physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-and-reduced-mass-l-2-10-0laqr-jjjxy

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- $(1,1) \equiv 1 \hat{i} + 1 \hat{j}=\hat{i}+\hat{j}$

- $\equiv 1 \hat{e}_x + 1 \hat{e}_y = \hat{e}_x + \hat{e}_y$

- $\vec{R}_{cm}=(0,0)$

- $\vec{R}_{cm}=\frac{2(1/2,1/2)+1(1/2,-1/2)}{6}$

- $\frac{+2(-1/2,-1/2)+1(-1/2,1/2)}{6}$

- $=(6,6)$

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<footer style = "font-size: 18px"> $\rightarrow$ Motion of Centre of Mass Relative Motion and Reduced Mass $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Center of Mass $\rightarrow$ Center of Mass </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Center of Mass </primary>
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- $m_1 + m_2 +...+m_n=M$

- $\vec{R} \equiv \vec{R}_{cm}$

- $= \frac{\sum_{i=1}^{n} m_i \vec r_{i}} {\sum_{i=1}^{n} m_i}$

- $=\frac{\sum_{i=1}^{n} m_i \vec{r}_i}{M}$

- $M \vec{R}= m_1 \vec{r}_1 + m_2 \vec{r}_2 + .. + m_n \vec{r}_n$

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<footer style = "font-size: 18px">Motion of Centre of Mass Relative Motion and Reduced Mass $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Center of Mass </span> $\rightarrow$ Center of Mass $\rightarrow$ Velocity of Center of Mass </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Center of Mass </primary>
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- <span style="color:green">Differentiate w.r.t 't' on both sides</span>

- $M \frac{d \vec{R}}{dt}=m_1 \frac{d \vec{r}_1}{dt}+m_2 \frac{d \vec{r}_2}{dt}+..+ m_n \frac{d \vec{r}_n}{dt}$

- $M \vec{v}= m_1 \vec{v}_1 + m_2 \vec{v}_2 + .. + m_n \vec{v}_n$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-3.jpg)

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<footer style = "font-size: 18px">Problem $\rightarrow$ Center of Mass $\rightarrow$ <span style = "color:blue"> Center of Mass </span> $\rightarrow$ Velocity of Center of Mass $\rightarrow$ Force of Center of Mass </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Velocity of Center of Mass </primary>
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- $\vec{V}=\vec{V}_{cm}=$ velocity of centre of mass

- $M \vec{A}=m_1 \vec{a}_1 + m_2 \vec{a}_2 + .. + m_n \vec{a}_n$

- $\vec{A}=\frac{d \vec{v}}{dt}; \vec{a}_i = \frac{d \vec{v}_i}{dt}$

- $M \vec{A} = \vec{F}_1 + \vec{F}_2 + .. + \vec{F}_n$

- $= \vec{F}_{ext}$

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<footer style = "font-size: 18px">Center of Mass $\rightarrow$ Center of Mass $\rightarrow$ <span style = "color:blue"> Velocity of Center of Mass </span> $\rightarrow$ Force of Center of Mass $\rightarrow$ Force of Center of Mass </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Force of Center of Mass </primary>
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- $M \vec{A}=m_1 \vec{a}_1 + m_2 \vec{a}_2 + .. + m_n \vec{a}_n$

- $\vec{A}=\frac{d \vec{v}}{dt}; \vec{a}_i = \frac{d \vec{v}_i}{dt}$

- $M \vec{A} = \vec{F}_1 + \vec{F}_2 + .. + \vec{F}_n$

- $= \vec{F}_{ext}$

- $M \vec{A}= \vec{F}_{ext}$ Covering equation

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-6.jpg)

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<footer style = "font-size: 18px">Velocity of Center of Mass $\rightarrow$ Force of Center of Mass $\rightarrow$ <span style = "color:blue"> Force of Center of Mass </span> $\rightarrow$ Linear Momentum of a System of Particles $\rightarrow$ Linear Momentum of a System of Particles </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Linear Momentum of a System of Particles </primary>
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- Forces

- 1) External

- 2) Internal

- Linear momentum of a system of particles

- $\vec{p}=m \vec{v}, \frac{d \vec{p}}{dt}=\vec{F}$

- $\vec{P} = \vec{p}_1 + \vec{p}_2 + .. + \vec{p}_n$

- $M \vec{V} = \frac{d \vec{P}}{dt}= \vec{F}_{ext}$

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<footer style = "font-size: 18px">Force of Center of Mass $\rightarrow$ Force of Center of Mass $\rightarrow$ <span style = "color:blue"> Linear Momentum of a System of Particles </span> $\rightarrow$ Linear Momentum of a System of Particles $\rightarrow$ Linear Momentum of a System of Particles </footer>

- Suppose, $\vec{F}_{ext}=0$

- $\frac{d \vec{P}}{dt} = 0 = \vec{P} = \vec{c}$, a constant vector

- $\Rightarrow$ The linear momentum of the system of particle remains the same.

- $\Rightarrow \vec{P}$ is a constant of motion.

- $\vec{P}$ is constant.

- i.e., the $\vec{v}_{CM}$ remains constant.

