit.