physics-class-11-unit-07-chapter-04-torque-angular-momentum-system-of-particles-rotational-m-l-4-10-o23psvervxa

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Rigid Body </primary>
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- Linear velocity: $ \vec{v}=\vec{w} \times \vec{n}$
 
- Transverse acceleration: $\vec{A}_t=\vec{\alpha} \times \vec{r} $
 
- Radial acceleration: $\vec{A}_n=\vec{\omega} \times(\vec{\omega} \times \vec{n})$

- Position vector: $ \vec{r}=r \hat{e}_r, $

- Differentiating with respect to time,

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

- $ =v_r \hat{e}_r+v_\theta \hat{e}_\theta$

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<footer style = "font-size: 18px"> $\rightarrow$ Torque and Angular Momentum  System of Particles and Rotational Motion $\rightarrow$ <span style = "color:blue"> Rigid Body </span> $\rightarrow$ Rigid Body $\rightarrow$ Rigid Body </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Rigid Body </primary>
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- For a particle fixed, $ \dot{r}=0 $, $ v_0 = 0$  and $ v_\theta=r \dot \theta$.

- $ \vec{\omega}=\omega \hat{e} _ z$,  $ \alpha_ 2 \vec{r}=r \hat{e}_r,$

- $ \therefore \vec{\omega} \times \vec{r}=\left(\omega \hat{e} _ z\right) \times\left(r \hat{e}_r\right)$

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

- $=\left(\ddot{r}- r\dot{\theta}^2\right) \hat{e_ r}+( r \ddot{\theta}+2 \dot{r} \dot{\theta}) \hat{e_{\theta}} $

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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-04-torque-angular-momentum-system-of-particles-rotational-o23psvervxa-03a.jpg)

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<footer style = "font-size: 18px">Torque and Angular Momentum  System of Particles and Rotational Motion $\rightarrow$ Rigid Body $\rightarrow$ <span style = "color:blue"> Rigid Body </span> $\rightarrow$ Rigid Body $\rightarrow$ Torque and Momentum </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Rigid Body </primary>
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- For a fixed particles, $\dot{r}=0, \ddot{r}=0$

- $\vec{A}_t  =\vec{\alpha} \times \vec{r} $

- $ =\left(\ddot{\theta} \hat{e}_ z\right) \times\left(r \hat{e}_n\right)  =r \ddot{\theta} \hat{e} _\theta$

- $\vec{A}_ {\mu }=\vec{\omega} \times(\vec{\omega} \times \vec{r})=\dot{\theta} \hat{e}_ z \times\left(r\dot{\theta} \hat{e}_{\theta}\right)$

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

- $ =A_r \hat{e} _ r + A _ t \hat{e} _ {\theta} =r \dot{\theta}^2 \hat{e} _r$

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<footer style = "font-size: 18px">Rigid Body $\rightarrow$ Rigid Body $\rightarrow$ <span style = "color:blue"> Rigid Body </span> $\rightarrow$ Torque and Momentum $\rightarrow$ Moment of a Force on a Single Particle </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Moment of a Force on a Single Particle </primary>
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- Torque $=\vec{\tau} = \vec{r} \times \vec{F} $, given by right hand screw rule. 
 
- $\vec{\tau} = r F \operatorname{sin} \theta=r F_{\perp} $
 
- $\vec{\tau} = F r \sin \theta=F r_{\perp}$
 
- $ [\vec \tau]$ has the dimensions of energy.
 
- $ \vec\tau=0$, if $\vec{F}  =0 $
 
- $ \vec{r}  =0 $, $ \theta  =0 \text { or } 180^{\circ}\$

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<footer style = "font-size: 18px">Rigid Body $\rightarrow$ Torque and Momentum $\rightarrow$ <span style = "color:blue"> Moment of a Force on a Single Particle </span> $\rightarrow$ Example $\rightarrow$ Angular Momentum of a Particle </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $\tau=r m g \sin \theta = r m g(\frac{x}{r})=x m g $

- $  \vec{\tau}=\vec{r} \times \vec{f}=$

- $\left|\begin{array}{lll}\hat{\imath} & \hat{\jmath} & \hat{k} \\\ x & -y & 0 \\\ 0 & mg & 0 \end{array}\right|$

- $ =\hat{k}(xmg) = mgx\hat{k}$

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<footer style = "font-size: 18px">Torque and Momentum $\rightarrow$ Moment of a Force on a Single Particle $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Angular Momentum of a Particle $\rightarrow$ Relation between Torque and Angular Momentum </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
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- let us consider a particle of mass $m$, and its position it is a with respect coordinate system it is at a particular position vector $\vec{r}$ and its momentum is $\vec{p}$ then its orbital angular momentum,

- $ \vec{l}=\vec{r} \times \vec{p} $

- $ =r p \sin \theta=r p_\perp $

- $ =p(r \sin \theta)=p r_\perp $

- $ \vec{l}=0, \vec{r}=0 \text { or, } \vec{p}=0 \quad\text { the }  \theta=0 \text { or } 180^0 $

- $\vec{\tau}$ = Torque (or moment of a force ) is rotational analogue of a force.

