physics-class-11-unit-07-chapter-03-vector-products-angular-velocity-and-angular-acceleration-l-3-10-afmammcc6rg

### <Success style="font-size:25px"> Vector Products Angular Velocity And Angular Acceleration L-3 </Success>
### <primary style="font-size:24px"> Vector Products </primary>
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- We have two vectors A and B.

- <span style="color:green">The 'dot' Product</span> : $\vec{A}.\vec{B} = |\vec{A}||\vec{B}| \cos \theta$

- Suppose a force is acting on a particle. Then the work done by the force,

- W = $\int_A^B \vec{F}.\vec{ds}$.

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<footer style = "font-size: 18px"> $\rightarrow$ Vector Products, Angular Velocity, Angular Acceleration $\rightarrow$ <span style = "color:blue"> Vector Products </span> $\rightarrow$ Cross Product $\rightarrow$ Properties of Cross Product </footer>

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### <primary style="font-size:24px"> Cross Product </primary>
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- $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}| \sin \theta \hat{n}$

- $'\theta'$ is taken through the smaller angle ($<180^0$)

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<footer style = "font-size: 18px">Vector Products, Angular Velocity, Angular Acceleration $\rightarrow$ Vector Products $\rightarrow$ <span style = "color:blue"> Cross Product </span> $\rightarrow$ Properties of Cross Product $\rightarrow$ Unit Vector </footer>

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### <primary style="font-size:24px"> Properties of Cross Product </primary>
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- **1.** $\vec{a} \times \vec{b} \neq \vec{b} \times \vec{a}$ = - $\vec{b} \times \vec{a}$

- **2.** Reflection  $\vec{a} \rightarrow -\vec{a}$

- $\vec{b} \rightarrow -\vec{b}$

- $\vec{a} \times \vec{b} = (-\vec{a}) \times (-\vec{b})$

- **3.** $\vec{a} \times \vec{c} = 0$

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<footer style = "font-size: 18px">Vector Products $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Properties of Cross Product </span> $\rightarrow$ Unit Vector $\rightarrow$ Cross Product </footer>

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### <primary style="font-size:24px"> Unit Vector </primary>
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- ($\hat{i}, \hat{j}, \hat{k}$) $\equiv$ ($\hat{e_x}, \hat{e_y}, \hat{e_z}$)

- $\hat{i} \times \hat{j} = \hat{k},$

- $\hat{j} \times \hat{k} = \hat{i},$

- $\hat{i} \times \hat{i} = 0$

- $\hat{j} \times \hat{i} = \hat{-k},$

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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Properties of Cross Product $\rightarrow$ <span style = "color:blue"> Unit Vector </span> $\rightarrow$ Cross Product $\rightarrow$ Example </footer>

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### <primary style="font-size:24px"> Cross Product </primary>
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- $\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k},$

- $\vec{b} = b_x \hat{i} + b_y \hat{j} +  b_z \hat{k}$

- $\left|\begin{array}{lll}\hat{\imath} & \hat{\jmath} & \hat{k} \\\ a_z & a_y & a_z \\\ b_x & b_y & b_z\end{array}\right|$

- $\vec{a} \times \vec{b}$ = $\hat{i} (a_y b_z - b_y a_z) - \hat{j} (a_x b_z - a_z b_x) + \hat{k} (a_x b_y - b_x a_y)$

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<footer style = "font-size: 18px">Properties of Cross Product $\rightarrow$ Unit Vector $\rightarrow$ <span style = "color:blue"> Cross Product </span> $\rightarrow$ Example $\rightarrow$ Average Angular Velocity </footer>

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### <primary style="font-size:24px"> Example </primary>
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- $\vec{a} = \hat{i} + 2 \hat{j} + 3 \hat{k},$

- $\vec{b} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}$

- $\vec{c} = \vec{a} \times \vec{b}$ =

- $\left|\begin{array}{lll}\hat{\imath} & \hat{\jmath} & \hat{k} \\\ 1 & 2 & 3 \\\ 2 & 3 & 4\end{array}\right|$

