physics-class-11-unit-08-chapter-06-keplers-law-centripital-forces-gallean-gravitational-law-l-6-7-yp0ydlndf0y

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Gravitational Laws </primary>
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<footer style = "font-size: 18px"> $\rightarrow$  $\rightarrow$ <span style = "color:blue"> Gravitational Laws </span> $\rightarrow$ Gravitational Laws $\rightarrow$ Gravitation </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Gravitational Laws </primary>
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- Ptolemy placed the Earth at the centre of his geocentric model.
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<footer style = "font-size: 18px"> $\rightarrow$ Gravitational Laws $\rightarrow$ <span style = "color:blue"> Gravitational Laws </span> $\rightarrow$ Gravitation $\rightarrow$ Freely Falling Bodies </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Gravitation </primary>
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- Important concept: Principle of Equivalence

- Gravitational Charge Mass = Inertial Mass

- Acceleration of a body in the earth's gravitational field is independent of its mass.

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<footer style = "font-size: 18px">Gravitational Laws $\rightarrow$ Gravitational Laws $\rightarrow$ <span style = "color:blue"> Gravitation </span> $\rightarrow$ Freely Falling Bodies $\rightarrow$ Planetary Motion </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Planetary Motion </primary>
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- <span style="color:green">Kepler's Laws </span>

- **The First Law**: Closed Elliptical Orbits.

- **The Second Law**: Equal Areas in Equal Durations.

- **The Third Law**: $\frac{T^2}{R^3}=$ Constant for all planets.

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<footer style = "font-size: 18px">Gravitation $\rightarrow$ Freely Falling Bodies $\rightarrow$ <span style = "color:blue"> Planetary Motion </span> $\rightarrow$ Ptolemaic Model $\rightarrow$ Heliocentric Model </footer>

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### <primary style="font-size:24px"> Ptolemaic Model </primary>
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- Ptolemy placed the Earth at the centre of his geocentric model.

- Saturn may be appearing to move in opposite direction to the earth.

- The apparent motion of a planet in a direction opposite to that of other bodies within its system: Retrograde motion.

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<footer style = "font-size: 18px">Freely Falling Bodies $\rightarrow$ Planetary Motion $\rightarrow$ <span style = "color:blue"> Ptolemaic Model </span> $\rightarrow$ Heliocentric Model $\rightarrow$ Kepler's First Law </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Kepler's Second Law </primary>
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- The angle subtended is the same in equal intervals of time.

- Equal areas are swept in equal intervals of time.

- <span style="color:blue">$A_1=A_2$</span>

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<footer style = "font-size: 18px">Heliocentric Model $\rightarrow$ Kepler's First Law $\rightarrow$ <span style = "color:blue"> Kepler's Second Law </span> $\rightarrow$ Third Law $\rightarrow$ Third Law </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Universal Patterns </primary>
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- **Excellent Observations**
	
- Universal patterns common to all planetary motions.

- How many universal patterns do we see?

- 3 patterns + 1 Results

- Move in elliptical orbits.

- Equal area in equal time.

- $\frac{T^2}{R^3}=$ constant.

- All these are independent of the mass of the planet.

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<footer style = "font-size: 18px">Third Law $\rightarrow$ Third Law $\rightarrow$ <span style = "color:blue"> Universal Patterns </span> $\rightarrow$ Force $\rightarrow$ Universality </footer>

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### <primary style="font-size:24px"> Force </primary>
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- Combine Kinematical Results with dynamics.

- In the first law, there is a force acting. If there were no force acting, all the planets would have been moving with straight line orbits.

- Newton's second law: $\frac{d \vec{P}}{d t}=\vec{F}$

- Third law of motion: $ \overrightarrow{F}_ { A \rightarrow B}=-\vec{F}_{B \rightarrow A}$

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<footer style = "font-size: 18px">Third Law $\rightarrow$ Universal Patterns $\rightarrow$ <span style = "color:blue"> Force </span> $\rightarrow$ Universality $\rightarrow$ Formulation </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Universality </primary>
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- Independent of the mass of the moving bodies (Heliocentric model).

