physics-class-11-unit-05-chapter-08-problem-solving-newtons-second-law-lecture-8-8-bhg5yruubxc

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Newton's Second Law </primary>
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- <span style="color:green"> $\vec{F}$=m $\vec{a}$</span>
 
- $\vec{a}$ is as measured from an inertial reference frame

- $\vec{F}$: Information is going to come from the free body diagram.

- $\vec{a}$: <span style="color:green">Kinematics</span>
	
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<footer style = "font-size: 18px"> $\rightarrow$ Problem Solving Newton's Second Law $\rightarrow$ <span style = "color:blue"> Newton's Second Law </span> $\rightarrow$ Multiple Bodies $\rightarrow$ Multiple Bodies </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Multiple Bodies </primary>
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- For example: hanging pulley

- Top pulley may be fixed, the bottom pulley, may be moving.

- We could have complex cases like this example.

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<footer style = "font-size: 18px">Problem Solving Newton's Second Law $\rightarrow$ Newton's Second Law $\rightarrow$ <span style = "color:blue"> Multiple Bodies </span> $\rightarrow$ Multiple Bodies $\rightarrow$ Newton's 3rd Law </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Newton's Law for Rod </primary>
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- <span style="color:green"> $ \Sigma \vec{F}$=m $\vec{a}$</span>

- Rod is light, we are assuming mass is nearly <span style="color:green">0</span>.

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<footer style = "font-size: 18px">Multiple Bodies $\rightarrow$ Newton's 3rd Law $\rightarrow$ <span style = "color:blue"> Newton's Law for Rod </span> $\rightarrow$ Newton's Law for Rod $\rightarrow$ Friction </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Newton's Law for Rod </primary>
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- <span style="color:blue">$\vec{F_1}= \vec{F_2}$</span>

- Case:1 Rod is in compression.

- Case:2 Rod is in tension.

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<footer style = "font-size: 18px">Newton's 3rd Law $\rightarrow$ Newton's Law for Rod $\rightarrow$ <span style = "color:blue"> Newton's Law for Rod </span> $\rightarrow$ Friction $\rightarrow$ Friction </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Friction </primary>
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- No slip then friction f is an unknown force.

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<footer style = "font-size: 18px">Newton's Law for Rod $\rightarrow$ Newton's Law for Rod $\rightarrow$ <span style = "color:blue"> Friction </span> $\rightarrow$ Friction $\rightarrow$ Friction </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Friction </primary>
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- Impending slip, $\rightarrow$ f=$\mu_s N$

- Actual slip, $\rightarrow$ f=$\mu_k N$

- Friction $\propto$ N

- <span style="color:green">**1) Assume no slip**</span>

- When there is no movement acceleration of body is 0.

- Find the value of f, from equation.

- $\Sigma F =0$.

- No slip then <span style="color:green">friction</span> f is an unknown force.

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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-08-problem-solving-newtons-second-law-lecture-8_8-bhg5yruubxc-02.jpg)

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<footer style = "font-size: 18px">Newton's Law for Rod $\rightarrow$ Friction $\rightarrow$ <span style = "color:blue"> Friction </span> $\rightarrow$ Friction $\rightarrow$ Friction </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Friction </primary>
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- a $\rightarrow$ <span style="color:blue">Unknown, solve for 'a'</span>

- **Check:**
 
- Is f <$\mu_s N$

- f <$\mu_k N$

- You will have to go to the y direction equations get the value of N.

- $f<\mu_s N=$ <span style="color:green">Assumption</span>

- Else resolve the problem assume slip, $f=\mu_k N$

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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-08-problem-solving-newtons-second-law-lecture-8_8-bhg5yruubxc-198-0933.6.jpg)
		
![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-08-problem-solving-newtons-second-law-lecture-8_8-bhg5yruubxc-04.jpg)

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<footer style = "font-size: 18px">Friction $\rightarrow$ Friction $\rightarrow$ <span style = "color:blue"> Friction </span> $\rightarrow$ Strings and pulleys $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Strings and pulleys </primary>
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- Pulley $P_1$ (fixed) on end of string, ties to mass $m_1$, other end to mass $m_2$.

- Assumed that the <span style="color:green">pulleys</span> are frictionless and light.

- String is inextensible which means the length of the string is constant.

- Find acceleration $a_1$ and acceleration $a_2$ of masses $m_1$ and $m_2$.