- = $\hat{i} (8-9) - \hat {j} (4-6) + \hat{k} (3-4)$ = $\hat{i} + 2 \hat{j} - \hat{k}$

- $\vec{c} . \vec{a} = (- \hat{i} + 2 \hat{j} - \hat{k}) . (\hat{i} + 2 \hat{j} + 3 \hat{k}) $ = -1+4-3 = 0

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<footer style = "font-size: 18px">Unit Vector $\rightarrow$ Cross Product $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Average Angular Velocity $\rightarrow$ Instantaneous Angular Velocity </footer>

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- $\Delta \theta$ : Angular Displacement

- $\Delta t$: time period

- $\vec{\omega}$: The direction of $\vec{\omega}$ is specified by

- The average angular velocity over the interval $\Delta t = \frac{\Delta \theta }{ \Delta t}$

- $\ {lim_{\Delta t \rightarrow 0 }}  \frac {\Delta \theta }{\Delta t} = \frac{d \theta}{dt} =$ Instantaneous angular velocity.

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<footer style = "font-size: 18px">Cross Product $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Average Angular Velocity </span> $\rightarrow$ Instantaneous Angular Velocity $\rightarrow$ Instantaneous Angular Velocity </footer>

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### <primary style="font-size:24px"> Instantaneous Angular Velocity </primary>
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- $\vec{\omega} \times \vec{r} = \vec{\omega} \times (\vec{OC} + \vec{CP})$

- = $(\vec{\omega} \times \vec{OC}) + (\vec{\omega} \times \vec{cp})$
 
- $\vec{\omega} \times \vec{r}$ = $(\vec{\omega} \times \vec{cp})$ = $|\vec{\omega}||\vec{CP}|$

- = $\omega r_{\perp}$
 
 - = $\vec{\omega} \times \vec{r} = \vec{v}$

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<footer style = "font-size: 18px">Average Angular Velocity $\rightarrow$ Instantaneous Angular Velocity $\rightarrow$ <span style = "color:blue"> Instantaneous Angular Velocity </span> $\rightarrow$ Angular Acceleration $\rightarrow$ Angular Acceleration </footer>

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- **Diffrention both side w.r.t 't'**

- $\frac{dv}{dt}$ = $\frac{dr}{dt} \omega + r \frac{d \omega }{dt}$ = $r \alpha$

- $\alpha  = |\vec{\alpha}|$

- $A_t = \frac{dv}{dt} = r \alpha$

- $\vec{A_t} = \vec{\alpha} \times \vec{r}$ (compare this with $\vec{v} = \vec{\omega} \times \vec{r}$)

- $A_r = \frac{v^2}{r} = \frac{(r\omega)^2}{r} = \omega (r \omega)$

- $\vec{A_r} = \vec{\omega} \times (\vec{\omega} \times \vec{r})$

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<footer style = "font-size: 18px">Instantaneous Angular Velocity $\rightarrow$ Angular Acceleration $\rightarrow$ <span style = "color:blue"> Angular Acceleration </span> $\rightarrow$ Particle in Two Dimension $\rightarrow$ For a rigid body motion </footer>

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- $\vec{r}  =r \hat{e}_r $

- $\vec{v}=\frac{d \vec{r}}{d t} =(\frac{d r}{d t} \hat{r_r})+r \frac{d \hat{e}_r}{d t} $

- $\vec{v}  =\dot{r} \hat{e}_r+r \dot{\theta} \hat{e}_\theta $

- $\vec{v} =(v_r) \hat{e}_{r+} (\vec{r}_\theta) \hat{e}_\theta$

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<footer style = "font-size: 18px">Angular Acceleration $\rightarrow$ Angular Acceleration $\rightarrow$ <span style = "color:blue"> Particle in Two Dimension </span> $\rightarrow$ For a rigid body motion $\rightarrow$ For a rigid body motion </footer>

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### <primary style="font-size:24px"> For a rigid body motion </primary>
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- $y=\sin \theta $

