Gravitation for NEET

Gravitation is one of the more formula-driven chapters in NEET mechanics, and that makes it a good scoring chapter. Most questions are ratio problems: what happens to g, the escape velocity or a satellite's period when the radius, mass or height changes. If you know how each quantity depends on r, you can often answer without a calculator-style calculation.

The main trap is using an approximate formula outside its range, especially the "h much smaller than R" formula for g at height.

Newton's law of gravitation

Every two point masses attract each other with a force

F = G m₁m₂ / r², where G = 6.67 × 10⁻¹¹ N m² kg⁻².

• The force acts along the line joining the masses and is always attractive.
• It obeys the third law: the Earth pulls you exactly as hard as you pull it.
• A uniform spherical shell (or solid sphere) attracts an outside mass as if all its mass were at its centre. Inside a uniform shell, the net force from the shell is zero.

G is a universal constant. g is not: it depends on where you are.

Acceleration due to gravity and its variation

At the surface, g = GM/R², about 9.8 m s⁻². Weight (mg) changes with g; mass does not.

The graph in words: going outward from the centre, g increases in a straight line from zero to its maximum at the surface, then falls off as 1/r² outside. So g is greatest at the surface, whether you go up or down.

For small h, the decrease going up a height h equals the decrease going down a depth d = 2h.

Potential energy and potential

Gravitational potential energy of mass m at distance r (≥ R) from the Earth's centre is U = −GMm/r, taking U = 0 at infinity. It is negative because work must be done to pull the mass away to infinity.

Gravitational potential is PE per unit mass: V = −GM/r.

The familiar mgh is an approximation for small heights. The exact change in PE when a mass is raised from the surface to height h is ΔU = GMm h / [R(R + h)] = mgh / (1 + h/R). For h ≪ R, this becomes mgh.

Escape velocity

The minimum launch speed needed to escape the Earth's gravity: set KE equal to the energy needed to reach infinity.

½mv_e² = GMm/R, so v_e = √(2GM/R) = √(2gR), about 11.2 km s⁻¹ for the Earth.

It does not depend on the mass of the body or the direction of launch (ignoring air resistance).

Satellites: speed, period and energy

For a satellite in a circular orbit of radius r = R + h, gravity supplies the centripetal force: GMm/r² = mv²/r.

So KE = −E and PE = 2E. The binding energy (energy needed to free the satellite) is GMm/2r. Near the surface, escape velocity is √2 times the orbital speed.

A geostationary satellite orbits above the equator, west to east, with a period of 24 hours, so it stays over one point. Its height is about 36,000 km.

Kepler's laws

1. Orbits: each planet moves in an ellipse with the Sun at one focus.
2. Areas: the line from Sun to planet sweeps equal areas in equal times. This follows from conservation of angular momentum, because gravity exerts no torque about the Sun. So a planet moves fastest at its closest point (perihelion).
3. Periods: T² ∝ a³, where a is the semi-major axis. For circular orbits, T² = (4π²/GM) r³.

Worked numericals

Example 1: where does g fall to a quarter?

• Height: (R/(R + h))² = 1/4, so R/(R + h) = 1/2 and h = R.
• Depth: 1 − d/R = 1/4, so d = 3R/4.

Example 2: energy to put a satellite in orbit

How much energy is needed to launch a satellite of mass m from the surface into a circular orbit at height R (so r = 2R)? Ignore the Earth's rotation.

• Energy at the surface (at rest): −GMm/R.
• Total energy in orbit: −GMm/(2 × 2R) = −GMm/4R.
• Energy needed = −GMm/4R − (−GMm/R) = 3GMm/4R. Since GM = gR², this is ¾ mgR.

Example 3: escape velocity on another planet

A planet has 8 times the Earth's mass and twice its radius. Find its escape velocity.

• v_e ∝ √(M/R) = √(8/2) = 2.
• v_e = 2 × 11.2 = 22.4 km s⁻¹.

Example 4: Kepler's third law

A satellite's orbital radius is made 4 times larger. What happens to its period and speed?

• T ∝ r³ᐟ² (that is, T² ∝ r³): (√4)³ = 2³ = 8 times.
• v ∝ 1/√r: speed becomes half. Check: T = 2πr/v, so the period scales as 4 ÷ ½ = 8 ✓.

Practice MCQs

1. Two masses attract with force F. Each mass is doubled and the distance is halved. The new force is: (a) F (b) 4F (c) 8F (d) 16F
2. A planet has the same density as the Earth but twice its radius. Its escape velocity is: (a) half the Earth's (b) the same (c) twice the Earth's (d) four times the Earth's
3. A body's weight at a depth of R/2 compared with its surface weight is: (a) a quarter (b) half (c) the same (d) double
4. The value of g at a height R/2 above the surface is: (a) 0 (b) g/2 (c) 4g/9 (d) 2g/3
5. A satellite has total energy −E. Its kinetic energy is: (a) E (b) −E (c) 2E (d) E/2
6. The ratio of orbital speeds of satellites at radii r and 4r is: (a) 1 : 2 (b) 2 : 1 (c) 1 : 4 (d) 4 : 1
7. If the Earth shrank so its radius fell by 1% with the same mass, g at the surface would: (a) fall by 1% (b) rise by 1% (c) rise by about 2% (d) not change
8. Which does not depend on the mass of the projected body? (a) escape velocity (b) kinetic energy needed to escape (c) weight (d) gravitational PE

Answers

1. (d) F ∝ m₁m₂/r²: 2 × 2 × 4 = 16.
2. (c) M ∝ ρR³, so v_e ∝ √(M/R) ∝ R√ρ. Twice R gives twice v_e.
3. (b) g(1 − ½) = g/2.
4. (c) g × (R/1.5R)² = 4g/9. The approximate formula wrongly gives 0.
5. (a) KE = −(total energy) = E.
6. (b) v ∝ 1/√r, so v₁/v₂ = √4 = 2.
7. (c) g ∝ 1/R², so Δg/g ≈ −2ΔR/R = +2%.
8. (a) v_e = √(2GM/R) has no m in it.

What to do next

• Write a single table of how g, v_o, v_e, T and the satellite energies scale with r, and revise it until you can reproduce it.
• Practise ten ratio questions where you change M, R, ρ or r, and solve each by proportion, not substitution.
• Solve previous-year NEET gravitation questions, timed at about 60 to 90 seconds each.
• Revise angular momentum from rotational motion for NEET before Kepler's second law questions.

Next in the syllabus: properties of solids and liquids.