physics-class-11-unit-07-chapter-08-rotational-motion-abouta-fixed-axis-angular-momentum-system-l-8-nyhe3y8dqve

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Angular Momentum - Rotation About a Fixed Axis </primary>
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- (1) Expression for the orbital angular momentum.

- (2) Principle of conservation of angular momentum

- (3) Rotation, roling and slipping

- (4) An example

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<footer style = "font-size: 18px"> $\rightarrow$ Rotational motion about a fixed axis-angular momentum  System of Particles and Rotational Motion $\rightarrow$ <span style = "color:blue"> Angular Momentum - Rotation About a Fixed Axis </span> $\rightarrow$ Angular Momentum $\rightarrow$ Angular Momentum </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Angular Momentum </primary>
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- $\vec{l} =\vec{r} \times \vec{p} \quad \vec{p}=n \vec{v} $

- $\vec{r}=\vec{op} =\overrightarrow{o c}+\vec{cp} $

- $\vec{l} =(\overrightarrow{o c}+\vec{cp}) \times m \vec{v} $

- =$(\vec{oc} \times m \vec{v})+(\vec{cp} \times m \vec{v}) $

- $\vec{l} =(\vec{oc} \times m \vec{v})+(r_{\perp}^2 m \omega \hat{k}) \vec{l}_z $

- $\vec{l} =\vec{l}_z+\overrightarrow{o c} \times m \vec{v}$

- $\vec{l_z} \parallel \hat{k}$

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<footer style = "font-size: 18px">Rotational motion about a fixed axis-angular momentum  System of Particles and Rotational Motion $\rightarrow$ Angular Momentum - Rotation About a Fixed Axis $\rightarrow$ <span style = "color:blue"> Angular Momentum </span> $\rightarrow$ Angular Momentum $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Angular Momentum </primary>
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- In general $\vec{\imath} \text { and } \vec{\omega}$ must not be parallel.

- $\vec{L}=\sum_i \vec{l_i}=\sum \vec{l}_{i_z}+\sum_i \overrightarrow{o c}_i \times m \vec{v}_i$

- $\vec{L}=\vec{L_z}+\vec{L_\perp}, \vec{L_z}$

- $=\sum \vec{l_{i_z}}=\left(\sum_i m_i r_{\perp_i}^2\right) \omega \hat{k}$

- $\vec{L}_z=I \omega \hat{k} \quad(p=m v)$

- So, $\vec{L_{\perp}}$=0,

- $ \vec{L}=\vec{L}_z= I \omega \hat{k}$

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<footer style = "font-size: 18px">Angular Momentum - Rotation About a Fixed Axis $\rightarrow$ Angular Momentum $\rightarrow$ <span style = "color:blue"> Angular Momentum </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Example 1 :- Circular Disc

- $\vec{L}=(I \omega) \hat{k}=\left(\frac{M R^2}{2} \omega\right) \hat{k}$

- $\vec{L}=2 \omega \hat{k} ; \frac{d \vec{L}}{d t}=I \frac{d \omega}{d t} \hat{k}=(I \alpha) \hat{k}=\tau \hat{k}$

- similarly   $\vec{L}=\hat{L_z}+\vec{L_{\perp}} ; \frac{d \vec{L_z}}{d t}=\tau \hat{k} ; \frac{d \vec{L_{\perp}}}{d t}$ =0

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<footer style = "font-size: 18px">Angular Momentum $\rightarrow$ Angular Momentum $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Example $\rightarrow$ Principle of Conservation of Angular Momentum (PCAM) </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Example </primary>
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- Example 2 :-

- $\vec{L}=(I \omega) \hat{k} $

- $\left(\frac{M R^2}{2}+M \alpha^2\right) \omega \hat{k}$

- <span style="color:blue">"Parallel axis" Theorem</span>

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<footer style = "font-size: 18px">Angular Momentum $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Principle of Conservation of Angular Momentum (PCAM) $\rightarrow$ Illustration </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Illustration </primary>
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- The line of  motion of bullet perpendicular to the axis of the cylinder

- Before collision: $l_i=m v_o d$

- After the collision: $l_f = I \omega$

- $(I_\text{{solid cylinder}} + I_\text{{ projectile}} ) \omega$

- $\left(\frac{M R^2}{2}+m R^2\right) \omega=m v_0 d$

- $\omega =\frac{mv_0d}{\left(\frac{M R^2}{2}+m R^2\right)}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Principle of Conservation of Angular Momentum (PCAM) $\rightarrow$ <span style = "color:blue"> Illustration </span> $\rightarrow$ Illustration 2 $\rightarrow$ Rotation, Rolling and Slipping </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Illustration 2 </primary>
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- Question :- $\omega_c = ?$

