physics-class-11-unit-07-chapter-09-problem-s1-motion-of-system-of-particles-and-rigid-bodies-l-9-10-pvvxz7zi7qi

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Question 1 </primary>
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- A bead is slide along the parabola without friction. The wire is accelerated parallel to the x axis with acceleration ' $a$ '. Find the new equilibrium position.

- $ N \cos \theta=m g-(1) $

- $ N \sin \theta=m a-(2)$

- $  \tan \theta=\frac{a}{g}$

- $ y=k x^2$, $\frac{d y}{d x}=2 k x={\tan \theta}$

- $x= \frac{a}{2kg}$

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<footer style = "font-size: 18px"> $\rightarrow$ Problem session-1 Motion of system of particles and rigid bodies $\rightarrow$ <span style = "color:blue"> Question 1 </span> $\rightarrow$ Question 2 $\rightarrow$ Uniform Circular Disk </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Question 2 </primary>
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- A particle is going around the circle with angular velocity $ \omega=5 \mathrm{rad} / \mathrm{sec} $, and the radius of the circle is $ r=4 {m}$. To calculate, the speed of the foot of the perpendicular. When OP sweeps $ \theta=30^{\circ}$.

- $ O P^{\prime}=x=r \cos \theta $

- $ \frac{d x}{d t}=-r \sin \theta(\frac{d \theta}{d t})=-r \omega \sin \theta $

- $|v\vec{p^{\prime}}|=(-4 \times 5 \times \sin 30)=10 \mathrm{r} / \mathrm{s}$

- Angular velocity of $P$ about $Q$

- The required angle $\angle MQP$ $=\frac{\theta}{2}=\frac{5}{2}=2.5 \mathrm{rad} / \mathrm{sec}$

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<footer style = "font-size: 18px">Problem session-1 Motion of system of particles and rigid bodies $\rightarrow$ Question 1 $\rightarrow$ <span style = "color:blue"> Question 2 </span> $\rightarrow$ Uniform Circular Disk $\rightarrow$ Multiple Choice Question </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Uniform Circular Disk </primary>
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- To calculated the $\mathrm{CM}$ of the remaining portion.
- $X_{C M}=\frac{m_1 x_1+m_2 x_2}{m_1+m_2}$

- $\sigma^{\prime}$ : mass per unit area of the disk

- $x_1  =\pi\left(\frac{l}{2}\right)^2 \sigma ; x_1=R / 2$

- $m_2  =\pi\left[R^2-\left(\frac{R}{2}\right)^2\right] \sigma  =\frac{3 \pi R^2}{4}$

- $X_{C M}=\frac{(\pi \frac{R^2}{4} \sigma \frac{R}{2}+\pi(\frac{3 \pi R^2}{4} \sigma) x}{\pi R^2 \rho}=0 $

- $\frac{R}{8}+\frac{3 x}{4}=0 \Rightarrow x=-\frac{R}{6} $

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<footer style = "font-size: 18px">Question 1 $\rightarrow$ Question 2 $\rightarrow$ <span style = "color:blue"> Uniform Circular Disk </span> $\rightarrow$ Multiple Choice Question $\rightarrow$ Problem on Moment of Inertia </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Multiple Choice Question </primary>
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- Problem involving moment of inertia, $ \omega$, linear velocity, rotational kinetic energy, orbital angular momentum, etc.

- Given a symmetric body is rotating about axis.

- Rotational kinetic energy: $ K E_{\text {Rot }}=K E=\frac{1}{2} I \omega^2-(1)$

- $ l=I \omega $

- $ \frac{K E}{l}=\frac{\omega}{2} \Rightarrow \omega=\frac{2 K E}{l} $

- $ I=\frac{l}{\omega}=\frac{l^2}{2 K E}$

- **$\omega$  is** (A)$\frac{{2KE}}{l}$, (B)$\frac{K E}{l}$, (C)$ \frac{K E}{2 l}$, (D)$ \frac{K E}{\sqrt{2 l}} $

- $ \omega=\frac{K E}{l} $
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<footer style = "font-size: 18px">Question 2 $\rightarrow$ Uniform Circular Disk $\rightarrow$ <span style = "color:blue"> Multiple Choice Question </span> $\rightarrow$ Problem on Moment of Inertia $\rightarrow$ Problem on Moment of Inertia </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Problem on Moment of Inertia </primary>
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- Three rods each of length L, joined to form an equilateral triangle.

