Gravitation Formulas
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Gravitation Formulae Sheet
1. Newton’s law of gravitation
F = \(\frac{\mathrm{Gm}_{1} \mathrm{m}_{2}}{\mathrm{r}^{2}}\)
G = 6.67 × 10-11 Nm2/kg2
\(\overrightarrow{\mathrm{F}}_{12}=-\overrightarrow{\mathrm{F}}_{21}\)
Always attraction in nature
2. Acceleration due to gravity
g = \(\frac{G M}{R^{2}}=\frac{4}{3}\)π GRρ
If M = const; g ∝ \(\frac{1}{R^{2}}\)
If ρ = const; g ∝ R
ρ → density of earth; M → mass of earth ; R → Radius of earth
3. Variation of g
(a) Altitude (height) effect g’ = g \(\left(1+\frac{h}{R}\right)^{-2}\)
if h << R then g’ = g \(\left(1-\frac{2 h}{R}\right)\)
(b) effect of depth g” = g \(\left(1-\frac{\mathrm{d}}{\mathrm{R}}\right)\)
4. Intensity of gravitational field
\(\overrightarrow{\mathrm{E}}_{g}=\frac{\mathrm{GM}}{\mathrm{r}^{2}}(-\hat{\mathrm{r}})\)
for earth
Eg = g = 9.86 m/s2
| Solid | Hollow |
| 1. r > R, Eg = –\( \frac{\mathrm{GM}}{\mathrm{r}^{2}}\) | Eg = –\( \frac{\mathrm{GM}}{\mathrm{r}^{2}}\) |
| 2. r = R, Eg = –\( \frac{\mathrm{GM}}{\mathrm{R}^{2}}\) | Eg = –\( \frac{\mathrm{GM}}{\mathrm{R}^{2}}\) |
| 3. r < R, Eg = –\( \frac{\mathrm{GMr}}{\mathrm{R}^{3}} \) | Eg = 0 |
5. Gravitational potential due to solid sphere /hollow sphere
Vg = –\(\int_{\infty}^{r} \vec{E}_{g} \cdot \vec{d} r\)
For points out side (r > R)
Vg = –\(\frac{\mathrm{GM}}{\mathrm{r}}\), Vg = –\(\frac{\mathrm{GM}}{\mathrm{r}}\)
For points on the surface (r = R)
Vg = –\(\frac{\mathrm{GM}}{\mathrm{R}}\), Vg = –\(\frac{\mathrm{GM}}{\mathrm{R}}\)
For points inside it r < R
Vg = -GM\(\left[\frac{3 \mathrm{R}^{2}-\mathrm{r}^{2}}{2 \mathrm{R}^{3}}\right]\), Vg = –\(\frac{\mathrm{GM}}{\mathrm{R}}\)
6. Gravitational P.E.
Ug = mVg
Change in P.E. on Going height h above the surface
ΔUg = mgh if h << Re In general ΔUg = \(\frac{m g h}{\left(1+\frac{h}{R}\right)}\)
7. Orbital velocity of a satellite
\(\frac{m v_{o}^{2}}{r}=\frac{G M m}{r^{2}}\)
v0 = \(\sqrt{\frac{\mathrm{GM}}{\mathrm{R}+\mathrm{h}}}\) (r = h + R)
if h << R
v0 = \(\sqrt{\frac{\mathrm{GM}}{\mathrm{R}}}=\sqrt{\mathrm{gR}}\) ≃ 8Km/sec.
8. Velocity of projection
loss of K.E. = gain in P.E.
\(\frac{1}{2} \mathrm{mv}_{\mathrm{p}}^{2}=-\frac{\mathrm{GMm}}{(\mathrm{R}+\mathrm{h})}-\left(-\frac{\mathrm{GMm}}{\mathrm{R}}\right)\)
vp = \(\left[\frac{2 \mathrm{GMh}}{\mathrm{R}(\mathrm{R}+\mathrm{h})}\right]^{1 / 2}=\left[\frac{2 \mathrm{gh}}{1+\frac{\mathrm{h}}{\mathrm{R}}}\right]^{1 / 2}\) (∵ GM = gR2)
9. Period of revolution
T = \(\frac{2 \pi r}{v_{0}}=\frac{2 \pi(R+h)^{3 / 2}}{R \sqrt{g}}\)
or T2 = \(\frac{4 \pi^{2} r^{3}}{G M}\)
If h << R
T = \(\frac{2 \pi R^{3 / 2}}{R \sqrt{g}} \simeq 1 \frac{1}{2}\) HR.
10. K.E. of satellite
K.E. = \(\frac{G M m}{2 r}=\frac{1}{2}\)mv02
11. P.E. of satellite
U = –\(\frac{\mathrm{GMm}}{\mathrm{r}}\)
12. Binding energy of satellite
BE = \(\frac{1}{2} \frac{\mathrm{GMm}}{\mathrm{r}}\)
13. Escape velocity
ve = \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}}=\sqrt{2 \mathrm{g} \mathrm{R}}=\mathrm{R} \sqrt{\frac{8 \pi \mathrm{Gd}}{3}}\)
ve = v0 \(\sqrt{2}\)
14. Effective weight in a satellite
W = 0
satellite behaves like a free fall body
15. Kepler’s laws for planetary motion
- Elliptical orbit with sun at one focus
- Areal velocity constant dA/dt = constant = \(\frac{L}{2 m}\)
- T2 ∝ r3, r = (r1 + r2)/2
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