- $l=I \omega$ and $l_i=l_f$

- $I_i=I_{c p}+I_m=\frac{M R^2}{2}+m R^2 $

- $I_f=I_{c p}+I_m \quad=\frac{M R^2}{2}+m x^2 $

- PCAM : $I_i \omega_i = I_f \omega_f$

- $(\frac{M R^2}{2}+m R^2) \omega = (\frac{M R^2}{2}+m x^2) \omega_c$

- $\omega_c=\frac{\left(\frac{M R^2}{2}+m R^2\right)}{\left(\frac{M R^2}{2}+m x^2\right)} \omega$ where $\omega_c > \omega$

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<footer style = "font-size: 18px">Principle of Conservation of Angular Momentum (PCAM) $\rightarrow$ Illustration $\rightarrow$ <span style = "color:blue"> Illustration 2 </span> $\rightarrow$ Rotation, Rolling and Slipping $\rightarrow$ Rotation, Rolling and Slipping </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Rotation, Rolling and Slipping </primary>
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- The rotating disc is moves gently on the table frictionless.

- $v_A=R \omega_0, v_B  =R \omega_0 $

- $v_c  =\frac{R}{2} \omega_0$

- Disc only rotates ; it will not roll !

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<footer style = "font-size: 18px">Illustration $\rightarrow$ Illustration 2 $\rightarrow$ <span style = "color:blue"> Rotation, Rolling and Slipping </span> $\rightarrow$ Rotation, Rolling and Slipping $\rightarrow$ Kinetic Energy of Rolling motion </footer>

- At $P_0:\left|\vec{v_{P_0}}\right|=\vec{v_{C M}}_=R \omega_0$

- $P_0$ is at instantaneous rest:

- $v_{CM} = R\omega_0$

- $\left|\vec{v_{P_1}}\right|=v_{C M}+R M=2 v_{C M}$

- Condition for rolling withouts slipping

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<footer style = "font-size: 18px">Illustration 2 $\rightarrow$ Rotation, Rolling and Slipping $\rightarrow$ <span style = "color:blue"> Rotation, Rolling and Slipping </span> $\rightarrow$ Kinetic Energy of Rolling motion $\rightarrow$ Kinetic Energy of a system of particles </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Kinetic Energy of Rolling motion </primary>
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- $K =K E \text { of a rolling body} $

- $=K E \text { of translation }+K E \text { of rotation}$

- $\text { KE of a system of particles} $

- $=K E |+K E \text { rotational motion about the CM } $

- $k=\frac{1}{2} m v_{c_m}^2+\frac{1}{2} I \omega^2$

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<footer style = "font-size: 18px">Rotation, Rolling and Slipping $\rightarrow$ Rotation, Rolling and Slipping $\rightarrow$ <span style = "color:blue"> Kinetic Energy of Rolling motion </span> $\rightarrow$ Kinetic Energy of a system of particles $\rightarrow$ Kinetic Energy of a system of particles </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Angular Momentum System L-8 </Success>
### <primary style="font-size:24px"> Kinetic Energy of a system of particles </primary>
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- $=K E |+K E$ rotational motion about the CM

- $k=\frac{1}{2} m v_{c_m}^2+\frac{1}{2} I \omega^2$

- $v_{CM} = R\omega$

- $I=mk^2, ' k ':$ radius of gyration

- $k=\frac{1}{2} m k^2 \frac{v_{CM}^2}{R^2}+\frac{1}{2} m v_{C M}^2$

- $k=\frac{1}{2} m v_{c m}^2\left(1+\frac{k^2}{R^2}\right)$

- The $K E$ of a rolling body

- =$KE$ of translation + $KE$ of rotation

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<footer style = "font-size: 18px">Rotation, Rolling and Slipping $\rightarrow$ Kinetic Energy of Rolling motion $\rightarrow$ <span style = "color:blue"> Kinetic Energy of a system of particles </span> $\rightarrow$ Kinetic Energy of a system of particles $\rightarrow$ Thank You </footer>

- a ring, a solid cylinder, a sphere
- $\operatorname{mgh}=\frac{m v^2}{2}\left(1+\frac{k^2}{R^2}\right) \Rightarrow v=\left(\frac{2gh}{1+\frac{k^2}{R^2}}\right)^{1 / 2}$
- Greatest velocity

| Object | $k$  | $v$ |
| -------- | -------- | -------- |
| Circular ring| R    | $\sqrt{gh}$ |
| $0^r$ disc  | $\frac{R}{\sqrt{2}}$   | $\sqrt{\frac{4}{3}gh}$ |
| Sphere (solid)   | $\sqrt{\frac{2}{5}}R$    | $\sqrt{\frac{10}{7}gh}$ |

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<footer style = "font-size: 18px">Kinetic Energy of Rolling motion $\rightarrow$ Kinetic Energy of a system of particles $\rightarrow$ <span style = "color:blue"> Kinetic Energy of a system of particles </span> $\rightarrow$ Thank You $\rightarrow$  </footer>