physics-class-11-unit-07-chapter-07-rotational-motion-abouta-fixed-axis-kinematics-dynamics-l-7-10-pdqi4hz644u

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Rotational Motion </primary>
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- Angular Displacement: $\omega$=$\frac{dQ}{dt}$

- Angular acceleration :$\alpha = \frac{d \omega}{dt}$

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/373fd082-8bdb-4f9b-d63d-de9fbb9a7000/public)

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<footer style = "font-size: 18px"> $\rightarrow$ Rotational Motion a Fixed Axis - Kinematics and Dynamics $\rightarrow$ <span style = "color:blue"> Rotational Motion </span> $\rightarrow$ Linear and Rotational Motion $\rightarrow$ Equation of Motion </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Linear and Rotational Motion </primary>
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- $ \alpha: \text { constant } $

- $\frac{d \omega}{d t}=\alpha$= constant

- Integrate: $\omega$ = $\alpha t +c $

- at  $ t=t_0 , \omega=\omega_0 \Rightarrow c=w_0$

- $\omega=\omega_0+\alpha t $

- t = $\frac {\omega-\omega_0}{\alpha }$

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| Linear Motion | Rotational motion |
| -------- | -------- |
| $ v=u+a t $    | $\omega$=$w_0+ \alpha t$|
| $ s=s_0+u t+\frac{1}{2} a^2 $   | $\theta = \theta_0 + \omega_0 t +\frac{1}{2} \alpha t^2$  |
| $ v^2=u^2+2 a s $   |$\omega^2 = \omega_0^2+ 2 \alpha \theta $    |

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<footer style = "font-size: 18px">Rotational Motion a Fixed Axis - Kinematics and Dynamics $\rightarrow$ Rotational Motion $\rightarrow$ <span style = "color:blue"> Linear and Rotational Motion </span> $\rightarrow$ Equation of Motion $\rightarrow$ Relation between angular and linear quantities </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Equation of Motion </primary>
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- $\frac {d\theta }{dt} = \omega_0 + \alpha t$

- Integrate: $\theta = \omega_0 t + \frac{\alpha t^2}{2}$ + c (c: a-constant)

- $\theta (t=0) = \theta_0 \Rightarrow c= \theta_0$

- $\theta = \theta_0 + \omega_0 t + \frac {\alpha t^2}{2}$

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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-07-rotational-motion-fixed-axis-kinematics-dynamics-l-7_10-pdqi4hz644u-01.jpg)
 
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<footer style = "font-size: 18px">Rotational Motion $\rightarrow$ Linear and Rotational Motion $\rightarrow$ <span style = "color:blue"> Equation of Motion </span> $\rightarrow$ Relation between angular and linear quantities $\rightarrow$ Tangential Acceleration </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Relation between angular and linear quantities </primary>
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- v = r $\omega$

- where: $(\vec{v} = \vec{\omega}  \times \vec{r})$

- v = $\frac{dr}{dt} = r \frac{d \theta}{dt}$

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/ef8de616-c8d2-4fae-39a8-ed1675b9bd00/public)

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<footer style = "font-size: 18px">Linear and Rotational Motion $\rightarrow$ Equation of Motion $\rightarrow$ <span style = "color:blue"> Relation between angular and linear quantities </span> $\rightarrow$ Tangential Acceleration $\rightarrow$ Radial acceleration </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Tangential Acceleration </primary>
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- $\vec {PQ} = \vec{a} (\vec{A})$

- $a_t = \frac{dv}{dt} = \frac{d}{dt} (r \omega)$

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

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

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/8eeeecd1-047c-441f-3036-53f549870800/public)

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<footer style = "font-size: 18px">Equation of Motion $\rightarrow$ Relation between angular and linear quantities $\rightarrow$ <span style = "color:blue"> Tangential Acceleration </span> $\rightarrow$ Radial acceleration $\rightarrow$ Relation Between Torque and Angular Acceleration </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Radial acceleration </primary>
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- $a_r = \frac {v^2}{r}$ (centripetal acceleration)

- $ = \frac{(r \omega)^2}{r} = r \omega^2 = r (\frac {d \theta }{dt})^2 = r \theta ^2$

- $\vec{A}_r=-r \dot{\theta}^2 \hat{e}_r $

- $\vec{a} = \vec{A} = A r \hat{e_r} + A_t \hat{e_\theta}$ :

- a = $|\vec{a}|$ = $\sqrt {a_t^2 + a_r^2}$ = $\sqrt {(r \alpha)^2 + (r \omega^2 )^2}$

- a = r$\sqrt{ {\alpha}^2+ {\omega}^4}$

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<footer style = "font-size: 18px">Relation between angular and linear quantities $\rightarrow$ Tangential Acceleration $\rightarrow$ <span style = "color:blue"> Radial acceleration </span> $\rightarrow$ Relation Between Torque and Angular Acceleration $\rightarrow$ Rigid body of any Shape Rotating About a Fixed Axis </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Relation Between Torque and Angular Acceleration </primary>
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- $F_ t=m a_t$

- The torque about the origin due to force $F_t$

- $\tau={F_ t}r=(m a_t) r $

- $ a_t=r \alpha $

- $\therefore  tau=(m r^2) \alpha= I \alpha $

- $\tau=I \alpha \Rightarrow \tau \propto \alpha$

- I: Proportionality constant

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<footer style = "font-size: 18px">Tangential Acceleration $\rightarrow$ Radial acceleration $\rightarrow$ <span style = "color:blue"> Relation Between Torque and Angular Acceleration </span> $\rightarrow$ Rigid body of any Shape Rotating About a Fixed Axis $\rightarrow$ Work and Energy in Rotational Motion </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Rigid body of any Shape Rotating About a Fixed Axis </primary>
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- $d {F_ t} = (dm) a_t$