- $\vec{l}$ = Angular momentum is the rotational analogue of linear momentum.

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<footer style = "font-size: 18px">Moment of a Force on a Single Particle $\rightarrow$ Example $\rightarrow$ <span style = "color:blue"> Angular Momentum of a Particle </span> $\rightarrow$ Relation between Torque and Angular Momentum $\rightarrow$ Relation between Torque and Angular Momentum </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Relation between Torque and Angular Momentum </primary>
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- $\vec{l}  =\vec{r} \times \vec{p} $

- $\frac{d}{d t}(\vec{l})  =\frac{d}{d t}(\vec{r} \times \vec{p})=\frac{d \vec{r}}{d t} \times \vec{p}+\vec{r} \times \frac{d \vec{p}}{d t}=\vec{r} \times \vec{F}=\vec{\tau}$

- $\frac{d \vec{l}}{d t} =\vec{\tau}$

- Compare,

- $\frac{d \vec{p}}{d t}=\vec{F}$

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<footer style = "font-size: 18px">Example $\rightarrow$ Angular Momentum of a Particle $\rightarrow$ <span style = "color:blue"> Relation between Torque and Angular Momentum </span> $\rightarrow$ Relation between Torque and Angular Momentum $\rightarrow$ Conservation of angular momentum </footer>

- $ \vec{L}  =\vec{l}_{+} \vec{l}_2+\cdots+\vec{l}_n$,

- $\vec{l}_ i = \vec{r} _ i \times \vec{p} _ i $, $(\vec{p}_ i= \vec{m}_ i \vec{l}_i )$

- $\vec{L}=\Sigma _ {i=1}^n \vec{l} _ i= \Sigma _ {i=1}^n(\vec{r} _ i \times \vec{p}_i)$

- $\frac{d \vec{L}}{d t}=\frac{d}{d t}(\Sigma _ {i=1}^n \vec{l}_ i)=\Sigma_ {i=1}^n \frac{d \overrightarrow{l_ i}}{d t}=\Sigma_ i \vec{\tau}_ i = \vec\tau_i$

- $\vec\tau_ i = \vec{r}_ i \times \vec{F}_ i = \vec{r} _ c \times(\vec{F _ i}^{E x t}+\vec{F_i} ^{Int})$

- $\vec\tau = \vec\tau_ {Ext}$, $\frac{d\vec{L}}{d t}=\vec{\tau}_{E x t}$, and $ \frac{d \vec{p}}{d t}=\vec{F} _{\text {Ext }}$

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<footer style = "font-size: 18px">Angular Momentum of a Particle $\rightarrow$ Relation between Torque and Angular Momentum $\rightarrow$ <span style = "color:blue"> Relation between Torque and Angular Momentum </span> $\rightarrow$ Conservation of angular momentum $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Torque Angular Momentum System Of Particles Rotational M L-4 </Success>
### <primary style="font-size:24px"> Conservation of angular momentum </primary>
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- The time rate of total orbital angular momentum (OAM) If a system of particles about a pt (or about a line) is equal lo the sum of the external torques.i.e, the torques due to the external forces alone).

- If $\vec{\tau}_{\text {ext }}=0$, $\frac{d \vec{L}}{d u}=0 \Rightarrow \vec{L}$ is a constant of motion.

- $\frac{d \vec{p}}{d t}=\vec{F}, $

- If $ \vec{f}=0$ , then ${\vec{p}}$ is a constant of the motion.

- $P$ is conserved.

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<footer style = "font-size: 18px">Relation between Torque and Angular Momentum $\rightarrow$ Relation between Torque and Angular Momentum $\rightarrow$ <span style = "color:blue"> Conservation of angular momentum </span> $\rightarrow$ Example $\rightarrow$ Thank You </footer>

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- Let's consider a particle which is moving with a constant velocity.

- $\vec{v}$ is a constant.

- Momentum: $ \vec{p}=m \vec{v}$

- The direction of $\vec{p}$ into the page.

- $\vec{l}= \vec{r} \times m \vec{v}=r  m v \sin \hat{\theta}, \quad $

- $r=|\vec{r}|, v=|\vec{v}| $

- $ \sin \theta=\frac{O M}{r} $

- $ l=  (r \sin \theta)(m v) =  (OM)(m v) \Rightarrow $

- l is constant of motion.

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<footer style = "font-size: 18px">Relation between Torque and Angular Momentum $\rightarrow$ Conservation of angular momentum $\rightarrow$ <span style = "color:blue"> Example </span> $\rightarrow$ Thank You $\rightarrow$  </footer>