“Negative means trapped”
Why gravitational energy is negative
Zero energy is defined at infinity, where the body is free. Anything bound has less than that, so its total energy is negative. Reaching zero is exactly what escaping means.
Class 11 Physics · Gujarat Board, CBSE, GUJCET and JEE
Gravitation formulas
Gravitation is where the mechanics of Class 11 meets the sky. It supplies the satellite results asked in every GUJCET and JEE paper, and it is the first place where potential energy is negative, which is the idea most students never fully accept.
18 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
The law and g
| What | Formula | Symbols and notes |
|---|---|---|
| Newton's law of gravitation | F = Gm₁m₂/r² | G = 6.674 × 10−11 N m² kg−2 |
| Acceleration due to gravity | g = GM/R² | independent of the falling body's mass |
| At a height h | gh = g(1 − 2h/R) | valid only when h is much smaller than R |
| At a depth d | gd = g(1 − d/R) | g is zero at the centre of the Earth |
| Effect of rotation | g' = g − ω²R cos²λ | λ is the latitude, so g is least at the equator |
Satellites
| What | Formula | Symbols and notes |
|---|---|---|
| Orbital velocity | vo = √(GM/r) = R√(g/r) | for a circular orbit of radius r from the centre |
| Time period | T = 2π√(r³/GM) | this is Kepler's third law written out |
| Escape velocity | ve = √(2GM/R) = √(2gR) ≈ 11.2 km s−1 | does not depend on the escaping body's mass |
| Escape and orbital | ve = √2 × vo | about 41% faster is all it takes to leave |
| Geostationary satellite | T = 24 h, height ≈ 36,000 km | equatorial orbit, moving west to east |
Energy in orbit
| What | Formula | Symbols and notes |
|---|---|---|
| Gravitational potential energy | U = −GMm/r | negative, and zero only at infinity |
| Kinetic energy in orbit | KE = GMm/2r | half the magnitude of the potential energy |
| Total energy in orbit | E = −GMm/2r | negative, which is what bound means |
| Binding energy | BE = GMm/2r | the energy needed to free the satellite |
| Potential | V = −GM/r | potential energy per unit mass |
Kepler's laws
| What | Formula | Symbols and notes |
|---|---|---|
| First law | Orbits are ellipses with the Sun at one focus | not circles |
| Second law | Equal areas are swept in equal times | a planet moves fastest when nearest the Sun |
| Third law | T² ∝ r³ | r is the semi-major axis |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Root two to be free”
Escape velocity from orbital velocity
Escape velocity is √2 times orbital velocity at the same radius. Roughly 8 km per second to orbit, 11.2 to leave.
“Height halves twice as fast as depth”
Variation of g
Going up, g falls as (1 − 2h/R). Going down, it falls as (1 − d/R). The factor 2 appears only in the height formula.
Where marks go
The mistakes that cost the most.
- Using g = GM/R² with r measured from the Earth's surface. Distances in gravitation are measured from the centre.
- Writing potential energy as mgh for a satellite. mgh works only near the surface where g is roughly constant.
- Dropping the minus sign in U = −GMm/r, which then makes every energy answer wrong in sign.
- Thinking astronauts float because there is no gravity. They are in free fall, and g at that height is still about 8.7 m s−2.
- Forgetting that escape velocity does not depend on the direction of projection or on the mass thrown.
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