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<footer style = "font-size: 18px">Friction $\rightarrow$ Friction $\rightarrow$ <span style = "color:blue"> Strings and pulleys </span> $\rightarrow$ Free Body Diagram $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free Body Diagram </primary>
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- FBD of body 2

- $m_2 g-T=m_2 a_2$

- <span style="color:green">FBD of body 1 :</span>

- $\Sigma F_y $: $N_1=m g$

- $T=m_1 a_1$

- $a_1 = a_2 $

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<footer style = "font-size: 18px">Friction $\rightarrow$ Strings and pulleys $\rightarrow$ <span style = "color:blue"> Free Body Diagram </span> $\rightarrow$ Free Body Diagram $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free Body Diagram </primary>
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- i) When a string passes over a fixed pulley then the accelerations (magnitudes) of the bodies tied on the two ends are equal.

- ii) The tension in the string is equal.

- <span style="color:green">$T_1 = T_2$.</span>

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<footer style = "font-size: 18px">Strings and pulleys $\rightarrow$ Free Body Diagram $\rightarrow$ <span style = "color:blue"> Free Body Diagram </span> $\rightarrow$ Free Body Diagram $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free Body Diagram </primary>
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- If we add friction on the table for <span style="color:green">mass 1</span>

- $m_2 g-T =m_2 a $

- $N_1 =m_1 g $

- $T-f  =m_1 a$

- Where: $f = \mu_k N_1$

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<footer style = "font-size: 18px">Free Body Diagram $\rightarrow$ Free Body Diagram $\rightarrow$ <span style = "color:blue"> Free Body Diagram </span> $\rightarrow$ Fixed Pulley $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Fixed Pulley </primary>
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- Pulley $P_1$ (Fixed)

- At the two ends the magnitude of accelerations will be equal.

- The tension will be the same throughout the string.

- We assume frictionless contacts at the slope as well as at the table.

- Draw the free body diagram of body 1.

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<footer style = "font-size: 18px">Free Body Diagram $\rightarrow$ Free Body Diagram $\rightarrow$ <span style = "color:blue"> Fixed Pulley </span> $\rightarrow$ Free Body Diagram $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free Body Diagram </primary>
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- <span style="color:green"> $N_1=m_1 g$</span>

- <span style="color:green"> $T=m_1 a$</span>

- There are two unknowns, only one equation.

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<footer style = "font-size: 18px">Free Body Diagram $\rightarrow$ Fixed Pulley $\rightarrow$ <span style = "color:blue"> Free Body Diagram </span> $\rightarrow$ Free Body Diagram $\rightarrow$ Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free Body Diagram </primary>
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- $N_2=m_2 g \cos \theta$

- $m_2 g \sin \theta -T = m_2 a$

- There are two unknowns <span style="color:green">T</span> and <span style="color:green">a</span>.

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<footer style = "font-size: 18px">Fixed Pulley $\rightarrow$ Free Body Diagram $\rightarrow$ <span style = "color:blue"> Free Body Diagram </span> $\rightarrow$ Pulley $\rightarrow$ Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Pulley </primary>
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- Accelerations of blocks connected to the ends of same string passing over a fixed pulley have the same magnitude.

- Fixed or moving pulleys

- The pulley is light and frictionless: <span style="color:green">$T_1=T_2$</span>

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<footer style = "font-size: 18px">Free Body Diagram $\rightarrow$ Free Body Diagram $\rightarrow$ <span style = "color:blue"> Pulley </span> $\rightarrow$ Pulley $\rightarrow$ Movable Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Pulley </primary>
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- Problems with multiple accelerating pulleys and string, set up equations relating accelerations of various blocks/ bodies connected by string.

- Done by relating coordinate of moving blocks and pulleys to the length of the string which is fixed.

- Using the length of the string is fixed, find the accelerations and relation between them.

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<footer style = "font-size: 18px">Free Body Diagram $\rightarrow$ Pulley $\rightarrow$ <span style = "color:blue"> Pulley </span> $\rightarrow$ Movable Pulley $\rightarrow$ Movable Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Movable Pulley </primary>
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- Pulley is not fixed.

-  $m_1$, $m_2$ and F are given.

- Frictionless contact

- Coordinate of mass $m_1$ (reference should be a fixed )

- <span style="color:green">$ x_1$</span> = coordinate of $m_1 $

- <span style="color:green">$x_2$</span>= centre of pulley.

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

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Movable Pulley </primary>
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- **What is the length of the string going on the pulley**

* Express length in terms of  <span style="color:green">$x_1$</span> and <span style="color:green">$x_2$</span>.
* $L=2 x_2-x_1$
* Differentiate with respect to time.
* $0=2 \dot{x}_2-\dot{x}_1$
* <span style="color:blue"> Differentiate $2^{\text {nd }}$ time</span>
* $0=2 \ddot{x}_2-\ddot{x}_1 $
* ${x} _1=2 \ddot{x} _2$

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<footer style = "font-size: 18px">Pulley $\rightarrow$ Movable Pulley $\rightarrow$ <span style = "color:blue"> Movable Pulley </span> $\rightarrow$ Movable Pulley $\rightarrow$ Free Body Diagram </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Movable Pulley </primary>
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- $a_1=2 a_2$

- <span style="color:blue"><ins>Generalize:</ins></span> If one end of a string passing over a moving pulley is fixed, acceleration of the other end is twice the accelerator of pulley.

