physics-class-11-unit-07-chapter-10-problem-s02-motion-of-system-of-particles-rigid-bodies-l-10-10-lh-xm6ct2ey

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem on Principle of Conservation of Angular Momentum </primary>
<hr>
<main>

- Two small spheres of mass m each are attached gently to the two ends of the rod. what is the final angular frequency of the system.

- There are no external torques so principle of conservation of angular momentum applies.

- $l_i=I_i \omega_i=\frac{M_L^2}{1^2} \omega_.$
 
- $l_f=I_f \omega_f$
 
- $=\left(\frac{M L^2}{12}+m L^2+m L^2\right) \omega_f$
 
- $\omega_f=\left(\frac{M L^2 / 12}{\frac{M L^2}{1 2}+2ML^2}\right) \omega_1$

</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/84a9b29f-97b0-4360-1021-e3c1cbeed600/public)

</div>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Problem Session-2 Motion of System of Particles and Rigid Bodies $\rightarrow$ <span style = "color:blue"> Problem on Principle of Conservation of Angular Momentum </span> $\rightarrow$ Problem on Principle of Conservation of Angular Momentum $\rightarrow$ Problem on Principle of Conservation of Angular Momentum </footer>

- <span style="color:blue">Case 1 :</span>
 
-  the connecting and is hight.

-  taking moments about p.
 
 - $(m \times l) \times x$
 
 - $=4 m l(2 l-x)$

- $x=\frac{8l}{5}$

</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c1b3c861-992f-4a3c-8baa-3afe37bf3500/public)

</main>
<hr>
<footer style = "font-size: 18px">Problem Session-2 Motion of System of Particles and Rigid Bodies $\rightarrow$ Problem on Principle of Conservation of Angular Momentum $\rightarrow$ <span style = "color:blue"> Problem on Principle of Conservation of Angular Momentum </span> $\rightarrow$ Problem on Principle of Conservation of Angular Momentum $\rightarrow$ Problem </footer>

- <span style="color:green">case 2 : </span>
 
-  The connecting and has the same mass few unit length (m).
 
 - $m l x=2 lm (l-x)+4 lm (2 l-x)$
 
 - $x=\frac{10l}{7}$
 
</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/c1b3c861-992f-4a3c-8baa-3afe37bf3500/public)

</main>
<hr>
<footer style = "font-size: 18px">Problem on Principle of Conservation of Angular Momentum $\rightarrow$ Problem on Principle of Conservation of Angular Momentum $\rightarrow$ <span style = "color:blue"> Problem on Principle of Conservation of Angular Momentum </span> $\rightarrow$ Problem $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem </primary>
<hr>
<main>

- (2) To calculate the angular velocity of the cm
- $l_i = l_f$

- $l_i = (2m v a ) + ( m \times 2v \times 2a) = 6m v a$

- $l_f = I w$

- $=\left[2 m \times a^2+m \times(2a)^2 +\left(8 m \times \frac{(6 a)^2}{12}\right)\right] \omega  = 30 m a^2 w$

- $30 m a^2 w = 6mvx$

- $\omega = \frac{v}{5a}$
 
</div>

![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-10-problem-s02-motion-of-particles-rigid-bodies-l-10_10-lh_xm6ct2ey-05.jpg)

</main>
<hr>
<footer style = "font-size: 18px">Problem on Principle of Conservation of Angular Momentum $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Problem $\rightarrow$ Problem Involving Slipping </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem </primary>
<hr>
<main>

- (3) Kinetic energy due to rotation.

- $KE_{rotation} = \frac{1}{2} Iw^2$

- $ = \frac{1}{2} (30ma^2)(\frac{v}{5x})^2$

- $ = \frac{3}{5}mv^2$
 
</div>

</main>
<hr>
<footer style = "font-size: 18px">Problem $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Problem Involving Slipping $\rightarrow$ Problem Involving Slipping </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem Involving Slipping </primary>
<hr>
<main>

- $\mu$ : the coefficient of friction.

- The rod rotation about A with a constant angular acceleration $(\alpha)$

- $\alpha$ is constant ; $\therefore$ then $\omega = \alpha t$
 
- Linear acceleration of the head: a $=$ l $\alpha$
 
- The reaction force on the bead due to the rod = N = ma = mL $\alpha$

</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/f9d9a444-f290-458d-4ff2-fe35dd1b4400/public)

</main>
<hr>
<footer style = "font-size: 18px">Problem $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Problem Involving Slipping </span> $\rightarrow$ Problem Involving Slipping $\rightarrow$ Problem Involves Slipping Toppling </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem Involving Slipping </primary>
<hr>
<main>

- Centripetal force on the bead : $m\left(m \theta^2\right)$

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

- $=m L(\alpha t)^2=m L \alpha^2 t^2$
 
-  the limiting frictional force $=\mu N=\mu m L \alpha$
 
-  for slipping $\mu ml \alpha=m l\alpha^2 t^2$
 
 - $t=\left(\frac{\mu}{\alpha}\right)^{1 / 2}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Problem $\rightarrow$ Problem Involving Slipping $\rightarrow$ <span style = "color:blue"> Problem Involving Slipping </span> $\rightarrow$ Problem Involves Slipping Toppling $\rightarrow$ Problem Involving Atomic Physics </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem Involves Slipping Toppling </primary>
<hr>
<main>

- Cubical block resting in a rough horizontal surface. The cofficient of friction is high such that the block does not slide before tripping . Calculate the $F_{min.}$ for the block to topple.
 
