physics-class-11-unit-05-chapter-02-forces-on-bodies-problems-involving-multiple-bodies-lecture-2-8-hxfwwafxxry

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Multiple Bodies Connected by Strings </primary>
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- $m_1=10 \mathrm{kg}$

- $m_2=20 \mathrm{kg}$

- $m_3=30 \mathrm{kg}$

- $m_3$ is bring pulled to right by a force $T_3$=100N

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<footer style = "font-size: 18px"> $\rightarrow$ Forces on Bodies Problems Involving Multiple Bodies $\rightarrow$ <span style = "color:blue"> Multiple Bodies Connected by Strings </span> $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ Multiple Bodies Connected by Strings </footer>

- Find acceleration of blocks?

- Force in string connecting $m_1$ and $m_2$ and the string connecting $m_2$ and $m_3$.

- **<span style="color:green">Solution:</span>**

- Strings are inextensible, lengths are constant. distance moved by block 1, 2 and 3 are same.

- $a_1=a_2=a_3$(Kinematical constraint)

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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-05-chapter-02-forces-on-bodies-problems-involving-multiple-bodies-2_8-hxfwwafxxry-044-0270.0.jpg)

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<footer style = "font-size: 18px">Forces on Bodies Problems Involving Multiple Bodies $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ <span style = "color:blue"> Multiple Bodies Connected by Strings </span> $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ Multiple Bodies Connected by Strings </footer>

|FBD | Equation |
| -------- | -------- |
| Body 3   | $T_3-T_2=m_3 a_3$     |
| Body 2    | $T_2-T_1=m_2 a_2$     |

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<footer style = "font-size: 18px">Multiple Bodies Connected by Strings $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ <span style = "color:blue"> Multiple Bodies Connected by Strings </span> $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ Multiple Bodies Connected by Strings </footer>

- $T_1=m_1 a_1$

- $T_3-T_2  =m_3 a_3$

- $T_2-T_1  =m_2 a_2$

- $T_1  =m_1 a_1$

- $a_1=a_2=a_2=a$

- $T_3=\left(m_1+m_2+m_3\right) a$

- $a=\frac{T_3}{m_1+m_2+m_3}$

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- $T_3-T_2=m_3a$

- $T_2=T_3-m_3a$

- $T_1=m_1a$

- $a=\frac{5}{3} \mathrm{m} / \mathrm{s}^2$

- $T_1=10a$

- $T_2=30a$

- $T_3=60a$

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- Like on engine pulling train comparment

- If masses $m_1,m_2, m_3$ are equal.

- $T_3=3 ma$

- $T_2=2 ma$

- $T_1= ma$

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<footer style = "font-size: 18px">Multiple Bodies Connected by Strings $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ <span style = "color:blue"> Multiple Bodies Connected by Strings </span> $\rightarrow$ Body in Uniform Circular Motion $\rightarrow$ Body in Uniform Circular Motion </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body in Uniform Circular Motion </primary>
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- $\vec{v}$ has a constant magnitute =v

- Component of acceleration pointing towards centre.

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<footer style = "font-size: 18px">Multiple Bodies Connected by Strings $\rightarrow$ Multiple Bodies Connected by Strings $\rightarrow$ <span style = "color:blue"> Body in Uniform Circular Motion </span> $\rightarrow$ Body in Uniform Circular Motion $\rightarrow$ Conical Pendulum </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body in Uniform Circular Motion </primary>
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- $a_r=\frac{v^2}{R}$

- <span style="color:blue">This acceleration has to be provided bynsome force on the body.</span>

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<footer style = "font-size: 18px">Multiple Bodies Connected by Strings $\rightarrow$ Body in Uniform Circular Motion $\rightarrow$ <span style = "color:blue"> Body in Uniform Circular Motion </span> $\rightarrow$ Conical Pendulum $\rightarrow$ Conical Pendulum </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Conical Pendulum </primary>
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- Pendulum

- Simple pendulum

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<footer style = "font-size: 18px">Body in Uniform Circular Motion $\rightarrow$ Body in Uniform Circular Motion $\rightarrow$ <span style = "color:blue"> Conical Pendulum </span> $\rightarrow$ Conical Pendulum $\rightarrow$ Conical Pendulum </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Conical Pendulum </primary>
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- Conical pendulum moves in a circle of radius R

