physics-class-11-unit-08-chapter-01-tidal-forces-energy-conservation-lecture-1-7-fwawgkpepx4

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Planetary Laws of Motion - Kepler's Law </primary>
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- Kepler shifted from the earth centred frame to the heliocentric frame.

- There he got an excellent fitting with elliptic orbits.
 
- Able to enumerate the three laws.

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<footer style = "font-size: 18px">TIDAL FORCES AND ENERGY CONSERVATION $\rightarrow$ Recap $\rightarrow$ <span style = "color:blue"> Planetary Laws of Motion - Kepler's Law </span> $\rightarrow$ Galilean law of falling Bodies $\rightarrow$ Newton's Formulation of Law of Gravitation </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Newton's Formulation of Law of Gravitation </primary>
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- Galilean Law of freely falling bodies
  
- Orbit of the moon around the Earth
  
- Orbits of planets around the sun.

- Leads to the formulation of <span style="color:green">universal law of gravitation</span>. 
 
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<footer style = "font-size: 18px">Planetary Laws of Motion - Kepler's Law $\rightarrow$ Galilean law of falling Bodies $\rightarrow$ <span style = "color:blue"> Newton's Formulation of Law of Gravitation </span> $\rightarrow$ Cavendish Experiment $\rightarrow$ Phenomena of Tides </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Cavendish Experiment </primary>
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- Determination of the gravitational constant.

- $\vec{F}_{1 \rightarrow 2}=-\frac{G m_1 m_2}{r^2}\hat{r}$
 
 - $\vec{F_{2 \rightarrow 1}}=-\vec{F_{1 \rightarrow 2}}$

- g is related to the mass of the earth through the acceleration due to gravity given by the Galilean law falling body.

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<footer style = "font-size: 18px">Galilean law of falling Bodies $\rightarrow$ Newton's Formulation of Law of Gravitation $\rightarrow$ <span style = "color:blue"> Cavendish Experiment </span> $\rightarrow$ Phenomena of Tides $\rightarrow$ Earth-Moon System </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Phenomena of Tides </primary>
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- The height of the water level rises or falls shows a regular pattern depending on the phase of the moon.

- Radius of earth = $R_e$

- Distance between the moon and the earth = $d_m$

- Distance between the moon and the earth = $d_s$

- For any point on the surface of the Earth the true distance varies between $\left(d_{s, m} \pm R_m\right)$.
 
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<footer style = "font-size: 18px">Newton's Formulation of Law of Gravitation $\rightarrow$ Cavendish Experiment $\rightarrow$ <span style = "color:blue"> Phenomena of Tides </span> $\rightarrow$ Earth-Moon System $\rightarrow$ Earth-Moon System </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Earth-Moon System </primary>
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- Distance between the moon and the earth = $d_m$

- Radius of earth = $R_e$

- Mass of earth = $M_e$

- Mass of moon = $M_m$

- The center of mass is in the rest frame of the earth.

- $F_1=\frac{G M_e M_m}{\left(d_m-R_E\right)^2}$

- $F_1^{\prime}=\frac{G M_e M_m}{\left(d_m+R_E\right)^2}$

- $\Delta F_1=F_1-F^{\prime}_1$ and $d \gg R_E$

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<footer style = "font-size: 18px">Phenomena of Tides $\rightarrow$ Earth-Moon System $\rightarrow$ <span style = "color:blue"> Earth-Moon System </span> $\rightarrow$ Earth-Moon System $\rightarrow$ Earth-Moon System </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Earth-Moon System </primary>
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- $F_1=\frac{K}{\left(d_m-R_E\right)^2}=\frac{K}{d_m^2\left[1-\frac{R_1}{d_m}\right]^2}$
 
 - $F_1=F_0\left[\frac{1}{1-\left(\frac{2 R_1}{d m}-\frac{R_1^2}{d m^2}\right)}\right]\equiv F_0 \frac{1}{1-x}$
 
- where, $K=G M_{\mathrm{e}} M_m$
 
- For calculations we make a binomial expansion.
 
