Maharashtra State Board · Class 10 · Physics · Chapter 1

Gravitation — Formula Sheet

Board Formulas

9 formulas

1.Universal Law of Gravitation (Newton, 1687)★
$F = G \frac{m _{1} m _{2}}{r ^{2}}$
$F$ : Gravitational force (N) · $G$ : Universal gravitational constant = 6.67 × 10⁻¹¹ N·m²/kg² · $m_{1}, m_{2}$ : Masses of the two bodies (kg) · $r$ : Distance between their centres (m)
Every mass attracts every other mass along the line joining their centres. Attractive only — no repulsion. Inverse-square law.
2.Acceleration due to Gravity (surface)★
$g = \frac{GM}{R ^{2}}$
$g$ : Acceleration due to gravity (m/s²) · $M$ : Mass of Earth ≈ 6 × 10²⁴ kg · $R$ : Radius of Earth ≈ 6.4 × 10⁶ m
Acceleration experienced by a freely falling body near Earth's surface. On Earth, g ≈ 9.8 m/s².
3.Weight★
$W = m g$
$W$ : Weight (N) · $m$ : Mass (kg) — scalar, constant everywhere · $g$ : Local acceleration due to gravity (m/s²)
Weight = gravitational force on the body. Unit: newton (N). Weight varies with location (g varies); mass does not.
4.g at Altitude h
$g_{h} = g (\frac{R}{R + h})^{2}$
g decreases with altitude. At h = R (one Earth radius up), g becomes g/4.
5.Orbital Velocity
$v_{orb} = \frac{GM}{r}$
$v_{o} r b$ : Orbital velocity (m/s) · $r$ : Orbital radius from Earth's centre (m)
Minimum tangential speed to keep a satellite in a circular orbit of radius r from Earth's centre.
6.Escape Velocity★
$v_{e} = \frac{2 GM}{R} = 2 g R$
$v_{e}$ : Escape velocity (m/s) · $R$ : Radius of Earth (m)
Minimum speed to escape Earth's gravity forever. For Earth: v_e ≈ 11.2 km/s. Note: v_e = √2 × v_orb (at same r).
7.Kepler's First Law (orbits)
$Each planet moves in an ellipse with the Sun at one focus$
Law of orbits. Orbits are ellipses, not circles.
8.Kepler's Second Law (equal areas)
$\frac{d A}{d t} = constant$
Line joining planet to Sun sweeps equal AREAS in equal times ⇒ planet moves faster when nearer the Sun. Consequence of conservation of angular momentum.
9.Kepler's Third Law (period-radius)★
$T^{2} \propto r^{3} i.e. \frac{T ^{2}}{r ^{3}} = \frac{4 π ^{2}}{GM}$
Square of orbital period is proportional to the cube of the mean orbital radius. Used by Newton to guess the inverse-square law.