physics-class-11-unit-06-chapter-06-potential-and-potential-energy-lecture-6-6-mq0p4bp-2e4

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Potential and Potential energy </primary>
<hr>
<main>

- <span style="color:green">Potential energy $V=\frac{1}{2} k x^2$</span>

- 'x': displacement from the equilibrium position.

- <span style="color:green">Kinetic energy $T=\frac{1}{2} m v^2$.</span>

- E=T+V = constant in time

- $\frac{d E}{d k}=0 \Rightarrow$ ma= - kx

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-06-chapter-06-potential-and-potential-energy-lecture-6_6-mq0p4bp_2e4-068-0795.6.jpg)

</main>
<hr>
<footer style = "font-size: 18px">Potential and Potential energy $\rightarrow$ Potential and Potential energy $\rightarrow$ <span style = "color:blue"> Potential and Potential energy </span> $\rightarrow$ Potential and Potential energy $\rightarrow$ Hooke's Law </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Potential and Potential energy </primary>
<hr>
<main>

- Potential energy increases when the ball rises and decreases when it
- falls.

- Obtain a general form for potential energy.

</div>

</div>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Hooke's Law </primary>
<hr>
<main>

- $F= - kx$
 
- $V=-\int F \frac{d x}{d t} d t\quad =\frac{1}{2} k x^2$

- $F=-\frac{G M m}{r^2}$ : <span style="color:green">Integrate along the line direction of the 
- force.</span>

- **Verify that:**

* $V=-\frac{G M m}{r}$, <span style="color:green">where:  M>>m</span>

- Is a good candidate for potential energy.

* $F=-\frac{d v}{d r} \hat{r}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Potential and Potential energy $\rightarrow$ Potential and Potential energy $\rightarrow$ <span style = "color:blue"> Hooke's Law </span> $\rightarrow$ Hooke's Law $\rightarrow$ Hooke's Law </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Hooke's Law </primary>
<hr>
<main>

- Look at the limit r  $\rightarrow \infty$

- $E=\frac{1}{2} m \vec{v}^2-\frac{G M m}{r}+V_0$

- At $r=\infty$ body at rest has zero energy.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Potential and Potential energy $\rightarrow$ Hooke's Law $\rightarrow$ <span style = "color:blue"> Hooke's Law </span> $\rightarrow$ Hooke's Law $\rightarrow$ Collection of Masses   N bodies </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Hooke's Law </primary>
<hr>
<main>

* $E=\frac{1}{2} m \vec{v}^2-\frac{G M m}{r}$

- **Verify that:**

* $\frac{d E}{d t}=0 \Rightarrow$  <span style="color:blue">Newtonian law of gravitation</span>.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Hooke's Law $\rightarrow$ Hooke's Law $\rightarrow$ <span style = "color:blue"> Hooke's Law </span> $\rightarrow$ Collection of Masses   N bodies $\rightarrow$ Collection of Masses   N bodies </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Collection of Masses   N bodies </primary>
<hr>
<main>

- 3N equation of Motion

- Now each of these bodies interact with every other body through the gravitational force.

- $M_1 \frac{d \vec{v}_1}{dt}$ = force = (N-1) terms.

</div>

</main>
<hr>
<footer style = "font-size: 18px">Hooke's Law $\rightarrow$ Hooke's Law $\rightarrow$ <span style = "color:blue"> Collection of Masses   N bodies </span> $\rightarrow$ Collection of Masses   N bodies $\rightarrow$ Collection of Masses   N bodies </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Collection of Masses   N bodies </primary>
<hr>
<main>

- Distance between $M_i$ and $M_j$ <span style="color:green">(i $\neq$ j)</span> is denoted by $R_{ij}$.
</div>

</main>
<hr>
<footer style = "font-size: 18px">Hooke's Law $\rightarrow$ Collection of Masses   N bodies $\rightarrow$ <span style = "color:blue"> Collection of Masses   N bodies </span> $\rightarrow$ Collection of Masses   N bodies $\rightarrow$ Conservation of energy </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Collection of Masses   N bodies </primary>
<hr>
<main>

- If there are N bodies of masses

- **$M_i$ .... , $M_n$, Then**

- $E_{T}=\frac{1}{2} \sum_{i=1}^N M_i \vec{v}_i^2(t)$

- $-\frac{1}{2} \sum_{i \neq j} \frac{G M_i M_j}{r_{i j}(t)}$

- <span style="color:blue">$E_{T}$ independent of time</span>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Collection of Masses   N bodies $\rightarrow$ Collection of Masses   N bodies $\rightarrow$ <span style = "color:blue"> Collection of Masses   N bodies </span> $\rightarrow$ Conservation of energy $\rightarrow$ Analysis </footer>

### <Success style="font-size:25px"> Potential And Potential Energy L-6 </Success>
### <primary style="font-size:24px"> Conservation of energy </primary>
<hr>
<main>

* Law of Conservation of energy is a good book keeper.

- <span style="color:green">Three body problem</span>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Collection of Masses   N bodies $\rightarrow$ Collection of Masses   N bodies $\rightarrow$ <span style = "color:blue"> Conservation of energy </span> $\rightarrow$ Analysis $\rightarrow$ Escape velocity </footer>