---
# System prepended metadata

title: Gravity
tags: [mechanics, "classic\_"]

---

# Gravity
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$$F = \frac{GMm}{r^2}$$

$$\vec F = \frac{GMm}{r^2}\vec r$$
## Surface gravity
$$F = ma$$

$$a_g = g = \frac{GM}{r^2}$$
* Consider centrifugal force
$$F_{net} = 0 = -mg+F_c+N$$

$$F_c = mv^2/r = m𝜔^2r$$		

$$N = mg-F_c = mg-m𝜔^2r$$
if $N = 0$
$$a = g-𝜔^2r$$
## Gravitational potential energy
$$W = Fx$$

$$W_g = -GMm\int_{r}^{\infty}\frac{1}{r^2}$$

$$= (-\frac{GMm}{\infty})-(-\frac{GMm}{r})$$

$$= 0-(-\frac{GMm}{r})$$
:::success
$$W_g= \frac{GMm}{r}$$
:::
## Gravity inside the earth
$$D = \frac{M_e}{\frac{4}{3}\pi R^3}$$

$$M = D\frac{4}{3}\pi r^3 = M_e\frac{r^3}{R^3}$$
:::success
$$F =\frac{GMm}{r^2} = \frac{GM_em}{R^3}r$$
:::
## Escape velocity
$$E = K+U = \frac{1}{2}mv^2-(\frac{GMm}{r}) = 0$$
:::success
$$v = \sqrt{\frac{2GM}{R}}$$
:::
## Kepler's laws of planetary motion
1. Orbit law: 
the sun is at the focus of the elliptical orbit of planetary motion 
The distance between the two foci of the ellipse $= a×e$
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* a : semi-axis length
* e : Eccentricity

2. Area law:
Equal area swept per unit time
$\frac{dA}{dt}$ is a constant
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$$\frac{dA}{dt} = \frac{1}{2}r^2\frac{d\theta}{dt} = \frac{1}{2}r^2𝜔$$

3. cycle law:
$$L = rp = rmv = r^2m𝜔$$

$$\frac{dA}{dt} = \frac{1}{2}r^2𝜔$$

$$\frac{dA}{dt} = L/2m$$

$$F = ma$$

$$\frac{GMm}{r^2} = mv^2/r = m𝜔^2r$$

$$𝜔 = 2\pi/T$$

$$\frac{GMm}{r^2} = m\frac{4\pi^2}{T^2}r$$
:::success
$$T^2 = (\frac{4\pi^2}{GM})r^3$$
:::
so $\frac{T^2}{r^3}$ is comstant

## Orbital energy
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$$U = -\frac{GMm}{r}$$

$$\frac{GMm}{r^2} = mv^2/r$$

$$K = \frac{1}{2}mv^2 = \frac{GMm}{2r}$$

$$K = -U/2$$

$$E = K+U =  \frac{GMm}{2r}-\frac{GMm}{r}$$
:::success
$$E = -\frac{GMm}{2r} = -K$$
:::
* Circular orbit $E = -\frac{GMm}{2r}$
radius :　$r$
* Elliptical orbit $E = -\frac{GMm}{2\alpha}$
longest diameter of ellipse : $2\alpha

## Gravity & Space
![image](https://hackmd.io/_uploads/SkZmg-eNxx.png)

1. gravity environment
2. A space whose acceleration is equal to the acceleration due to gravity
3. The acceleration of an object is equal to the acceleration due to gravity

The above three situations are the same

Curved spacetime
![Spacetime_curvature](https://hackmd.io/_uploads/rk5xDB7Vkx.png)

Einstein ring (gravitational lensing)
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