physics-class-11-unit-06-chapter-01-work-energy-basic-concepts-variable-forceskinetic-energy-l-1-6-8wv0qlipa48

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work and Energy </primary>
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- If due to the action of this force, the body undergoes a displacement d.

- <span style="color:green"> One Dimensional case:</span>

- Define the work done as the product of the force F and the displacement d.

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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-01-work-energy-concepts-variable-forceskinetic-energy-l-1_6-8wv0qlipa48-02.jpg)

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<footer style = "font-size: 18px">Work and Kinetic Energy $\rightarrow$ Work and Energy $\rightarrow$ <span style = "color:blue"> Work and Energy </span> $\rightarrow$ Work One Dimensional Case $\rightarrow$ Work One Dimensional Case </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work One Dimensional Case </primary>
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- The particle moves a distance d in the same direction.

- <span style="color:green"> W = F d</span>

- Work done by Force F is the product of magnitude of Force and magnitude of displacement. <span style="color:blue">(1D case)</span>

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<footer style = "font-size: 18px">Work and Energy $\rightarrow$ Work and Energy $\rightarrow$ <span style = "color:blue"> Work One Dimensional Case </span> $\rightarrow$ Work One Dimensional Case $\rightarrow$ Work One Dimensional Case </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work One Dimensional Case </primary>
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- We have a force $\vec{F}$ which is acting because of which particle undergoes a displacement $\vec{r}$.

- <span style="color:blue">1D case:</span> <span style="color:green">$W=|\vec{F}||\vec{r}|$</span>

- $n$ forces can act on the body simultaneously.

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<footer style = "font-size: 18px">Work and Energy $\rightarrow$ Work One Dimensional Case $\rightarrow$ <span style = "color:blue"> Work One Dimensional Case </span> $\rightarrow$ Work One Dimensional Case $\rightarrow$ Generalize Work </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Generalize Work </primary>
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- $\phi \rightarrow$ <span style="color:blue">The angle between</span> $\overrightarrow{F}$ and $\vec{r}$.

- W  = Work done by $\vec{F} $

- $W =\left|\overrightarrow{F}\right||\vec{r}| \cos \phi =\overrightarrow{F} \cdot \vec{r}$

- $ \vec{r}$= <span style="color:green">Displacement vector</span>

- $ \text { a) } \phi=90^{\circ} \Rightarrow W=0 $

- $ \text { b) } \phi>90^{\circ}, \quad W< 0 $

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<footer style = "font-size: 18px">Work One Dimensional Case $\rightarrow$ Work One Dimensional Case $\rightarrow$ <span style = "color:blue"> Generalize Work </span> $\rightarrow$ Generalize Work $\rightarrow$ Atomic Scale </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Generalize Work </primary>
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- Work done by a force $=\vec{F} \cdot \vec{r}$

- Sometimes, we use $\vec{d}$ for displacement

- $W=\vec{F} \cdot \vec{d}$

- $W \rightarrow$ <span style="color:green">Scalar, which can be either 0, positive, or negative.</span>

- <span style="color:blue">S.I unit of W = N.m = Joule</span>

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<footer style = "font-size: 18px">Work One Dimensional Case $\rightarrow$ Generalize Work $\rightarrow$ <span style = "color:blue"> Generalize Work </span> $\rightarrow$ Atomic Scale $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- <span style="color:green">Example:</span> A student lifts a rock of mass $10 \mathrm{kg}$ from the floor to raise it to a height of $2 \mathrm{m}$ (with negligible acceleration)

- <span style="color:green">Identify:</span>

- **a)** Forces on the rock

- **b)** Work done by each of these forces

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<footer style = "font-size: 18px">Generalize Work $\rightarrow$ Atomic Scale $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Problem $\rightarrow$ Problem </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- <span style="color:green"> Force by gravity = mg</span>

-  It is moving up, the displacement is = 2 meters.

- $\phi=180^{\circ}$

-  Work done by gravity $ =(\mathrm{mg})(2)(-1) =-196 \mathrm{J}$

- Work done by <span style="color:green">$F$</span>

- $ F=m g, d=2, \phi=0^{\circ}$

-  Force is acting in upward direction, displacement is in the same direction.