- Torque d $\tau$  due to the force $d{F_T}$ about the origin:

- $d\tau  = r d{F_ t} = (r dm) a_t$

- = (r dm) $(r \alpha)$
 
 - $\tau = \alpha \int {r^2 dm}$
 
 - $\Rightarrow \tau  = I \vec{\alpha}$

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<footer style = "font-size: 18px">Radial acceleration $\rightarrow$ Relation Between Torque and Angular Acceleration $\rightarrow$ <span style = "color:blue"> Rigid body of any Shape Rotating About a Fixed Axis </span> $\rightarrow$ Work and Energy in Rotational Motion $\rightarrow$ Instantaneous Power </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Work and Energy in Rotational Motion </primary>
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- $\vec{\tau} =\vec{r} \times \vec{F} $

- ${[\vec{\tau}] }$  = Dimension work or Energy

- $\vec{\tau}$ rotates the object $d \theta$, then the work done for this infinitedecimal rotation:

- $ d \omega=\tau d \theta(F d x) $

- $ \frac{d \omega}{d t}=\tau \frac{d \theta}{d t}=\tau \omega $

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<footer style = "font-size: 18px">Relation Between Torque and Angular Acceleration $\rightarrow$ Rigid body of any Shape Rotating About a Fixed Axis $\rightarrow$ <span style = "color:blue"> Work and Energy in Rotational Motion </span> $\rightarrow$ Instantaneous Power $\rightarrow$ Work Energy Theorem </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Work Energy Theorem </primary>
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- **Work energy theorem in rotational motion**

- we have

- $\tau = I \alpha  = I \frac{d \omega }{dt} =I \frac {d \omega }{dt} \frac {d \theta }{dt} = I \omega \frac {d \omega}{d \theta}$

- So, $\tau d \theta =  I \omega d \omega = d W $

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<footer style = "font-size: 18px">Work and Energy in Rotational Motion $\rightarrow$ Instantaneous Power $\rightarrow$ <span style = "color:blue"> Work Energy Theorem </span> $\rightarrow$ Work Energy Theorem $\rightarrow$ Work Energy Theorem </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Work Energy Theorem </primary>
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- **Integrate,**

- $w=\int_{\theta_0}^\theta \tau d \theta=\int_{\theta_0}^\theta I \omega d $ $\omega=\frac{I}{2}(\omega^2-\omega_0^2)$

- $\theta - \theta_0$ : the change in angular displacement.

- $\omega-\omega_0$ : the corresponding change in angular speed.

- W = $\frac{3}{2} ({\omega^2} - {\omega_0}^2)$

- W = $\frac{m}{2} ({v^2} - {\omega}^2)  \text{in linear motion }$

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<footer style = "font-size: 18px">Instantaneous Power $\rightarrow$ Work Energy Theorem $\rightarrow$ <span style = "color:blue"> Work Energy Theorem </span> $\rightarrow$ Work Energy Theorem $\rightarrow$ Rational Vs Linear motion </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Work Energy Theorem </primary>
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- **Rotational Motion**
	- 1. Angular velocity:  $ \omega  = \frac {d \theta}{dt} = \dot{\theta}$
	- 2. Angular acceleration: $\alpha  = \frac {d \omega} {dt} = \ddot{\theta}$
	- 3. Torque: $\tau  = I \alpha $

- **Linear Motion**
	- 1. Linear velocity: v= $\frac {dx}{dt}$
	- 2. Linear acceleration: a = $\frac {dv}{dt} = \ddot{x}$
	- 3. Force: $F = M a $

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<footer style = "font-size: 18px">Work Energy Theorem $\rightarrow$ Work Energy Theorem $\rightarrow$ <span style = "color:blue"> Work Energy Theorem </span> $\rightarrow$ Rational Vs Linear motion $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Rotational Motion Abouta Fixed Axis Kinematics Dynamics L-7 </Success>
### <primary style="font-size:24px"> Rational Vs Linear motion </primary>
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| Rotational Motion  | Linear Motion  |
| -------- | -------- |
|1. Angular velocity:  $ \omega  = \frac {d \theta}{dt} = \dot{\theta}$|1. Linear velocity: v= $\frac {dx}{dt}$|
|2. Angular acceleration: $\alpha  = \frac {d \omega} {dt} = \ddot{\theta}$|2. Linear acceleration: a = $\frac {dv}{dt} = \ddot{x}$|
|3. Torque  $\tau=I \alpha $|3. Force: $F = M a $|
|4. $\omega=\omega_0+\alpha t $|4. $v=u+a t $|
|  $\theta =\theta_0+\omega_0 t+\frac{1}{2} \alpha t^2 $|  $ s=x_0+u t+\frac{1}{2} a t^2 $|
|  $\omega^2=\omega_0 ^2+ 2 a (\theta-\theta_0) $ |$ v^2=u^2+2 a(s-x_0) $|
|5. Work done  (W) = $\int_{\theta_0}^\theta \tau d \theta $|5. Work done (W) =$\int_{x_0}^x F_x d x $|
|6. KE =$\frac{1}{2} I \omega^2 $|6. $ k E=\frac{1}{2} m v^2 $|
|7. Power (P) = $\tau \omega $ |7. Power  P = F v|
|8. Angular momentum (L)= $I \omega$|8. Linear momentum $p=m v$|
|9. $\tau = \frac{dL}{dt} \leftarrow$ newton's 2nd law|9. $F=\frac{dp}{dt}$|

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<footer style = "font-size: 18px">Work Energy Theorem $\rightarrow$ Work Energy Theorem $\rightarrow$ <span style = "color:blue"> Rational Vs Linear motion </span> $\rightarrow$ Thank You $\rightarrow$  </footer>