Gravitation — Newton's Law and Orbital Motion
Gravitation is a short chapter with unusually heavy arithmetic. The physics is one formula; the marks are lost in powers of ten, in using the wrong distance, and in confusing the two symbols G and g, which look almost identical on a hurried answer sheet. This guide gives the formulas, then works four examples in full — including deriving the value of g from scratch and finding how long a satellite takes to go round the Earth.
Newton's law of universal gravitation
Every pair of masses attracts along the line joining them, with a force proportional to both masses and inversely proportional to the square of the separation. The crucial detail in the fine print: r is measured centre to centre, not surface to surface.
The formulas that follow from it
Gravity at height h: g′ = G M / (R + h)² = g · R² / (R + h)²
Gravitational potential energy: U = − G M m / r
Orbital speed: v = √(G M / r)
Orbital period (Kepler's third law): T = 2π √(r³ / G M), so T² ∝ r³
Escape speed: ve = √(2 G M / R) = √(2 g R)
What each symbol means
| Symbol | Meaning | SI unit |
|---|---|---|
| G | Universal gravitational constant — the same everywhere in the universe | N m² kg⁻² |
| g | Acceleration due to gravity — a local value that depends on the planet | m s⁻² |
| M | Mass of the large body (planet or star) | kg |
| R | Radius of that body | m |
| h | Height above the surface | m |
| r | Centre-to-centre distance, equal to R + h for an orbit | m |
Reference values used below: mass of the Earth M = 5.972 × 10²⁴ kg, mean radius R = 6.371 × 10⁶ m. These are mean values — the Earth is not a perfect sphere, so the measured g varies slightly with latitude and altitude. The internationally agreed standard gravity is 9.80665 m s⁻²; textbooks round it to 9.8 or 9.81 m s⁻². Any of those is acceptable in an exam provided you say which one you used and stay consistent.
Worked example 1 — deriving g from G, M and R
g = G M / R²
Numerator: 6.674 × 10⁻¹¹ × 5.972 × 10²⁴
6.674 × 5.972 = 39.857, and 10⁻¹¹ × 10²⁴ = 10¹³
so G M = 39.857 × 10¹³ = 3.9857 × 10¹⁴ m³ s⁻²
Denominator: R² = (6.371 × 10⁶)² → 6.371² = 40.590, so R² = 4.0590 × 10¹³ m²
g = 3.9857 × 10¹⁴ ÷ 4.0590 × 10¹³ = 9.82 m s⁻²
That is the accepted surface value, obtained from three independently measured constants — a genuinely satisfying check that the inverse-square law is right.
Worked example 2 — escape speed from the Earth
ve = √(2 g R) = √(2 × 9.82 × 6.371 × 10⁶)
2 × 9.82 = 19.64
19.64 × 6.371 = 125.13, so the quantity under the root is 1.2513 × 10⁸ m² s⁻²
√(1.2513 × 10⁸) = 1.1186 × 10⁴ m s⁻¹
ve ≈ 11.2 km s⁻¹
Note what is missing from the formula: the mass of the escaping object. A cricket ball and a rocket need the same launch speed to escape — which is why the formula is written without m at all.
Worked example 3 — how long a satellite takes to circle the Earth
A satellite orbits at a height of 400 km. Find its time period.
First the orbital radius, centre to centre:
r = R + h = 6371 km + 400 km = 6771 km = 6.771 × 10⁶ m
Now r³: 6.771² = 45.846, and 45.846 × 6.771 = 310.43
so r³ = 310.43 × 10¹⁸ = 3.1043 × 10²⁰ m³
r³ ÷ G M = 3.1043 × 10²⁰ ÷ 3.9857 × 10¹⁴ = 7.7885 × 10⁵ s²
√(7.7885 × 10⁵) = 882.5 s
T = 2π × 882.5 = 5545 s ≈ 92.4 minutes
So a satellite in low Earth orbit goes round roughly once every hour and a half — which matches what is actually observed for spacecraft at that altitude.
Worked example 4 — why astronauts float, and it is not because gravity is gone
Find g at that same 400 km altitude.
g′ = g × R² / (R + h)² = 9.82 × (6371 ÷ 6771)²
6371 ÷ 6771 = 0.94092
0.94092² = 0.88534
g′ = 9.82 × 0.88534 = 8.69 m s⁻²
That is about 89% of surface gravity. Astronauts up there are not weightless because gravity has vanished — they float because the spacecraft and everything inside it are in continuous free fall around the Earth together. "Zero gravity" is a misleading phrase for what is really free fall.
How weak gravity really is between everyday objects
Two 1000 kg cars parked 10 m apart attract each other with
That is under a millionth of a newton — far too small to notice against friction. Gravity only becomes dominant when one of the masses is planet-sized, which is why the same law that is negligible in a car park holds the Moon in orbit.
Common mistakes that cost marks
- Using the height instead of the orbital radius. For a satellite at 400 km, r is 6771 km, not 400 km. This single slip produces answers wrong by orders of magnitude and is the most common error in the chapter.
- Forgetting to square r. The inverse-square is the whole law; F ∝ 1/r is simply a different, wrong physics.
- Confusing G with g. G is a universal constant of nature, 6.674 × 10⁻¹¹ N m² kg⁻². Small g is a local acceleration, about 9.8 m s⁻² on Earth and different on every other body.
- Dropping the minus sign in U = −GMm/r. Gravitational potential energy is negative because the zero is defined at infinite separation; a positive answer here means a sign was lost.
- Thinking escape speed depends on the escaping mass. It does not — and it is also independent of launch direction, provided nothing blocks the path.
- Sloppy powers of ten. Handle the mantissas and the exponents separately, as in example 1, and the arithmetic stops being frightening.
Where gravitation appears in exams
| Exam | Typical use |
|---|---|
| CBSE / ICSE Class 11 | Value of g, variation with height and depth, escape speed, Kepler's laws |
| JEE / NEET | Orbital and escape speed comparisons, satellite energy, geostationary orbits |
| Class 11–12 chemistry | The same inverse-square algebra reappears in Coulomb's law for charges |
| General aptitude sections | Powers-of-ten arithmetic and unit handling, tested through physics data |
The chemistry link is worth noticing. Coulomb's law, F = k q₁q₂ / r², has exactly the same shape as Newton's law — the only differences are the constant and the fact that charges can repel while masses only attract. Every algebraic move you practise here transfers directly to lattice energy and ionic-bonding questions later.
Do the powers-of-ten arithmetic without slips. Gravitation questions are mostly exponent handling, and the Scientific Calculator — the view that opens by default — has the xy, √ and exponent keys these problems need.
Open the ABC Chemistry Calculator Suite →Class 11–12 and struggling to keep physics-style numericals from eating your chemistry marks? ABC Chemistry runs Class 11–12 chemistry coaching at the Gurugram centre plus online classes across India — details at abcchemistry.in.