physics-class-11-unit-06-chapter-02-work-energy-impulse-principles-conservation-of-momentum-l-2-6-sd3zqspo7c8

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle (Example) </primary>
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- $ m_A=200 \mathrm{kg} $

- $m_B=300 \mathrm{kg}$

- Blocks A and B are connected by a light cable on a frictionless pulley (also massless).

- The system is released from state of rest.

- Find the velocity of block A after it has moved two meters.

- $\mu_k$: <span style="color:green">coefficient of friction</span> between A and table = 0.25

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<footer style = "font-size: 18px"> $\rightarrow$ Work Energy impulse Principles and Conservation of Momentum $\rightarrow$ <span style = "color:blue"> Work Energy Principle (Example) </span> $\rightarrow$ Work Energy Principle (Example) $\rightarrow$ Work Energy Principle (Example) </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle (Example) </primary>
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- Draw the free body diagrams of bodies A and B.

- The string is pulling the body A with a force T.

- If a force T is being applied to pull body A the string will pull body B with the same force T.

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<footer style = "font-size: 18px">Work Energy impulse Principles and Conservation of Momentum $\rightarrow$ Work Energy Principle (Example) $\rightarrow$ <span style = "color:blue"> Work Energy Principle (Example) </span> $\rightarrow$ Work Energy Principle (Example) $\rightarrow$ Work Energy Principle on block </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle (Example) </primary>
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- Forces which are acting:

- String force

- Weight

- The contact force, will consist of a normal reaction.

- Friction force

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<footer style = "font-size: 18px">Work Energy Principle (Example) $\rightarrow$ Work Energy Principle (Example) $\rightarrow$ <span style = "color:blue"> Work Energy Principle (Example) </span> $\rightarrow$ Work Energy Principle on block $\rightarrow$ Work Energy Principle on block </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- $a_y=0$

- $N_A=m_ag$

- $f=\mu_k N_A$

- $ f=\mu_k m_A g  =0.25 \times 200 \times 9.8 =490 \mathrm{N}$

- Work Energy Principle on block

- $ \Delta K+\Delta V$ = <span style="color:green">$W_{\text{other force}}$</span>

- $\text { state: } 1 \rightarrow \text { rest }$

- $\text { state 2} \rightarrow \{v}$

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<footer style = "font-size: 18px">Work Energy Principle (Example) $\rightarrow$ Work Energy Principle (Example) $\rightarrow$ <span style = "color:blue"> Work Energy Principle on block </span> $\rightarrow$ Work Energy Principle on block $\rightarrow$ Work Energy Principle on block </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- $K_2=\frac{1}{2} m_A v^2$

- $K_1=0 $

- This is for change in <span style="color:green">kinetic energy</span>, $ \Delta K$

- The block is moving in a horizontal plane due to gravitational potential energy, $ V_1=V_2$.

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<footer style = "font-size: 18px">Work Energy Principle (Example) $\rightarrow$ Work Energy Principle on block $\rightarrow$ <span style = "color:blue"> Work Energy Principle on block </span> $\rightarrow$ Work Energy Principle on block $\rightarrow$ Work Energy Principle on block </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- The block is moving in a horizontal plane due to gravitational potential energy, <span style="color:green">$ V_1=V_2$</span>.

- We call this as the reference state.

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<footer style = "font-size: 18px">Work Energy Principle on block $\rightarrow$ Work Energy Principle on block $\rightarrow$ <span style = "color:blue"> Work Energy Principle on block </span> $\rightarrow$ Work Energy Principle on block $\rightarrow$ Work Energy Principle on block </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- $K_2=\frac{1}{2} m_A v^2$

- $K_1=0 $

- This is for change in <span style="color:green">kinetic energy</span>, $ \Delta K$

- <span style="color:blue">$ V_1=V_2$</span> $\Rightarrow \Delta V=0$

- <span style="color:green">$W_{\text{other force}}:$</span> $\left(T-\mu_k m_A g\right).s$

- $s = 2 m$

- $\frac{1}{2} m_A v^2=(T-490) 2$

- Now, there are 2 unknowns V and T.

