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07Class 11 Physics

Gravitation formulas

Gravitational force, field, potential and orbit formulas. Keep track of whether you're above or below the surface, since the rules change.

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All Physics formulas

Force and acceleration

Newton's law of gravitation
F=Gm1m2r2\displaystyle F = \frac{Gm_1m_2}{r^2}
Acceleration due to gravity
g=GMR2\displaystyle g = \frac{GM}{R^2}
At height h
gh=g(1+h/R)2≈g(1−2hR)\displaystyle g_h = \frac{g}{(1 + h/R)^2} \approx g\left(1 - \frac{2h}{R}\right)

The approximation needs h much smaller than R.

At depth d
gd=g(1−dR)\displaystyle g_d = g\left(1 - \frac{d}{R}\right)

Energy and escape

Gravitational potential energy
U=−GMmr\displaystyle U = -\frac{GMm}{r}
Gravitational potential
V=−GMr\displaystyle V = -\frac{GM}{r}
Escape speed
ve=2GMR=2gR\displaystyle v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}

Orbits

Orbital speed
vo=GMr\displaystyle v_o = \sqrt{\frac{GM}{r}}

Close to the surface, vₑ = √2 × vₒ.

Time period
T=2πr3GM\displaystyle T = 2\pi\sqrt{\frac{r^3}{GM}}
Kepler's third law
T2∝r3\displaystyle T^2 \propto r^3
Energy of a satellite
E=−GMm2r,K=GMm2r,U=−GMmr\displaystyle E = -\frac{GMm}{2r}, \quad K = \frac{GMm}{2r}, \quad U = -\frac{GMm}{r}

Where students slip up

  • Using R when r = R + h is needed for satellites. r is measured from Earth's centre.
  • Dropping the minus sign on potential energy. Bound orbits have negative total energy.
  • Using the g(1 − 2h/R) approximation when h is comparable to R.

Try it on your weakest chapter.

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