- $  W=m g(2)=+196 \mathrm{J }$

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<footer style = "font-size: 18px">Atomic Scale $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Problem $\rightarrow$ Work Problem with Friction </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work Problem with Friction </primary>
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- Force <span style="color:green">$F$</span> applied on a crate/ block of mass m. The coefficient of kinetic friction between block and ground <span style="color:blue">$=\mu_k$</span>

- Block moves with a constant velocity and moves a distarce d.
	
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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-01-work-energy-concepts-variable-forceskinetic-energy-l-1_6-8wv0qlipa48-07.jpg)

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

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work Problem with Friction </primary>
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- **a)**Identify forces on the crate and their magnitude.

- **b)** Find work done by each of these forces.

- <span style="color:green">FBD of the block</span>

-  Since $\vec{a} =0, $  
 
 - $\sum F_x =0 $  and $\sum F_y =0 $

-  mg = N

-  F = f

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

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work Problem with Friction </primary>
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- Since, there is relative motion between the block and the surface,

- **<span style="color:green">Friction force</span>** $f=\mu_k N$ $=\mu_k m g$

- $F=f=\mu_k m g$

- **<span style="color:green">4 forces acting on block</span>**

- a) mg acting downwards.

- b) N =m g, normal reaction, acting upwards.

- c) $F=\mu_k m g$, applied in the forward direction.

- d) $f=\mu_k m g$, acting in the -x direction.

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

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work Problem with Friction </primary>
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-  b) Find the work done by each of these four forces.

- $\vec{d} = \vec{r} = d\hat{i}$

- <span style="color:green"> $mg \downarrow N \uparrow$</span>

- 'mg' and 'N' act in a direction perpendicular to the displacement d.

- Work done by 'N' and 'mg' = 0

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### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work Problem with Friction </primary>
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- Force in $x$ direction
 
- Work done by $F=F d \cos \left(0^{\circ}\right)=\mu_k m g d$

- Work done by $f=f d \cos \left(180^{\circ}\right)=-\mu_k m gd$

- Sum of all the four work done by different forces = 0.

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<footer style = "font-size: 18px">Work Problem with Friction $\rightarrow$ Work Problem with Friction $\rightarrow$ <span style = "color:blue"> Work Problem with Friction </span> $\rightarrow$ String Tied to Winch $\rightarrow$ String Tied to Winch </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> String Tied to Winch </primary>
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- Block of mass m is being pulled with a string on an incline.

- Block moves a distance d.

- Block being moved up a ramp on a frictionless surface with constant speed for a distance d.

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![image](https://imagedelivery.net/YfdZ0yYuJi8R0IouXWrMsA/599197d2-140c-4687-2fcc-80b22258d400/public)

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<footer style = "font-size: 18px">Work Problem with Friction $\rightarrow$ Work Problem with Friction $\rightarrow$ <span style = "color:blue"> String Tied to Winch </span> $\rightarrow$ String Tied to Winch $\rightarrow$ String Tied to Winch </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> String Tied to Winch </primary>
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- a) Find  force $F$ which cable must exert.

- b) Work done by cable force $F$ so that the block moves a distance $d$.

- c) Work done by gravity during this process.

- d) If the block were to move vertically by the same height $h=d$ sin$\theta$, what would be work done by gravity.

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<footer style = "font-size: 18px">Work Problem with Friction $\rightarrow$ String Tied to Winch $\rightarrow$ <span style = "color:blue"> String Tied to Winch </span> $\rightarrow$ String Tied to Winch $\rightarrow$ String Tied to Winch </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> String Tied to Winch </primary>
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- Draw the free body diagram of the block.

- Resolve mg along x and y.

- <span style="color:blue"> $N=m g \cos \theta$</span>

- <span style="color:blue"> $F=m g \sin \theta$</span>

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<footer style = "font-size: 18px">String Tied to Winch $\rightarrow$ String Tied to Winch $\rightarrow$ <span style = "color:blue"> String Tied to Winch </span> $\rightarrow$ String Tied to Winch $\rightarrow$ String Tied to Winch </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> String Tied to Winch </primary>
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- **a)** <span style="color:green">Work done by</span>

- F = $m g \sin \theta d = m g \sin \theta d$

- **b)** <span style="color:green">Work done by gravity:</span>

- $m g d \cos (90+\theta) = -m g d \sin \theta$

- **c)** <span style="color:green">Resolve mg along x and y</span>

- $N=m g \cos \theta$

- $F=m g \sin \theta$

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### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> General method to find work done </primary>
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- a) Draw FBD of particle, to identify forces acting on particle, and these forces will be along with their directions.
 
