Gravitation for NDA

Gravitation is a short chapter with a long reach. It explains why things fall, why the Moon stays in orbit, why astronauts float, why tides rise twice a day and why a satellite over the equator seems to hang still in the sky. NDA's GK section likes it because every one of those facts can be turned into a one-line question, and because the ratios (distance doubled, radius halved, depth R/2) are quick to test without a calculator.

Nearly every question here is one of two kinds. Either it asks what happens to g or to weight in some situation, or it asks a satellite fact: escape velocity, the geostationary period, why an orbiting astronaut feels weightless. Learn the handful of formulas below with their conditions and both kinds become easy.

Typical question types are:

• the effect of changing mass or distance on the gravitational force;
• where g is greatest or least, and how it changes with height and depth;
• mass versus weight, and weight on the Moon;
• weightlessness in orbit and in a freely falling lift;
• orbital and escape velocity, and what they depend on;
• geostationary and polar satellites;
• Kepler's laws, and the cause of tides.

Newton's law of gravitation

Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centres:

F = G m₁m₂ / r²

• G is the universal gravitational constant, about 6.67 × 10⁻¹¹ N m² kg⁻². It is the same everywhere and does not depend on the medium between the bodies.
• Gravity is always attractive and is the weakest of the fundamental forces, which is why we notice it only when at least one mass is huge.
• Because of the inverse square: double the distance and the force falls to ¼; halve it and the force becomes 4 times.
• By the third law, the Earth pulls an apple and the apple pulls the Earth with equal force. The apple visibly accelerates because its mass is tiny; the Earth's acceleration is negligible.

Distinguish g from G. G is a universal constant. g is the acceleration of a freely falling body at a particular place, and it varies.

Acceleration due to gravity

At the Earth's surface, g = GM/R², where M and R are the Earth's mass and radius. Its value is about 9.8 m/s² (often taken as 10). Notice that g does not contain the mass of the falling body, so in a vacuum a feather and a coin fall together.

Why g is less at the equator. Two reasons. The Earth bulges at the equator, so a point there is farther from the centre. And the Earth's rotation uses part of gravity to keep a body moving in its daily circle, and that effect is largest at the equator and zero at the poles. If the Earth stopped rotating, g would increase at the equator but stay the same at the poles. If it spun about 17 times faster than now, objects at the equator would feel weightless.

Mass and weight

On the Moon, g is about one-sixth of the Earth's value, so a body weighs about one-sixth as much, but its mass is unchanged. A beam balance gives the same reading on the Moon because it compares two masses under the same g; a spring balance reads less. The Moon has almost no atmosphere partly because its low gravity cannot hold gas molecules.

Weightlessness

"Weight" as you feel it is the push of the floor or seat on you. If you and your support fall with the same acceleration, that push becomes zero and you feel weightless.

• In a freely falling lift, the scale reads zero.
• Astronauts in an orbiting spacecraft feel weightless because they and the craft are falling freely around the Earth together. Gravity is still acting on them; at the height of a space station it is only about a tenth weaker than at the surface.
• In an orbiting spacecraft, a pendulum clock stops working because the bob has no effective weight to swing it back.

Satellites: orbital and escape velocity

A satellite stays in orbit because gravity supplies exactly the centripetal force it needs. It is falling towards the Earth all the time, but moving sideways fast enough that it keeps missing.

• Orbital velocity close to the surface is v₀ = √(gR), about 7.9 km/s. It does not depend on the satellite's mass, and it becomes smaller for higher orbits. Higher satellites also take longer to go round.
• Escape velocity is the minimum launch speed needed to leave the Earth's gravity for good: vₑ = √(2gR), about 11.2 km/s. It depends on the mass and radius of the planet, not on the mass of the body or the direction of launch.
• The two are linked: vₑ = √2 × v₀. A satellite whose speed is increased by about 41% will escape.

A geostationary satellite appears fixed over one point because it goes round once in the same time the Earth rotates, in the same direction. That is why a dish antenna can point at it permanently.

Kepler's laws and tides

1. Law of orbits: each planet moves in an ellipse with the Sun at one focus.
2. Law of areas: the line joining the Sun and a planet sweeps equal areas in equal times. So a planet moves fastest when closest to the Sun and slowest when farthest. (This is conservation of angular momentum.)
3. Law of periods: T² ∝ r³, where T is the period and r the mean distance from the Sun. Farther planets take much longer to go round.