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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-08-problem-solving-newtons-second-law-lecture-8_8-bhg5yruubxc-16.jpg)
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<footer style = "font-size: 18px">Movable Pulley $\rightarrow$ Movable Pulley $\rightarrow$ <span style = "color:blue"> Movable Pulley </span> $\rightarrow$ Free Body Diagram $\rightarrow$ Free body Diagram of the Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free body Diagram of the Pulley </primary>
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- $2T_1 = T_2$

- $a_1 = 2 a_2$

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/5132e74d-3c3c-4735-f6a4-aa492ee5fa00/public)

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<footer style = "font-size: 18px">Movable Pulley $\rightarrow$ Free Body Diagram $\rightarrow$ <span style = "color:blue"> Free body Diagram of the Pulley </span> $\rightarrow$ Free body Diagram of the Pulley $\rightarrow$ Fixed and Movable Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Fixed and Movable Pulley </primary>
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- Find <span style="color:green">$a_1, a_2, a_3 $</span> assume frictioless surface, pulleys light pulley. constant length String.

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/59af8c55-b3e0-4bbc-bf54-c5e7463fa000/public)

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<footer style = "font-size: 18px">Free body Diagram of the Pulley $\rightarrow$ Free body Diagram of the Pulley $\rightarrow$ <span style = "color:blue"> Fixed and Movable Pulley </span> $\rightarrow$ Fixed and Movable Pulley $\rightarrow$ Free body Diagram of the Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Fixed and Movable Pulley </primary>
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- Draw free body diagram of bodies 2 and 3.

- $m_{2} g-T_2=m_2 a_2$

- $m_{2} g-T_2=m_3 a_3$

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/eb70d0cd-feb0-4551-8b5c-462020b0f400/public)

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<footer style = "font-size: 18px">Free body Diagram of the Pulley $\rightarrow$ Fixed and Movable Pulley $\rightarrow$ <span style = "color:blue"> Fixed and Movable Pulley </span> $\rightarrow$ Free body Diagram of the Pulley $\rightarrow$ Fixed and Movable Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Free body Diagram of the Pulley </primary>
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- FBD of $P_2$

- $T_1 = 2T_2$

- FBD of $m_1$

- $2 T_2=m_1 a_1 $

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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-08-problem-solving-newtons-second-law-lecture-8_8-bhg5yruubxc-21.jpg)
		![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-08-problem-solving-newtons-second-law-lecture-8_8-bhg5yruubxc-19.jpg)

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<footer style = "font-size: 18px">Fixed and Movable Pulley $\rightarrow$ Fixed and Movable Pulley $\rightarrow$ <span style = "color:blue"> Free body Diagram of the Pulley </span> $\rightarrow$ Fixed and Movable Pulley $\rightarrow$ Fixed and Movable Pulley </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Fixed and Movable Pulley </primary>
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- $a_1 , a_2 , a_3 $, and $T_2 $

- We have four unknowns and we have three equations.

- Find relation between accelerations

- x $\leftarrow$ Distance of mass 1 form right .

- <span style="color:green">$x_1 + x_p = l_1$</span>

- Where: $l_1$ is length of the first string.

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![alt text](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/0d081ec5-e650-423f-82d0-c1814c43f500/public)

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<footer style = "font-size: 18px">Fixed and Movable Pulley $\rightarrow$ Free body Diagram of the Pulley $\rightarrow$ <span style = "color:blue"> Fixed and Movable Pulley </span> $\rightarrow$ Fixed and Movable Pulley $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Problem Solving Newtons Second Law L-8 </Success>
### <primary style="font-size:24px"> Fixed and Movable Pulley </primary>
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* <span style="color:green">$(x_2-x_p)+(x_3-x_p)=l_2$</span>
* $(x_2-x_p)+(x_3-x_p)=l_2 $
* $x_2+x_3-2 x_p=l_2 $
* $x_2+x_3 -2(l_1 - x_1)=l_2 $
* $ x_2 + x_3 + 2 x_1 = l_2 + 2 l_1 $
* <span style="color:green"> $l_2 + 2 l_1$ = Consant</span>
* $\ddot{x}_1 = -a_1$
* $a_2 + a_3 -2a_1 =0$