- b) Find displacement vector $\vec{d}$ or $\vec{r}$ of the particle

- c) Angle between $\vec{F}_i$ and $\vec{d}$ is  $\phi_i$

- Work by $F_i=\left|\vec{F}_i\right||\vec{d}| \cos \phi_i$

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<footer style = "font-size: 18px">String Tied to Winch $\rightarrow$ String Tied to Winch $\rightarrow$ <span style = "color:blue"> General method to find work done </span> $\rightarrow$ General method to find work done $\rightarrow$ One Dimensional Variable Force </footer>

- Sign of $W$

- $W$ can be 0 positive or negative.

- This definition of $W$ has been for assuming that force <span style="color:blue">$\vec{F}$</span> is a constant.

- We could have a case where <span style="color:green">$\vec{F}$</span> is not constant.

- <span style="color:green">$\vec{F}$</span> function of displacement.

- Generalize the concept of work done.

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<footer style = "font-size: 18px">String Tied to Winch $\rightarrow$ General method to find work done $\rightarrow$ <span style = "color:blue"> General method to find work done </span> $\rightarrow$ One Dimensional Variable Force $\rightarrow$ One Dimensional Variable Force </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> One Dimensional Variable Force </primary>
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- Particle under the influence of $F$ moves from $x_i$ to $x_f$.

- We divide region from $x_i$ to $x_f$.

- We divide in small displacements $\Delta x$.

- Let the average force on the region $\Delta x$ = $\bar{F(x)}$.

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<footer style = "font-size: 18px">General method to find work done $\rightarrow$ One Dimensional Variable Force $\rightarrow$ <span style = "color:blue"> One Dimensional Variable Force </span> $\rightarrow$ One Dimensional Variable Force $\rightarrow$ One Dimensional Variable Force </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> One Dimensional Variable Force </primary>
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- <span style="color:green">Work done</span> in this small interval $\Delta x$ = $\bar{F}$ $(x) \Delta x$.

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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-01-work-energy-concepts-variable-forceskinetic-energy-l-1_6-8wv0qlipa48-17.jpg)

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<footer style = "font-size: 18px">One Dimensional Variable Force $\rightarrow$ One Dimensional Variable Force $\rightarrow$ <span style = "color:blue"> One Dimensional Variable Force </span> $\rightarrow$ One Dimensional Variable Force $\rightarrow$ One Dimensional Variable Force Summary </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> One Dimensional Variable Force </primary>
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- Work done in this small interal $\Delta x$ = $\bar{F}$ $(x) \Delta x$

- $\delta W=\bar{F}(x) \Delta x $

- Total work Done =$\Sigma \delta W $

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<footer style = "font-size: 18px">One Dimensional Variable Force $\rightarrow$ One Dimensional Variable Force $\rightarrow$ <span style = "color:blue"> One Dimensional Variable Force </span> $\rightarrow$ One Dimensional Variable Force Summary $\rightarrow$ Integral Form of Work </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> One Dimensional Variable Force Summary </primary>
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- Work done in this small interal $\Delta x$ = $\bar{F}$ $(x) \Delta x$

- $\delta W=\bar{F}(x) \Delta x $

- Total work Done =$\Sigma \delta W $ = $\sum \bar{F}(x) \Delta x $

- In limit $\Delta x \rightarrow 0$, Summation becomes integral, $\sum \rightarrow \int$

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<footer style = "font-size: 18px">One Dimensional Variable Force $\rightarrow$ One Dimensional Variable Force $\rightarrow$ <span style = "color:blue"> One Dimensional Variable Force Summary </span> $\rightarrow$ Integral Form of Work $\rightarrow$ Spring Force </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Spring Force </primary>
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- $x=0$ is at the uncompressed position of spring.

- If we compress or stretch the spring, a force is applied by spring.

- Spring force F(x) is a function of x.

