physics-class-11-unit-07-chapter-01-center-of-mass-system-of-particles-and-rotational-motion-l1-10-4vpgfmdfo6a

### <Success style="font-size:25px"> Center Of Mass System Of Particles And Rotational Motion L-1 </Success>
### <primary style="font-size:24px"> Systems of particles and rotational motion </primary>
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- **<ins>Motivation</ins>**

- The motion of point particles.

- If one has extended objects like a football, in kinematics, football is  represented by a point particle.

-  Ideal rigid body is one which does not deform, and shape does not change.
 
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<footer style = "font-size: 18px"> $\rightarrow$ Introduction, Center of Mass System of Particles and Rotational Motion $\rightarrow$ <span style = "color:blue"> Systems of particles and rotational motion </span> $\rightarrow$ Two Kinds of Motions $\rightarrow$ Two Kinds of Motions </footer>

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### <primary style="font-size:24px"> Two Kinds of Motions </primary>
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- <span style="color:blue">1. Translational motion</span>

- <span style="color:green">Slipping</span>

- <span style="color:green">Skidding</span>

- Wheel rotating about an axis and moving, has got both rotational motion and translational motion.

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<footer style = "font-size: 18px">Introduction, Center of Mass System of Particles and Rotational Motion $\rightarrow$ Systems of particles and rotational motion $\rightarrow$ <span style = "color:blue"> Two Kinds of Motions </span> $\rightarrow$ Two Kinds of Motions $\rightarrow$ Precision </footer>

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### <primary style="font-size:24px"> Two Kinds of Motions </primary>
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-  All the particles of body have the same velocity $\vec{v}$: <span style="color:blue">Translational motion.</span>

- The instantaneous velocity is zero, so different points have different velocities.

- <span style="color:blue">Rotational motion:</span> rotation of rigid body about a fixed axis.

- Different points on rigid body, having different linear velocities.

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<footer style = "font-size: 18px">Systems of particles and rotational motion $\rightarrow$ Two Kinds of Motions $\rightarrow$ <span style = "color:blue"> Two Kinds of Motions </span> $\rightarrow$ Precision $\rightarrow$ Systems of particles and rotational </footer>

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### <primary style="font-size:24px"> Precision </primary>
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- Axis of the top processes about the vertical line this is known as <span style="color:blue">precision.</span>

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<footer style = "font-size: 18px">Two Kinds of Motions $\rightarrow$ Two Kinds of Motions $\rightarrow$ <span style = "color:blue"> Precision </span> $\rightarrow$ Systems of particles and rotational $\rightarrow$ Rotate a Door </footer>

- The ball goes without rotating about any axis: <span style="color:blue">Pure translation</span>

- Rotation about a fixed axis: <span style="color:blue">Rotational motion</span>

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<footer style = "font-size: 18px">Two Kinds of Motions $\rightarrow$ Precision $\rightarrow$ <span style = "color:blue"> Systems of particles and rotational </span> $\rightarrow$ Rotate a Door $\rightarrow$ Centre of Mass </footer>

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- Applying a normal force or some force which is making an angle to the door.

- If one apply the force at the hinges, the door is not going to rotate.

- Rotation is all about the point of application of a force.

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<footer style = "font-size: 18px">Precision $\rightarrow$ Systems of particles and rotational $\rightarrow$ <span style = "color:blue"> Rotate a Door </span> $\rightarrow$ Centre of Mass $\rightarrow$ Center of Mass </footer>

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- Standard example: Pedestal fan

- Blade rotates and you get the air.

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<footer style = "font-size: 18px">Systems of particles and rotational $\rightarrow$ Rotate a Door $\rightarrow$ <span style = "color:blue"> Centre of Mass </span> $\rightarrow$ Center of Mass $\rightarrow$ Position Vector </footer>

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- $X_{C M}=\frac{m_1 x_1+m_2 x_2}{m_1+m_2}$

- if  $m_1  =  m_2$

- $X_{C M}=\frac{x_{1+} x_2}{2}$

- **Several Particles System**

- $X_{c_M}=\frac{m_1 x_1+m_2 x_2+\cdot +m_n x_n}{\left(m_1+m_2+\cdot+m_n\right)}$

- $X_{C M}=\frac{\sum_{i=1}^n m_i x_i}{\sum_{i=1}^n m_i}$

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<footer style = "font-size: 18px">Rotate a Door $\rightarrow$ Centre of Mass $\rightarrow$ <span style = "color:blue"> Center of Mass </span> $\rightarrow$ Position Vector $\rightarrow$ Center of Mass Rigid Body </footer>

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- $\vec{r}=x \hat{\imath}+y \hat{\jmath}+z\hat{k}$

- The unit vectors, $\hat{e}_x, \hat{e}_y \operatorname{and} \hat{e}_z$

- $\vec{r}=x \hat{e}_x+y \hat{e}_y+z \hat{e}_z$

- $\vec{r}_{cm} $= $\frac{\sum m_1 \vec{r}_1}{\sum m_1}$

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<footer style = "font-size: 18px">Centre of Mass $\rightarrow$ Center of Mass $\rightarrow$ <span style = "color:blue"> Position Vector </span> $\rightarrow$ Center of Mass Rigid Body $\rightarrow$ Center of Mass Solar System </footer>

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- $X_{C M}=\frac{\sum_{i=1}^N\left(\Delta m_i\right) x_i}{\sum_{i=1}^N\left(\Delta m_i\right)} \rightarrow \frac{\int x d m}{\int d m}$

- $\vec{r}_{C M}=\frac{\int \vec{r} d m}{\int d m}$

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<footer style = "font-size: 18px">Center of Mass $\rightarrow$ Position Vector $\rightarrow$ <span style = "color:blue"> Center of Mass Rigid Body </span> $\rightarrow$ Center of Mass Solar System $\rightarrow$ Two-Dimensional Example </footer>

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- Choose the center of sun as the origin.

- $m_1$ = Mass of the Sun  = $2 \times 10^{30}$ kg
- $m_2$ = Mass of the Earth = $6 \times 10^{24}$ kg
- $r_1$ = Distance of the Sun from the origin = 0  (since the origin is the center of the Sun)
- $r_2$ = Distance of the Earth from the Sun = $1.5 \times 10^{11}$ meters

- $ R = \frac{m_1 \times r_1 + m_2 \times r_2}{m_1 + m_2}$

- $ R = \frac{(2 \times 10^{30} \times 0) + (6 \times 10^{24} \times 1.5 \times 10^{11})}{2 \times 10^{30} + 6 \times 10^{24}}$

- $R \approx 4.5 \times 10^{5} \text{ meters}$

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<footer style = "font-size: 18px">Position Vector $\rightarrow$ Center of Mass Rigid Body $\rightarrow$ <span style = "color:blue"> Center of Mass Solar System </span> $\rightarrow$ Two-Dimensional Example $\rightarrow$ Thank You </footer>

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- $(-1,1) =-1\hat{i}+1 \hat{j} =-\hat{i}+\hat{j}$

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

- $ =\frac{(0,0)}{6}=(0,0)$

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

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<footer style = "font-size: 18px">Center of Mass Rigid Body $\rightarrow$ Center of Mass Solar System $\rightarrow$ <span style = "color:blue"> Two-Dimensional Example </span> $\rightarrow$ Thank You $\rightarrow$  </footer>