 - $\sum F_y: \quad N=M_g$
 
 - $\sum F_x: \quad F=f$
 
- Torque equation about c
	
 - $\left(F \times \frac{L}{2}\right)+\left(f \times \frac{L}{2}\right)=N \times \frac{L}{2}$
 
 - $F + f = N$
 
 - $2 F=N \Rightarrow F=\frac{N}{2}=\frac{Mg}{2}$
 
</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5d737218-ae60-46d4-449f-a61abd3f1a00/public)

</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5489ca12-c9ef-42b8-b6a3-fb5f3086e800/public)
</div>

</main>
<hr>
<footer style = "font-size: 18px">Problem Involving Slipping $\rightarrow$ Problem Involving Slipping $\rightarrow$ <span style = "color:blue"> Problem Involves Slipping Toppling </span> $\rightarrow$ Problem Involving Atomic Physics $\rightarrow$ Problem Involving Atomic Physics </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem Involving Atomic Physics </primary>
<hr>
<main>

- Diatomic molecule, rotational frequency and quantum theory.

- x : reparation between the atoms

- Given oxygen atoms
 
- The separation between the atoms, $ x = 1.20 \times 10^{-10} m$

- Mass of the oxygen atoms $= 2.66\times 10^{-26} kg$

- Calculate the rotational frequency $\omega$?
 
- $I=\sum m_i r_i^2=m\left(\frac{x}{2}\right)^2+m\left(\frac{x}{2}\right)^2=\frac{m x^2}{2}$

- The fundamental unit of angular momentum (according to quantum theory) $=\hbar$

- $\hbar=1.054 \times 10^{-34} \frac{\mathrm{kg} \mathrm{m}^2}{s}.$
 
</div>
	
<div style='float: right; width:1%;'>

</main>
<hr>
<footer style = "font-size: 18px">Problem Involving Slipping $\rightarrow$ Problem Involves Slipping Toppling $\rightarrow$ <span style = "color:blue"> Problem Involving Atomic Physics </span> $\rightarrow$ Problem Involving Atomic Physics $\rightarrow$ Problem Involving Angular Momentum </footer>

- $I \omega \approx \hbar$

- $\omega=\frac{\hbar}{2}$

- =$\frac{1.054 \times 10^{34} kgm^2/s}{(\frac{2.66 \times 10^{-26}kg}{2})(1.20 \times 10^{-10}m)^2}$

- $\simeq 5.2 \times 10^{11} \mathrm{rad} / \mathrm{sec}$

</div>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Problem Involving Atomic Physics $\rightarrow$ <span style = "color:blue"> Problem Involving Atomic Physics </span> $\rightarrow$ Problem Involving Angular Momentum $\rightarrow$ Problem Involving Angular Momentum </footer>

### <Success style="font-size:25px"> Problem S02 Motion Of System Of Particles Rigid Bodies L-10 </Success>
### <primary style="font-size:24px"> Problem Involving Angular Momentum </primary>
<hr>
<main>

<div style='float: left; width:50%; text-align:left; font-size:27px;'>
 
- A rigid rod of mass m and length L rotates in a vertical plane about a frictionless pivot through the center.

- 1. The moment of inertia of the system

- $I=\frac{M L^2}{12}+m_1\left(\frac{L}{2}\right)^2+m_2\left(\frac{L}{2}\right)^2=\frac{L^2}{4}\left(\frac{M}{3}+M_1+M_2\right)$

- 2. One ' $\omega$ ' is known ' $l$ ' can be calculated

- $l=I \omega$
 
 - $=\frac{L^2}{4}\left(\frac{m}{3}+m_1+m_2\right) \omega$

</div>

![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/58b8d048-63a2-46d5-fb36-1b3e2b2f0700/public)

</main>
<hr>
<footer style = "font-size: 18px">Problem Involving Atomic Physics $\rightarrow$ Problem Involving Atomic Physics $\rightarrow$ <span style = "color:blue"> Problem Involving Angular Momentum </span> $\rightarrow$ Problem Involving Angular Momentum $\rightarrow$ Thank you </footer>

- 3. $\tau_1=m_1 g \frac{l}{2} \text { cos $\theta$ }(\odot \equiv \text { one of the plane })$
 
 - $\tau_2=m_2 g \frac{1}{2} \cos \theta(\otimes \equiv \text { into the phane })$
 
 - $\tau_{\text {Total }}=\frac{1}{2}\left(m_1-m_2\right) q l \cos \theta$
 
 - $\odot , if m_1 > m_2$
 
 - $\otimes if m_2 < m$
 
 - $\text { Since } I \alpha=l \text {, we can calculate } \alpha$
 
 - $\alpha=\frac{\tau_ {\text {Total }}}{I} =\frac{2\left(m_ 1-m_ 2\right) g \operatorname{cos \theta}}{\frac{M}{3}+m_ 1+m_2}$

</div>

</main>
<hr>
<footer style = "font-size: 18px">Problem Involving Atomic Physics $\rightarrow$ Problem Involving Angular Momentum $\rightarrow$ <span style = "color:blue"> Problem Involving Angular Momentum </span> $\rightarrow$ Thank you $\rightarrow$  </footer>