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<footer style = "font-size: 18px">One Dimensional Variable Force Summary $\rightarrow$ Integral Form of Work $\rightarrow$ <span style = "color:blue"> Spring Force </span> $\rightarrow$ Hooke's law $\rightarrow$ Spring Force </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Hooke's law </primary>
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- Hook's law: $F(x)=-k x$

- k is the spring constant, k>0

- Force F is proportional to the extension or the compression of the spring,
 
- F $\propto x$

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<footer style = "font-size: 18px">Integral Form of Work $\rightarrow$ Spring Force $\rightarrow$ <span style = "color:blue"> Hooke's law </span> $\rightarrow$ Spring Force $\rightarrow$ Work Done by the Spring </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Spring Force </primary>
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- F(x) =-k x

- x >0 <span style="color:blue">Extension</span>

- x <0 <span style="color:green">Compression</span>

- x =0 position of relaxed spring

- SI unts of <span style="color:green">$k \rightarrow \mathrm{N} / \mathrm{m}$ </span>

- If spring changs from $x_i$ to $x_f$ work done by spring froces.

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<footer style = "font-size: 18px">Spring Force $\rightarrow$ Hooke's law $\rightarrow$ <span style = "color:blue"> Spring Force </span> $\rightarrow$ Work Done by the Spring $\rightarrow$ Reference Frame </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Work Done by the Spring </primary>
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- W =$\int_{x_i}^{x_f} F_{s p} d x=\int_{x_i}^{x_4}-k x d x $

- = -k $\int_{x_i}^{x_f} x d x=-\frac{k}{2} x^2|_{x_i}^{x_f} $=$\frac{1}{2} k\left(x_i^2-x_f^2\right)$

- This is the work done by the spring.

- Length of spring  does not come into picture.

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<footer style = "font-size: 18px">Hooke's law $\rightarrow$ Spring Force $\rightarrow$ <span style = "color:blue"> Work Done by the Spring </span> $\rightarrow$ Reference Frame $\rightarrow$ Kinetic Energy </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Reference Frame </primary>
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- $\Rightarrow$ <span style="color:green">Work done</span> by a force

- =$|\vec{F}|| \vec{d} \mid \cos \phi$

- $\vec{d}$ is <span style="color:green">displacement.</span>

- The work done depends on the reference frame in which we are observing the movement of the particle.

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<footer style = "font-size: 18px">Spring Force $\rightarrow$ Work Done by the Spring $\rightarrow$ <span style = "color:blue"> Reference Frame </span> $\rightarrow$ Kinetic Energy $\rightarrow$ Kinetic Energy </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Kinetic Energy </primary>
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- Kinetic energy is a property of a body, which a body gets by virtue of its speed.

- If a body moves with a speed $v$,

- <span style="color:green">Kinetic Energy</span> $k \equiv \frac{1}{2} m v^2$.

- where, $m$ is the mass of the body.

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<footer style = "font-size: 18px">Work Done by the Spring $\rightarrow$ Reference Frame $\rightarrow$ <span style = "color:blue"> Kinetic Energy </span> $\rightarrow$ Kinetic Energy $\rightarrow$ Kinetic Energy </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Kinetic Energy </primary>
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- $\Rightarrow \frac{1}{2} m(\vec{v} \cdot \vec{v}) $

- $\vec{v}$ = <span style="color:green"> Velocity of particle</span>
 
 - $\{v}\rightarrow$ <span style="color:blue">Speed of particle</span>

- $ \{v^2} =\vec{v} \cdot \vec{v}$

- <span style="color:green">Kinetic Energy $\geqslant 0$</span>

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<footer style = "font-size: 18px">Reference Frame $\rightarrow$ Kinetic Energy $\rightarrow$ <span style = "color:blue"> Kinetic Energy </span> $\rightarrow$ Kinetic Energy $\rightarrow$ Thank You </footer>

### <Success style="font-size:25px"> Work Energy Basic Concepts Variable Forceskinetic Energy L-1 </Success>
### <primary style="font-size:24px"> Kinetic Energy </primary>
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- Unit of K = Same as unit of work done = <span style="color:green">J</span> = $\left(\operatorname{kg} \frac{m^2}{s^2}\right)$

- SI unit of kinetic energy is joules.

- Kinetic energy being dependent on the speed is a frame dependent quantity. So, kinetic energy is also dependent on the reference frame.

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<footer style = "font-size: 18px">Kinetic Energy $\rightarrow$ Kinetic Energy $\rightarrow$ <span style = "color:blue"> Kinetic Energy </span> $\rightarrow$ Thank You $\rightarrow$  </footer>