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### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- **Block:2**

- $\Delta K+\Delta V=\text W_{ other force. }$

- $ \Delta K=\left(\frac{1}{2} m_B v^2-0\right)$

- $\Delta V=\left(V_2-V_1\right)\$

- $V_1=0$

- $V_2 =-mgs$

- $\Delta V =-2 m g$

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### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- <span style="color:green">T </span>is acting upwards

- Work done by T on block B: - T(2)

- $\frac{1}{2} m_B v^2-m_B g(2)=-2T$

- (1) + (2)

- $\frac{1}{2} m_A v^2+\frac{1}{2} m_B v^2-m_B g(2)=-2 \times 490$

- $v=4.427 \mathrm{~m} / \mathrm{s}$

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### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Work Energy Principle on block </primary>
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- This problem illustrates that if we consider A and B, then the tension which acts on these bodies separately, but because the work done by tension on body A = -W by tension on body B.

- The work done by T cancels out.

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<footer style = "font-size: 18px">Work Energy Principle on block $\rightarrow$ Work Energy Principle on block $\rightarrow$ <span style = "color:blue"> Work Energy Principle on block </span> $\rightarrow$ Conservation of Energy $\rightarrow$ Momentum </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Conservation of Energy </primary>
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- $\Delta K + \Delta V$ = <span style="color:green">$W_{other force}$</span>

- Also referred to as principle of conservation of mechanical energy.

- This has been derived from Newton's second law.

- Rest: We calculate velocity or displacements, with respect to an inertial frame.

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<footer style = "font-size: 18px">Work Energy Principle on block $\rightarrow$ Work Energy Principle on block $\rightarrow$ <span style = "color:blue"> Conservation of Energy </span> $\rightarrow$ Momentum $\rightarrow$ Impulse </footer>

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### <primary style="font-size:24px"> Impulse </primary>
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- Impulse of a force $\vec{F}$ during time internal from $(t_1 $ to ${t_2})$ $\rightarrow$$\vec{I}$

- $\vec{I}=$ $\int_{t_1}^{t_2} \vec{F} d t$

- If <span style="color:green">$\vec{F}$ is constant</span>, $\vec{I}=\vec{F}\left(t_2-t_1\right)$.

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<footer style = "font-size: 18px">Conservation of Energy $\rightarrow$ Momentum $\rightarrow$ <span style = "color:blue"> Impulse </span> $\rightarrow$ Newton's second law $\rightarrow$ Impulse momentum principle </footer>

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- $\vec{F}=\frac{d \vec{P}}{d t}$ (rate of change of linear momentum $\vec{P}$).

- $\vec{F}dt=d\vec{P}$.

- $\int_{t_1}^{t_2}\vec{F}dt=\int_{P_1}^{P_2}d\vec{P}=\left(\overrightarrow{P_2}-\overrightarrow{P_1}\right) =\Delta \vec{p}\$

- $t_1 \rightarrow \overrightarrow{P_1}$

- $t_2 \rightarrow \overrightarrow{P_1}$

- Impulse of force $\vec{F}$ a particle from  $t_1 $ to ${t_2}$

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<footer style = "font-size: 18px">Momentum $\rightarrow$ Impulse $\rightarrow$ <span style = "color:blue"> Newton's second law </span> $\rightarrow$ Impulse momentum principle $\rightarrow$ Units of impulse </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Impulse momentum principle </primary>
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- $\vec{I}=\Delta\vec{P}$

- The change in linear momentum of a particle  is given by the impulse of the forces acting on the particle.

- Principle of impulse is useful if the force is a function of time.

- $\vec{F}=\vec{F}(t)$

- <span style="colorgreen">Some salient features</span>

- a) Impulse is a vector quantity. Vector equation can write as scalar components.