- Marry kinematics to Dynamics.

- Simplification

- Circular Orbits

- Centripetal Force

- The Third Law

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<footer style = "font-size: 18px">Universal Patterns $\rightarrow$ Force $\rightarrow$ <span style = "color:blue"> Universality </span> $\rightarrow$ Formulation $\rightarrow$ Circular orbit </footer>

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### <primary style="font-size:24px"> Formulation </primary>
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- $A\left(M_A\right) \rightleftarrows B\left(M_B\right)$

- <span style="color:blue">Use the Third Law</span>

- $F_{A \leftrightarrow B} \propto M_A M_B$

- <span style="color:blue">Use Centripetal Concept</span>

- $\vec{F_{A \leftrightarrow B}} \propto M_A M_B f(r) \hat{r_{A B}}$

- $\vec{F_{A \rightarrow B}}=-\vec{F_{B \rightarrow A}}$

- $\vec{F_{B \rightarrow A}} \propto M_A  =\vec{M}_A \vec{a}_A$

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<footer style = "font-size: 18px">Force $\rightarrow$ Universality $\rightarrow$ <span style = "color:blue"> Formulation </span> $\rightarrow$ Circular orbit $\rightarrow$ Circular Orbit </footer>

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### <primary style="font-size:24px"> Circular orbit </primary>
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- Kinematics of the circular orbit: velocity is tangential and the force  is radial inward.

- The acceleration is always in the direction of the force.

- Centrifugal force is an inertial force. It is a pseudo force.

- Centrifugal force will be outward radial.

- Centripetal force is inward.

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<footer style = "font-size: 18px">Universality $\rightarrow$ Formulation $\rightarrow$ <span style = "color:blue"> Circular orbit </span> $\rightarrow$ Circular Orbit $\rightarrow$ Determination of f(r) </footer>

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### <primary style="font-size:24px"> Circular Orbit </primary>
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- The center of the orbit at the origin.

- $\vec{F} \propto -\vec{r}$.

- Acceleration $\vec{a} \propto -\vec{r}$.

- $\vec{F}=-G M_A M_B \vec{r} f(r)$

- $f(r) > 0$

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<footer style = "font-size: 18px">Formulation $\rightarrow$ Circular orbit $\rightarrow$ <span style = "color:blue"> Circular Orbit </span> $\rightarrow$ Determination of f(r) $\rightarrow$ Determination of The Third Law </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
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- Let $f(r)=r^n$

- <span style="color:blue">Second Law of Kepler automatically satisfied</span>.

- $\frac{d}{d t}(\vec{r} \times \vec{p})=\vec{r} \times \vec{F}$.

- $m \omega^2 r=K m f(r) $

- $T=\frac{2 \pi}{\omega}=\frac{2 \pi \sqrt{r}}{\sqrt{K f(r)}}$

- Kepler's Third Law $\Longrightarrow$ $f(r)=\frac{1}{r^2}$

- Determination of f(r): $ f(r) \propto r^n $

- $F =m \omega^2 r= mMG f(r)$

- $\omega^2 r=K f(r)$

- $\omega =\frac{2 \pi}{T} \Rightarrow \omega^2=\frac{(2 \pi)^2}{T^2}$

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<footer style = "font-size: 18px">Circular orbit $\rightarrow$ Circular Orbit $\rightarrow$ <span style = "color:blue"> Determination of f(r) </span> $\rightarrow$ Determination of The Third Law $\rightarrow$ Determination of The Third Law </footer>

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### <primary style="font-size:24px"> Determination of The Third Law </primary>
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- $\omega^2 r=k f(r)$