Tides are caused mainly by the Moon's gravity, with a smaller share from the Sun. Spring tides (highest high tides) come at new moon and full moon, when the Sun, Earth and Moon are in line. Neap tides come near the first and last quarter, when the two pulls are at right angles.

Worked NDA-style MCQs

Q1. The distance between two bodies is halved. The gravitational force between them becomes:
(a) half (b) double (c) one-fourth (d) four times

F ∝ 1/r². Halving r multiplies F by 2² = 4. Answer: (d).

Q2. A body weighs 600 N on the Earth's surface. At a height equal to the Earth's radius, it weighs:
(a) 150 N (b) 300 N (c) 200 N (d) 600 N

g at h = R is gR²/(2R)² = g/4, so the weight is 600/4 = 150 N. Answer: (a).

Q3. An astronaut has a mass of 60 kg on the Earth. Taking g = 10 m/s² on the Earth and one-sixth of that on the Moon, the astronaut's mass and weight on the Moon are:
(a) 10 kg, 100 N (b) 60 kg, 600 N (c) 60 kg, 100 N (d) 10 kg, 600 N

Mass stays 60 kg. Weight = 60 × 10/6 = 100 N. Answer: (c).

Q4. If the Earth shrank to half its radius with its mass unchanged, g at its surface would become:
(a) g/2 (b) 2g (c) 4g (d) g/4

g = GM/R². Halving R multiplies g by 4. Answer: (c).

Q5. A body weighs 400 N at the Earth's surface. At a depth of half the Earth's radius, it weighs:
(a) 100 N (b) 200 N (c) 400 N (d) 800 N

g at depth d = g(1 − d/R) = g(1 − ½) = g/2. Weight = 200 N. Answer: (b).

Q6. A planet is 4 times as far from the Sun as the Earth. Its period of revolution is:
(a) 4 years (b) 8 years (c) 16 years (d) 64 years

T² ∝ r³, so T² = 4³ = 64 and T = 8 years. Answer: (b).

Common mistakes

• Confusing G with g. G never changes; g changes with place.
• Using the depth formula for height, or the reverse. Going up, g falls as 1/(R + h)²; going down, it falls in a straight line to zero at the centre.
• Saying mass changes on the Moon. Only weight changes.
• Thinking escape velocity depends on the body's mass. A cricket ball and a rocket need the same 11.2 km/s.
• Putting geostationary satellites over the poles. They must be above the equator.

Practice set

1. The value of g at the centre of the Earth is: (a) zero (b) 9.8 m/s² (c) maximum (d) infinite
2. A body weighs most at: (a) the equator (b) the poles (c) the centre of the Earth (d) the top of a mountain
3. The escape velocity from the Earth depends on: (a) the mass of the body (b) the direction of launch (c) the shape of the body (d) the mass and radius of the Earth
4. The period of a geostationary satellite is: (a) 12 hours (b) 27.3 days (c) 24 hours (d) 90 minutes
5. On the Moon, compared with the Earth, a spring balance reading of a given body would be: (a) the same (b) less (c) more (d) zero
6. If both the mass and the radius of a planet were double the Earth's, g on it would be: (a) g/2 (b) g (c) 2g (d) 4g
7. If orbital velocity close to the Earth's surface is v, the escape velocity is: (a) 2v (b) v/√2 (c) v² (d) √2 v
8. Tides in the sea are caused mainly by: (a) winds (b) the Earth's rotation alone (c) the Moon's gravitational pull (d) ocean currents

Answers:

1. (a). g(1 − d/R) is zero when d = R.
2. (b). g is largest at the poles.
3. (d). vₑ = √(2GM/R); the body's own mass cancels out.
4. (c). It matches the Earth's rotation.
5. (b). Weight falls to about one-sixth; a beam balance would read the same.
6. (a). GM/R² becomes G(2M)/(2R)² = ½ × GM/R².
7. (d). vₑ = √(2gR) = √2 × √(gR).
8. (c). The Sun's pull adds to it at new and full moon.

What to do next

• Copy the "variation of g" table from memory and solve one ratio question for each row.
• Make a satellite card: orbital velocity, escape velocity, geostationary height and period, and why astronauts float.
• Read space technology for NDA to link these facts to Indian satellites and launch vehicles.
• Revise work, energy and power, then continue to properties of matter.