- $I_x=P_{2, x}-P_{1, x}=m\left(v_{2, x}-v_{ 1 ,x})\right.$

- $I_y=P_{2, y}-P_{1, y}=m\left(v_{2,y}-v_{ 1 ,y})\right.$

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<footer style = "font-size: 18px">Impulse $\rightarrow$ Newton's second law $\rightarrow$ <span style = "color:blue"> Impulse momentum principle </span> $\rightarrow$ Units of impulse $\rightarrow$ Impulse momentum principle </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Units of impulse </primary>
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- The dimension of impulse is force: M $\frac{L}{T^2}.T$

- = $\frac{ML}{T}$

- SI units: <span style="color:blue">N.sec</span>

- For a single particle:

- Impulse methods are useful if <span style="color:green">$\vec{F}$</span> is a function of times.

- Impulse give us change in momentum.

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<footer style = "font-size: 18px">Newton's second law $\rightarrow$ Impulse momentum principle $\rightarrow$ <span style = "color:blue"> Units of impulse </span> $\rightarrow$ Impulse momentum principle $\rightarrow$  </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Impulse momentum principle </primary>
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- $\vec{I}=m(\vec{v_2}-\vec{v_1})$

- $m \overrightarrow {v_2 } =m \overrightarrow{v_1}+\vec{I} $

- $m \overrightarrow{v_1}$ = <span style="color:blue">Initial momentum</span>

- $m \overrightarrow {v_2 }$ = <span style="color:green">Final momentum</span>

- $\vec{I}$ = <span style="color:green">Impulse</span>

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<footer style = "font-size: 18px">Impulse momentum principle $\rightarrow$ Units of impulse $\rightarrow$ <span style = "color:blue"> Impulse momentum principle </span> $\rightarrow$ Area under the F-t curve $\rightarrow$ Conservation of Momentum </footer>

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### <primary style="font-size:24px"> Instantaneous Impulse </primary>
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- $\vec{I_{\text {inst. }}}=\int_{t_1}^{t_1+\epsilon} \vec{F} d t$

- limit $\epsilon\rightarrow{0}$

- $\vec{F}\rightarrow$ <span style="color:green">very large</span> $\rightarrow{\infty}$

- $F_{av.}(\Delta{t})\rightarrow$<span style="color:blue">$\{finite}$</span>

- $F_{av.}$= <span style="color:green">Large</span>

- $(\Delta{t})$= <span style="color:green">Very small</span>

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<footer style = "font-size: 18px">Conservation of Momentum $\rightarrow$ Instantaneous Impulse $\rightarrow$ <span style = "color:blue"> Instantaneous Impulse </span> $\rightarrow$ Instantaneous Impulse $\rightarrow$ Effect of Impulsive Force </footer>

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### <primary style="font-size:24px"> Instantaneous Impulse </primary>
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- eg. Roger Federer hitting a tennis ball.

- The time of contact between the ball and the racket is very small.

- <span style="color:green"> Contact force $\{\rightarrow}$ large.</span>

- Similar example: Virat Kohli hits the cricket ball with bat.

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<footer style = "font-size: 18px">Instantaneous Impulse $\rightarrow$ Instantaneous Impulse $\rightarrow$ <span style = "color:blue"> Instantaneous Impulse </span> $\rightarrow$ Effect of Impulsive Force $\rightarrow$ Impulsive Force </footer>

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- Because Impulsive Force is very large.

- <span style="color:green"> ($t_1$ to $t_1 +\epsilon$) $\longrightarrow$</span>

- During the time period ${\epsilon}$, we neglect the effect of the other finite forces.

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<footer style = "font-size: 18px">Instantaneous Impulse $\rightarrow$ Effect of Impulsive Force $\rightarrow$ <span style = "color:blue"> Impulsive Force </span> $\rightarrow$ Impulsive Force $\rightarrow$ Impulsive Force </footer>

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- Draw force versus time.

- During time period from <span style="color:green">$t_1$ to $t_1 +\epsilon$</span>, only the effect of the impulsive force is counted.