- $(2 \pi)^2 \frac{r}{T^2}=k f(r)$

- $r^2 \times \frac{r}{T^2}=K^{\prime} f(r) \times r^2$ = constant

- $\frac{r^3}{T^2} \Rightarrow f(r) r^2$ = Constant

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<footer style = "font-size: 18px">Circular Orbit $\rightarrow$ Determination of f(r) $\rightarrow$ <span style = "color:blue"> Determination of The Third Law </span> $\rightarrow$ Determination of The Third Law $\rightarrow$ Determination of The Third Law </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Determination of The Third Law </primary>
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- $f(r) \propto \frac{1}{r^2}  \Rightarrow$

- Gravitational Force $F_G=\frac{G M_A M_B}{r^2}$
	
- There is a constant of proportionality which can be absorbed in G.

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<footer style = "font-size: 18px">Determination of f(r) $\rightarrow$ Determination of The Third Law $\rightarrow$ <span style = "color:blue"> Determination of The Third Law </span> $\rightarrow$ Determination of The Third Law $\rightarrow$ Law of Gravitation </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Determination of The Third Law </primary>
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- $\vec{F} _ {A \rightarrow B}=-\frac{G M_A M_B}{r_{A B}^2} \hat{r} _{A B}$
	
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<footer style = "font-size: 18px">Determination of The Third Law $\rightarrow$ Determination of The Third Law $\rightarrow$ <span style = "color:blue"> Determination of The Third Law </span> $\rightarrow$ Law of Gravitation $\rightarrow$ Variation with Height </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Law of Gravitation </primary>
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- $A: \quad M_A ; \quad \vec{R}_A $

- $B: \quad M_B ; \quad \vec{R}_B $

- $\vec{R}  _ {A, B}=\vec{R} _ A-\vec{R} _ B$

- $\vec{F}  _ {A \rightarrow B}$ = $-\frac{G M_A M_B}{R_{A B}^3} \vec{R} _{A B}$

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<footer style = "font-size: 18px">Determination of The Third Law $\rightarrow$ Determination of The Third Law $\rightarrow$ <span style = "color:blue"> Law of Gravitation </span> $\rightarrow$ Variation with Height $\rightarrow$ Uniform mass density </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Variation with Height </primary>
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- Let a body be at a height $h$ above the Earth's surface.

- $\frac{1}{(R+h)^2}=\frac{1}{R^2\left(1+2 h / R+h^2 / R^2\right)}$

- Employ binomial expansion.

- $F=\frac{G M m}{R^2}\left$1-\frac{2 h}{R}\right$$

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<footer style = "font-size: 18px">Determination of The Third Law $\rightarrow$ Law of Gravitation $\rightarrow$ <span style = "color:blue"> Variation with Height </span> $\rightarrow$ Uniform mass density $\rightarrow$ Falling body </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Uniform mass density </primary>
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- <span style="color:green">Uniform Mass Density</span>

- All the mass is concentrated in the circle at the center of the sphere.

- $F=\frac{G M m}{r^2}$

- $M=\rho \int d V=\frac{4}{3} \pi R^3 \rho$

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<footer style = "font-size: 18px">Law of Gravitation $\rightarrow$ Variation with Height $\rightarrow$ <span style = "color:blue"> Uniform mass density </span> $\rightarrow$ Falling body $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Keplers Law Centripital Forces Gallean Gravitational Law L-6 </Success>
### <primary style="font-size:24px"> Falling body </primary>
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- <span style="color:green">Falling body</span>

- r is the radius of the earth.

- h is the height above the center.

- $a=+\frac{G M_E}{(R+h)^2} ; \quad {h \ll R}$

- Zeroth order: Take $h \approx 0$

- $\frac{h}{R} \approx 0$

- $a=\frac{G M}{R^2}=g $

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<footer style = "font-size: 18px">Variation with Height $\rightarrow$ Uniform mass density $\rightarrow$ <span style = "color:blue"> Falling body </span> $\rightarrow$ Thank You $\rightarrow$  </footer>