- mass of each rod : m

- length of each rod : l

- $\angle$ DQO $= {30^{\circ}} $

- $\frac{I}{z} = \frac{I}{{z^{\prime}}} + md^2$

- $ \tan 30^{\circ}$$=\frac{d}{l/ 2}=\frac{1}{\sqrt{3}} \Rightarrow d=\frac{l}{2 \sqrt{3}}$

- $ I_z =\frac{m l^2}{12} + m(\frac{l}{2 \sqrt{3}})^2=\frac{m l^2}{b}$

- $ I_{\text {system }}=3 \times \frac{3 L^2}{6}=\frac{m L^2}{2}$

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<footer style = "font-size: 18px">Uniform Circular Disk $\rightarrow$ Multiple Choice Question $\rightarrow$ <span style = "color:blue"> Problem on Moment of Inertia </span> $\rightarrow$ Problem on Moment of Inertia $\rightarrow$ Problem Involving Torque </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Problem on Moment of Inertia </primary>
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- Two solid sphers have the same mass, made of materials of different densities. Which will have larger MI about an axis passing through the center.

- $ I_1=\frac{2}{5} m r_1^2, I_2=\frac{2}{5} m r_2^2$

- $ \frac{I_1}{I_2}=\frac{r_1^2}{r_2^2} $

- $ m_1=m=\frac{4}{3} \pi r_1^3 s_1 \Rightarrow r_1^3=\frac{3 m}{4 \pi s_1} $

- $r_1^2=\left(\frac{8 m}{4 a^3}\right)^{2 / 3}$

- $ m_2=m=\frac{4}{3} \pi r_2^3 s_2 \Rightarrow r_2^3=\frac{3 m}{4 \pi s_2} $

- $r_2^2=\left(\frac{8 m}{4 a^3}\right)^{2 / 3}$

- $\frac{I_1}{I_2} \propto (\frac{s_2}{s_1})^{2/3} \Rightarrow I \propto $ $\frac{1}{\rho^{2/3}}$

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<footer style = "font-size: 18px">Multiple Choice Question $\rightarrow$ Problem on Moment of Inertia $\rightarrow$ <span style = "color:blue"> Problem on Moment of Inertia </span> $\rightarrow$ Problem Involving Torque $\rightarrow$ Shell Problem </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Problem Involving Torque </primary>
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- To find the point  of application of the force.

- <span style="color:green">Taking moments about c:</span>

- $ \tau_c=0 $

- $ \tau_D=30 \times 3=90 {L}$

- $ 90=10 \times x \Rightarrow x=9 {m}$

- In fact, one can put $\times N$ at $B$, then

- $90  =x(10) $

- $\Rightarrow$ X  =9 N

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<footer style = "font-size: 18px">Problem on Moment of Inertia $\rightarrow$ Problem on Moment of Inertia $\rightarrow$ <span style = "color:blue"> Problem Involving Torque </span> $\rightarrow$ Shell Problem $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Problem S1 Motion Of System Of Particles And Rigid Bodies L-9 </Success>
### <primary style="font-size:24px"> Shell Problem </primary>
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- Thin spherical shell, the shell rolls without slipping. To find the acceleration of the shell.

- Translational motion: F + f = ma ....(1)

- Rotational motion

- $F R-f R=I \alpha=I(\frac{a}{R})$  ....(2)

- $F-f=\frac{I a}{R^2}(3)$

- (l)+(2)

- $2 F=(m+\frac{I}{R^2}) a$

- $=(m+\frac{2 /3 m R^2}{R^2}) a \Rightarrow a=\frac{6 R}{5 m}$

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<footer style = "font-size: 18px">Problem on Moment of Inertia $\rightarrow$ Problem Involving Torque $\rightarrow$ <span style = "color:blue"> Shell Problem </span> $\rightarrow$ Thank You $\rightarrow$  </footer>