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-9.jpg)

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<footer style = "font-size: 18px">Force of Center of Mass $\rightarrow$ Linear Momentum of a System of Particles $\rightarrow$ <span style = "color:blue"> Linear Momentum of a System of Particles </span> $\rightarrow$ Linear Momentum of a System of Particles $\rightarrow$ Kinetic Energy of a System </footer>

- $\vec{F}_{ext}=0$

- Motion of various parts of a system is separated (or spit) into:

- (1) motion of CM

- (2) motion about the CM

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<footer style = "font-size: 18px">Linear Momentum of a System of Particles $\rightarrow$ Linear Momentum of a System of Particles $\rightarrow$ <span style = "color:blue"> Linear Momentum of a System of Particles </span> $\rightarrow$ Kinetic Energy of a System $\rightarrow$ Kinetic Energy of a System </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Kinetic Energy of a System </primary>
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- Question: What about the total KE of a two particle system relative to CM?

- $v_{CM}=\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2}$

- Velocity of $m_1$ w r t CM, $\vec{v}_{1,CM}$

- $=\vec v _ {1} - \vec v_{CM}$

- $=\vec{v}_1 - (\frac{m_1 v_1 + m_2 v_2}{m_1 + m_2})$

- $=\frac{m_1 \vec{v}_1 - m_1 \vec{v}_1 + m_2 \vec{v}_1 - m_2 \vec{v}_2}{m_1 + m+2}$

- $\vec{v}_{1,CM}= \frac{m_2(\vec{v}_1 - \vec{v}_2)}{m_1 + m_2}$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-11.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-12.jpg)

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<footer style = "font-size: 18px">Linear Momentum of a System of Particles $\rightarrow$ Linear Momentum of a System of Particles $\rightarrow$ <span style = "color:blue"> Kinetic Energy of a System </span> $\rightarrow$ Kinetic Energy of a System $\rightarrow$ Kinetic Energy of a System </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Kinetic Energy of a System </primary>
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- $\vec{v}  _ {2,CM} = \vec{v}  _ 2 - \vec{v}_{CM}$

- $=\vec{v}_2 -(\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 +m_2})$

- $\vec{v}_{2,CM} = \frac{m_1(\vec{v}_2 - \vec{v}_1)}{m_1 + m_2}$

- $ KE| _ {m _ 1,CM} = \frac{1}{2} m _ 1 (\frac{m _ 2 \vec{v} _ {1 _ 2}}{m _ 1 + m_2})^2$

- $ = \frac{1}{2} m_1 \frac{m_2^2 |v_{1_2}|^2}{(m_1 + m_2)^2}$

- $ KE| _ {m _ 2,CM} = \frac{1}{2} m _ 2 (\frac{m _ 1 \vec{v} _ {2 _ 1}}{m _ 1 + m_2})^2$

- $ = \frac{1}{2} m_2 \frac{m_1^2 |v_{1_2}|^2}{(m_1 + m_2)^2}$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-13.jpg)

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<footer style = "font-size: 18px">Linear Momentum of a System of Particles $\rightarrow$ Kinetic Energy of a System $\rightarrow$ <span style = "color:blue"> Kinetic Energy of a System </span> $\rightarrow$ Kinetic Energy of a System $\rightarrow$ Kinetic Energy of a System </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
### <primary style="font-size:24px"> Kinetic Energy of a System </primary>
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- $ KE|_{system,CM} $

- $= \frac{1}{2} \frac{m_1 m_2 |v_{1,_2}|^2(m_2 + m_1)}{(m_1 + m_2)^2}$

- $KE |_{system,CM}$

- $=\frac{1}{2} (\frac{m_1 m_2}{m_1 + m_2}) |v_{1,2}|^2$

- Reduced mass

- $\mu_0 = \frac{m_1 m_2}{m_1 + m_2}$

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![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-07-chapter-02-motion-of-center-of-mass-relative-motion-reduced-mass-laqr-jjjxy-15.jpg)

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<footer style = "font-size: 18px">Kinetic Energy of a System $\rightarrow$ Kinetic Energy of a System $\rightarrow$ <span style = "color:blue"> Kinetic Energy of a System </span> $\rightarrow$ Kinetic Energy of a System $\rightarrow$ Kinetic Energy of a System </footer>

### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
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- $KE| _ {Total} = \frac{1}{2} \mu|\vec{v} _ {rel}|^2 + \frac{M}{2} v_{CM}^2$

- $KE|=\frac{1}{2} \mu |\vec{v}_{rel}|^2 + \frac{1}{2} (m_1 + m_2)(\frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{m_1 + m_2})^2$

- RHS $=\frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (\vec{v}_1 - \vec{v}_2)^2$

- $ = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (\vec{v}_1^2 + \vec{v}_2^2 - 2 \vec{v}_1 . \vec{v}_2)$

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### <Success style="font-size:25px"> Motion Of Center Of Mass Relative Motion And Reduced Mass L-2 </Success>
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- = $ + \frac{1}{2} \frac{(m_1 + m_2)}{(m_1 + m_2)^2} (m_1^2 v_1^2 + m_2^2 v_2^2 + 2m_1 m_2 \vec{v}_1 . \vec{v}_2) + \frac{1}{2} \frac{1}{(m_1 + m_2)} (m_1^2 v_1^2 + m_2^2 v_2^2 + 2 m_1 m_2 \vec{v}_1 . \vec{v}_2)$

- $=  \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$

- $KE|_{Total} = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2$

- $\mu = M = \frac{m_1 m_2}{m_1 + m_2}$

- $\frac{1}{\mu}=\frac{1}{m_1} + \frac{1}{m_2}$

- $\mu \leq m_1, \mu \leq m_2$

- Reduced mass is always less or equal to mass of each body.

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