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<footer style = "font-size: 18px">Effect of Impulsive Force $\rightarrow$ Impulsive Force $\rightarrow$ <span style = "color:blue"> Impulsive Force </span> $\rightarrow$ Impulsive Force $\rightarrow$ Uses of Principle of Momentum </footer>

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### <primary style="font-size:24px"> Impulsive Force </primary>
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- Average contact force <span style="color:green">$\overrightarrow{F_{a v}}$</span>.

- Time of contact: <span style="color:blue">$\Delta{t}$</span>.

- $\vec{F}_{\text {av. }} \Delta t=\Delta \vec{p}$

- If know $\vec{P_1}$ and $\overrightarrow{P_2}$, we can find $\vec{F}_{av. }$

- $\vec{P_1}=m\vec{v_{1}}$

- $\vec{P_2}=m\vec{v_{2}}$

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<footer style = "font-size: 18px">Impulsive Force $\rightarrow$ Impulsive Force $\rightarrow$ <span style = "color:blue"> Impulsive Force </span> $\rightarrow$ Uses of Principle of Momentum $\rightarrow$ Conservation of Momentum </footer>

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### <primary style="font-size:24px"> Conservation of Momentum </primary>
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- **Post collision**

- The second type of problem is when we have one body which is moving and it suddenly breaks into two or more parts.

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<footer style = "font-size: 18px">Impulsive Force $\rightarrow$ Uses of Principle of Momentum $\rightarrow$ <span style = "color:blue"> Conservation of Momentum </span> $\rightarrow$ Conservation of Momentum $\rightarrow$ Conservation of Momentum </footer>

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### <primary style="font-size:24px"> Conservation of Momentum </primary>
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- **Case of two particles**

- We treat both the particles as one system.

- If no external force acts on the two particles then the momentum of both these particles together as a system is conserved.

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<footer style = "font-size: 18px">Uses of Principle of Momentum $\rightarrow$ Conservation of Momentum $\rightarrow$ <span style = "color:blue"> Conservation of Momentum </span> $\rightarrow$ Conservation of Momentum $\rightarrow$ Impulse Momentum Principle </footer>

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### <primary style="font-size:24px"> Conservation of Momentum </primary>
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- **<span style="color:green">FBD of A:</span>** $\vec{f_{A B}}$, the force exerted on particle A by particle B.

- **<span style="color:green">FBD of B:</span>** $\vec{f_{BA}}$, Force exerted on B by A.

- When we add up the 2 systems

- <span style="color:blue">Newtons third law:</span> $\vec{f_{A B}}=-\vec{f_{B A}}$.

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<footer style = "font-size: 18px">Conservation of Momentum $\rightarrow$ Conservation of Momentum $\rightarrow$ <span style = "color:blue"> Conservation of Momentum </span> $\rightarrow$ Impulse Momentum Principle $\rightarrow$ Law of Conservation of Momentum </footer>

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### <primary style="font-size:24px"> Impulse Momentum Principle </primary>
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- <span style="color:green">**Impulse momentum principle for particle A**</span>

- $\int \vec{F_A} d t+\int \vec{f}_{A B} d t=\Delta \vec{P_A}$

- <span style="color:green">**Impulse momentum principle for particle B**</span>

- $\int \vec{F_B} d t+\int \vec{f}_{BA} d t=\Delta \vec{P_B}$

- <span style="color:blue">**Add:**</span> $\int\vec{(F_A}+\vec{F_B)}dt+O=\Delta\vec{P_A}+\Delta\vec{P_B}$

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<footer style = "font-size: 18px">Conservation of Momentum $\rightarrow$ Conservation of Momentum $\rightarrow$ <span style = "color:blue"> Impulse Momentum Principle </span> $\rightarrow$ Law of Conservation of Momentum $\rightarrow$ Collision Problem </footer>

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- If $\vec{F_A}$  and $\vec{F_B}=O$

- $\Delta\vec{P_A}+\Delta\vec{P_B}=O$

- $\{(m_A\vec{v_A}}+{m_B\vec{v_B})}_1={(m_A\vec{v_A}}+{m_B\vec{v_B})}_2$

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

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### <primary style="font-size:24px"> Collision Problem </primary>
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- During the collision period collision forces >> other finite forces

- For the period of collision if both particles treated as a system, momentum of the system is conserved.