- $\dot{r}=\frac{d}{d t}(r) $

- $\hat{e_r}=\cos \theta \hat{e}_x+\sin \theta \hat{e}_y$

- $\frac{d}{d t} (\hat{e_r})=(-\sin \theta \hat{e_t})+ (\cos \theta \hat{e}_y) \dot{\theta}$     =      $\hat{e}_\theta \dot{\theta}$

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<footer style = "font-size: 18px">Particle in Two Dimension $\rightarrow$ For a rigid body motion $\rightarrow$ <span style = "color:blue"> For a rigid body motion </span> $\rightarrow$ Angular Acceleration in Two Dimension $\rightarrow$ Angular Acceleration in Two Dimension </footer>

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- $\vec{A}=\frac{d \vec{v}}{d t}=\frac{d}{d t} (\dot{r} \hat{e}_r + r \dot{\theta} \hat{c}_\theta )$

- =$\ddot{r} \hat{e}_r+\dot{r} \frac{d}{d t} (\hat{e}_r)+\frac{d r}{d t} \dot{\theta} \hat{e}_\theta+r \ddot{\theta} \hat{e}_\theta+r \dot{\theta} \frac{d}{d t} \hat{e}_0) $

- $ = \ddot{r} \hat e_{r}  + \dot{r} (\hat{e}_{\theta} \dot{\theta})$
 
- $ + \dot{r} \dot{\theta} \hat e_{\theta} + r \dot{\theta} \hat e_{\theta}$

- $ + r \dot{\theta} (-\dot{\theta} \hat{e}_{r}) $

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<footer style = "font-size: 18px">For a rigid body motion $\rightarrow$ For a rigid body motion $\rightarrow$ <span style = "color:blue"> Angular Acceleration in Two Dimension </span> $\rightarrow$ Angular Acceleration in Two Dimension $\rightarrow$ Angular Velocity in Two Dimension </footer>

- $ =(\ddot{r}-r \dot{\theta}^2) \hat{e_r}+(r \dot{\theta}+2 \dot{r} \dot{\theta}) \hat{e}_\theta $

- = $A_r \hat{e_r} + A_{\theta} \hat{e{\theta}}$

- $A_r = \frac{v^2}{r} = \frac{(r {\omega})^2} {r} = \omega (r \omega)$

- $\vec{A}_r = \vec{\omega} \times (\omega \times \vec{r})$

- For a rigid body (n=0)

- $A_r = - r \dot{\theta}^2$

- $A_{\theta} = r \dot{\theta }$

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<footer style = "font-size: 18px">For a rigid body motion $\rightarrow$ Angular Acceleration in Two Dimension $\rightarrow$ <span style = "color:blue"> Angular Acceleration in Two Dimension </span> $\rightarrow$ Angular Velocity in Two Dimension $\rightarrow$ Angular Velocity in Two Dimension </footer>

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- $x = r \cos \theta,$

- $y = r \sin \theta$

- $|\vec{r}|=1 $

- $x=\cos \theta $

- $y=\sin \theta $

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<footer style = "font-size: 18px">Angular Acceleration in Two Dimension $\rightarrow$ Angular Acceleration in Two Dimension $\rightarrow$ <span style = "color:blue"> Angular Velocity in Two Dimension </span> $\rightarrow$ Angular Velocity in Two Dimension $\rightarrow$ Thank You </footer>

- $\dot{r}=\frac{d}{d t}(r) $

- $\hat{e_r}=\cos \theta \hat{e}_x+\sin \theta \hat{e}_y$

- $\frac{d}{d t} (\hat{e_r})=(-\sin \theta \hat{e_t})+ (\cos \theta \hat{e}_y) \dot{\theta}$

- = $\hat{e}_\theta \dot{\theta}$

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<footer style = "font-size: 18px">Angular Acceleration in Two Dimension $\rightarrow$ Angular Velocity in Two Dimension $\rightarrow$ <span style = "color:blue"> Angular Velocity in Two Dimension </span> $\rightarrow$ Thank You $\rightarrow$  </footer>