- $R=L \sin \theta$

- Height of pendulum from the plane of motion of ball
- $ =L \cos \theta$

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<footer style = "font-size: 18px">Body in Uniform Circular Motion $\rightarrow$ Conical Pendulum $\rightarrow$ <span style = "color:blue"> Conical Pendulum </span> $\rightarrow$ Conical Pendulum $\rightarrow$ Conical Pendulum </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Conical Pendulum </primary>
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- **FBD of ball**

- $\overrightarrow{a_r}=\frac{v^2}{R^2}$

- z axis: $T \cos \theta-m g=0$

- r axis: $T \sin \theta  =ma_r =m v^2 / R$

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<footer style = "font-size: 18px">Conical Pendulum $\rightarrow$ Conical Pendulum $\rightarrow$ <span style = "color:blue"> Conical Pendulum </span> $\rightarrow$ Conical Pendulum $\rightarrow$ Conical Pendulum </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Conical Pendulum </primary>
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- $\tau=\frac{\text { circular distance }}{\text { velecity }}$

- $\quad =\frac{2 \pi R}{v}=\frac{2 \pi R}{\sqrt{g R \tan \theta}}$

- $R=L \sin \theta$

- $\tau=2 \pi \sqrt{\frac{L \cos \theta}{g}}$

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<footer style = "font-size: 18px">Conical Pendulum $\rightarrow$ Conical Pendulum $\rightarrow$ <span style = "color:blue"> Conical Pendulum </span> $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ Body Moving on a Circular Track </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body Moving on a Circular Track </primary>
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- We will assume that the body is moving with a constant speed.

- Radial component of the acceleration is what is provided by the friction.

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/681c7826-0b24-4c6e-5d23-69c6e69ba600/public)

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<footer style = "font-size: 18px">Conical Pendulum $\rightarrow$ Conical Pendulum $\rightarrow$ <span style = "color:blue"> Body Moving on a Circular Track </span> $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ Body Moving on a Circular Track </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body Moving on a Circular Track </primary>
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- We will assume that the friction force which is in the direction of the motion of the car.

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<footer style = "font-size: 18px">Conical Pendulum $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ <span style = "color:blue"> Body Moving on a Circular Track </span> $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ Body Moving on a Circular Track </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body Moving on a Circular Track </primary>
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- Vertical direction is z

- FBD of the particle, and we will look at r-z plane.

- z is coming out of paper.

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<footer style = "font-size: 18px">Body Moving on a Circular Track $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ <span style = "color:blue"> Body Moving on a Circular Track </span> $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ Body Moving on a Circular Track </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body Moving on a Circular Track </primary>
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- $N=m g \quad\left(a_3=0\right)$

- $f=\frac{m v^2}{R}$

- (r direction)

- Friction force between road and the particle in the r direction provides the centripetal acceleration.

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### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body Moving on a Circular Track </primary>
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- f has a limiting value,

- And maximum value of f $=\mu_s N$

- Once the body moves $f=\mu_k N$

- $N=m g  \left(a_3=0\right)$

- $f=\frac{m v^2}{R}$  (r direction)

- Friction force between raod and particle in the r direction provided centripetal acceleration.

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### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Body Moving on a Circular Track </primary>
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- $f=\frac{m v^2}{R}$

- N=mg

- f = $\mu_s N=\mu_s m g$

- If particle moves along the circle with a large v or less R

- It is possible f < $\frac{m v^2}{R}$.

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<footer style = "font-size: 18px">Body Moving on a Circular Track $\rightarrow$ Body Moving on a Circular Track $\rightarrow$ <span style = "color:blue"> Body Moving on a Circular Track </span> $\rightarrow$ Uniform Circulation Motion $\rightarrow$ Uniform Circulation Motion </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Banking of the Road </primary>
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- To reduce skidding, the highway is tilted at an angle.

- Banking of the road: the vehicle on road is tilted at an angle.

- N is at an angle

- $mg= \cos \theta$

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<footer style = "font-size: 18px">Uniform Circulation Motion $\rightarrow$ Uniform Circulation Motion $\rightarrow$ <span style = "color:blue"> Banking of the Road </span> $\rightarrow$ Banking of the Road $\rightarrow$ Banking of the Road </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Banking of the Road </primary>
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- Component of N $(N \sin \theta)$

- This provides the acceleration in the r direction.