 - $\frac{R_{\mathrm{E}}}{d_m} \ll 1$

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<footer style = "font-size: 18px">Earth-Moon System $\rightarrow$ Earth-Moon System $\rightarrow$ <span style = "color:blue"> Earth-Moon System </span> $\rightarrow$ Earth-Moon System $\rightarrow$ Earth-Moon System </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Earth-Moon System </primary>
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- Binomial Expansion:

- $\frac{1}{1-x}=1+x+{x}^2+O\left(x^3\right)$
 
 - $x=\left(\frac{2 R_E}{d m}-\frac{R_E^2}{d m^2}\right)$, $\frac{R_E}{d m}={r}$
 
 - $x =\left(2 r-r^2\right)$
 
 - $\frac{1}{1-x} = 1+\left(2 r-r^2\right)+\left(2 r-r^2\right)^2+\cdots \cdot$
 
 - $=1+\left(2 r-r^2\right)+4 r^2+O\left(r^3\right)$
 
 - $=1+2 r+3 r^2$
 
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### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Earth-Moon System </primary>
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- $F_1=F_0\left[1+2 r+3 r^2\right]$
 
- $F'_1=F_0\left[1-2 r+3 r^2\right]$
 
- $\Delta F_1=4 F_0\left(\frac{R_E}{d_m}\right)$
 
- $r$ is a dimension quantity
 
- $\Delta F_2 \rightarrow$ Force on the Earth due to the sun at diametrically opposite points.

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<footer style = "font-size: 18px">Earth-Moon System $\rightarrow$ Earth-Moon System $\rightarrow$ <span style = "color:blue"> Earth-Moon System </span> $\rightarrow$ Earth-Sun System $\rightarrow$ Earth-Sun and Moon System </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Earth-Sun System </primary>
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- $\Delta F_2=4 F_0^{\prime}\left(\frac{R_E}{d_s}\right)$.
 
 - $\Delta F_1=\frac{4 G M_e M_m}{d_m^2} \frac{R_E}{d_m}$
 
 - $\Delta F_2=\frac{4 G M_c M_s}{d_s^2} \frac{R_E}{d_s}$
 
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<footer style = "font-size: 18px">Earth-Moon System $\rightarrow$ Earth-Moon System $\rightarrow$ <span style = "color:blue"> Earth-Sun System </span> $\rightarrow$ Earth-Sun and Moon System $\rightarrow$ Period of Tides </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Earth-Sun and Moon System </primary>
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- We calaulate the ration:

- $\frac{\Delta F_1}{\Delta F_2}=\frac{M_m}{M_s}\left(\frac{d_s}{d_m}\right)^3$ $\approx3.5$
 
 - $M_m=7.3 \times 10^{22} \mathrm{~kg}$
 
 - $M_m=2 \times 10^{30} \mathrm{~kg}$
 
 - $d_S=150 \times 10^{6} \mathrm{~kg}$
 
 - $d_m=0.3 \times 10^{6} \mathrm{~kg}$
 
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<footer style = "font-size: 18px">Earth-Moon System $\rightarrow$ Earth-Sun System $\rightarrow$ <span style = "color:blue"> Earth-Sun and Moon System </span> $\rightarrow$ Period of Tides $\rightarrow$ New Moon Phase </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Gravitational Potential Energy </primary>
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- There is a spring and a mass and this spring is compressed.

- As soon as the stop is removed, the block moves.

- It acquires energy.

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<footer style = "font-size: 18px">Period of Tides $\rightarrow$ New Moon Phase $\rightarrow$ <span style = "color:blue"> Gravitational Potential Energy </span> $\rightarrow$ Potential Energy $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Tidal Forces Energy Conservation L-1 </Success>
### <primary style="font-size:24px"> Potential Energy </primary>
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- From Hooke's law: $F=-k x$
 
- Stored Energy: $\frac{1}{2} k x^2$ is potential energy.

- If this block is oscillating about its equilibrium position,

- $E_T=\frac{1}{2} m v^2+\frac{1}{2} k x^2$ = constants
 
- $\frac{d E}{d t}=0 \Rightarrow mv\frac{dv}{d t} +kxv=0$

- Newton says in his law of gravitation that all forces behave in the same fashion.
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<footer style = "font-size: 18px">New Moon Phase $\rightarrow$ Gravitational Potential Energy $\rightarrow$ <span style = "color:blue"> Potential Energy </span> $\rightarrow$ Thank You $\rightarrow$  </footer>