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

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- Angular Momentum <span style="color:blue">(Momentum of Momentum)</span> of the particle about point 0

- $\equiv\vec{r}\times{m \vec{v}}$ = $\vec{H_o}$

- $\{m \vec{v}}$= <span style="color:green">Linear Momentum
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<footer style = "font-size: 18px">Collision Problem $\rightarrow$ Angular Momentum $\rightarrow$ <span style = "color:blue"> Angular Momentum </span> $\rightarrow$ Angular Momentum $\rightarrow$ Angular Momentum </footer>

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- $\vec{H_0}=\vec{r} \times m \vec{v}$

- $\frac{d \vec{H_0}}{d t}=\frac{d}{d t}(\vec{r} \times m \vec{v})$

- $\frac{d \vec{r}}{d t} \times m \vec{v}+\vec{r} \times  {m}\frac{d \vec{v}}{d t}$

- $\vec{v}=\frac{d \vec{r}}{d t} $

- $\vec{r} \times{m}  \frac{d \vec{v}}{d t}=\vec{r}\times{m}\vec{a}$

- $\vec{V}\times{m\vec{V}}=0$

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

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- $\frac{d \vec{H_0}}{d t}=\vec{r} \times m\vec{a}$

- $m\vec{a}=\vec{F}_{\text{ext.on particle}}=\vec{F}$

- $\frac{d \vec{H_0}}{d t}=\vec{r} \times \vec{F}$

- $\vec{r} \times \vec{F}$= <span style="color:green">Moment of force about point O.</span>

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- $\frac{d \overrightarrow{H_0}}{d t}=\overrightarrow{m}_0$

- If $\vec{m_0}=0$

- $\frac{d \overrightarrow{H_0}}{d t}=0$

- $\vec{H_0}$=Constant

- $\vec{(H_0)_1}=\vec{(H_0)_2}$

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<footer style = "font-size: 18px">Angular Momentum $\rightarrow$ Angular Momentum $\rightarrow$ <span style = "color:blue"> Angular Momentum </span> $\rightarrow$ Law of Conservation of angular Momentum $\rightarrow$ Law of Conservation of angular Momentum </footer>

### <Success style="font-size:25px"> Work Energy Impulse Principles Conservation Of Momentum L-2 </Success>
### <primary style="font-size:24px"> Law of Conservation of angular Momentum </primary>
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- <span style="color:green">Gravitational force </span>=$\frac{-Gm_1m_2}{r^2}$

- It acts towards the center of the planet.

- For the motion of satellite about planet <span style="color:green">$\vec{H_0}$ will be constant. </span>

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<footer style = "font-size: 18px">Angular Momentum $\rightarrow$ Law of Conservation of angular Momentum $\rightarrow$ <span style = "color:blue"> Law of Conservation of angular Momentum </span> $\rightarrow$ Law of Conservation of angular Momentum $\rightarrow$ Thank You </footer>

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- If particle is under the motion of a force which always point towards fixed point O,

- $(\vec{r} \times m \vec{v})_1=(\vec{r} \times m \vec{v})_2$

- where, <span style="color:green">$\vec{r}$</span> is the position vector with respect to O.

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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-06-chapter-02-work-energy-impulse-principles-conservation-of-momentum-l-2_6-sd3zqspo7c8-35.jpg)

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<footer style = "font-size: 18px">Law of Conservation of angular Momentum $\rightarrow$ Law of Conservation of angular Momentum $\rightarrow$ <span style = "color:blue"> Law of Conservation of angular Momentum </span> $\rightarrow$ Thank You $\rightarrow$  </footer>