- Assuming friction force f=O

- $N \sin \theta=\frac{m v^2}{R}$

- $N\cos\theta=mg$

- $\angle \theta$ : for a well designed road

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<footer style = "font-size: 18px">Uniform Circulation Motion $\rightarrow$ Banking of the Road $\rightarrow$ <span style = "color:blue"> Banking of the Road </span> $\rightarrow$ Banking of the Road $\rightarrow$ Friction Force </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Friction Force </primary>
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- <span style="color:blue">Friction force on a body in contact with fluid (liquid or gas) airplane travelling in air </span>

- Layer of oil between block and the table.

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<footer style = "font-size: 18px">Banking of the Road $\rightarrow$ Banking of the Road $\rightarrow$ <span style = "color:blue"> Friction Force </span> $\rightarrow$ Contact Between Two Solids $\rightarrow$ Fluid Friction </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Contact Between Two Solids </primary>
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| $f \leqslant \mu N$  | Friction Force |
| -------- | -------- |
| $f<\mu N$     | Do not move     |
| $f=\mu N$     | Relative motion     |

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<footer style = "font-size: 18px">Banking of the Road $\rightarrow$ Friction Force $\rightarrow$ <span style = "color:blue"> Contact Between Two Solids </span> $\rightarrow$ Fluid Friction $\rightarrow$ Fluid Friction </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Fluid Friction </primary>
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- a=O

- Friction force due to air on this body (opposing the velocity v) =<span style="color:blue">Drag force</span>

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<footer style = "font-size: 18px">Friction Force $\rightarrow$ Contact Between Two Solids $\rightarrow$ <span style = "color:blue"> Fluid Friction </span> $\rightarrow$ Fluid Friction $\rightarrow$ Fluid Friction </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Fluid Friction </primary>
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- Drag forces is a function of velocity v

- Solid friction: f $\propto$ N $\leftarrow$ Force

- In case of fluid friction: Drag is function of velocity.

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<footer style = "font-size: 18px">Contact Between Two Solids $\rightarrow$ Fluid Friction $\rightarrow$ <span style = "color:blue"> Fluid Friction </span> $\rightarrow$ Fluid Friction $\rightarrow$ Fluid Friction </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Fluid Friction </primary>
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- <span style="color:blue">Body moves at very slow speed</span>

- Drag(D) $\propto$ v

- <span style="color:blue">Body move at high speed</span>

- Drag(D) $\propto$ $v^2$

- <span style="color:blue">For high speed</span>

- $D=\frac{1}{2} C \rho_f v^2 A$

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<footer style = "font-size: 18px">Fluid Friction $\rightarrow$ Fluid Friction $\rightarrow$ <span style = "color:blue"> Fluid Friction </span> $\rightarrow$ Fluid Friction $\rightarrow$ Fluid Friction </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Fluid Friction </primary>
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- $A=\pi R^2$

- <span style="color:blue">Body falling in a fluid</span>

- $D=\frac{1}{2} C_D \rho_f v^2 A$

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### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- <span style="color:blue">Example:</span> Raindrop of radius R=1.5 mm, which is falling from a cloud of height h=1500m.

- Given:  C=0.6

- $\rho_w=1000 \frac{\mathrm{kg}}{\mathrm{m}^3}$,

- $\rho_{\text {air }}=1.2 \frac{\mathrm{kg}}{\mathrm{m}^3}$

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- $m g=D=\frac{1}{2} C \rho_f V_t^2 A$

- $m=\rho_w \cdot \frac{4}{3} \pi R^3$

- $A=\pi R^2$

- <span style="color:blue">Terminal velocity of raindrop</span>: $v_t=\sqrt{\frac{8 R \rho_w g}{3 C \rho_a}} \leftarrow 7.4 \mathrm{~m} / \mathrm{s}$

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<footer style = "font-size: 18px">Fluid Friction $\rightarrow$ Fluid Friction $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Terminal Velocity $\rightarrow$ Viscocity </footer>

### <Success style="font-size:25px"> Forces On Bodies Problems Involving Multiple Bodies L-2 </Success>
### <primary style="font-size:24px"> Terminal Velocity </primary>
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- <span style="color:blue">Galileo</span> leaning tower of pisa

- Free fall : stone /feather

- Reach ground at the same time

- Had D=0

- Falling a hight of 1500 m

- $v=\sqrt{2 \times g \times 1500}$

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<footer style = "font-size: 18px">Fluid Friction $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Terminal Velocity </span> $\rightarrow$ Viscocity $\rightarrow$ Components </footer>