physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4-6-irxoxk-b20e

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
<hr>
<main>

- Spring and mass: $m \ddot{x} = -k x$

- Simple pendulum: $mℓ \ddot{\theta}=-m g \theta$

- **In real life :**

- We ether apply force on a system because there is friction.

- The system comes to stop after some time.

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px"> $\rightarrow$ Forced Oscillation of Undamped Oscillator, Forced Free Motion of Damped Oscillator $\rightarrow$ <span style = "color:blue"> Simple Harmonic Motion </span> $\rightarrow$ Types of Oscillator $\rightarrow$ Free Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Types of Oscillator </primary>
<hr>
<main>

- <span style="color:blue">Free Oscillator</span> (No friction) under a periodically applied force

- <span style="color:blue">Forced Oscillator</span> : An Oscillator that experiences, frictional force

- <span style="color:blue">Forced damped oscillator</span>

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Forced Oscillation of Undamped Oscillator, Forced Free Motion of Damped Oscillator $\rightarrow$ Simple Harmonic Motion $\rightarrow$ <span style = "color:blue"> Types of Oscillator </span> $\rightarrow$ Free Oscillator $\rightarrow$ Free Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- An Oscillator without friction but being under the influence of a constant force.

- $m \ddot{x}= - k x + F$

- $\ddot{x}+\frac{k}{m} x-\frac{F}{m}=0$

- $\frac{k}{m}=\omega_0^2$

- $\omega_0=$ natural frequancy of the Oscillator

- $\ddot{x}+(\frac{k}{m} x-\frac{F}{m})=0$

- $m \ddot{x}+k(x-\frac{F}{k})=0$

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-3.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-4.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Simple Harmonic Motion $\rightarrow$ Types of Oscillator $\rightarrow$ <span style = "color:blue"> Free Oscillator </span> $\rightarrow$ Free Oscillator $\rightarrow$ Free Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $m \ddot{x}+k(x-\frac{F}{k}) = 0$

- Define $y=(x-\frac{F}{k})$

- $\dot{y}=\dot{x} \quad(\frac{d y}{d t}=\frac{d x}{d t})$

- $\ddot{y}=\ddot{x} \quad(\frac{d^2y}{dt^2}=\frac{d^2x}{dt^2})$

- $m \ddot{y}+k y=0$

- $y(t) = A \cos \omega_0 t + B \sin \omega_0 t$

- $x(t) - \frac{F}{k} = A \cos \omega_0 t + B \sin \omega_0 t$

- $x(t)=(\frac{F}{k})+A \cos \omega_0 t+B \sin \omega_0 t$

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-5.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-6.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Types of Oscillator $\rightarrow$ Free Oscillator $\rightarrow$ <span style = "color:blue"> Free Oscillator </span> $\rightarrow$ Free Oscillator $\rightarrow$ Free Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- **II:** We apply a periodic force F(t) = F cos $\omega t$.

- $m \ddot{x} = -k x + F \cos \omega t$

- $\ddot{x}+\frac{k}{m} x=\frac{F}{m} \cos \omega t$

- <span style="color:blue">If F = 0</span>, $\ddot{x}+\omega_0^2 x = 0$

- $x(t) = A \cos \omega_0 t + B \sin \omega_0 t$

- $F \neq 0 \quad \ddot{x}+\omega_0^2 x=\frac{F}{m} \cos \omega t$

- If we take $x \propto$ cos $\omega$ t

- $\ddot{x} \propto-\omega^2 \cos \omega t$

- We cannot have x $\propto$ sin $\omega t$ or x $\propto$ $t^n$.

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-5.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-6.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Free Oscillator $\rightarrow$ Free Oscillator $\rightarrow$ <span style = "color:blue"> Free Oscillator </span> $\rightarrow$ Free Oscillator $\rightarrow$ Free Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $\ddot{x}+\omega_0^2 x=\frac{F}{m} \cos \omega t$

- If we take x $\propto$ sin $\omega$t $\Longrightarrow$ x = C sin $\omega$t

- $\ddot{x} \propto-\omega^2 \sin \omega t=\ddot{x}=-\omega^2 C \sin \omega t$

- $-\omega^2 C \sin \omega t+\omega_0^2 C \sin \omega t=\frac{F}{m} \cos \omega t$

- L.H.S has sin $\omega$t dependence / RHS has cos $\omega$t dependence

- Will never match at all times.

- Similarly we show that x $\propto t^n$ cannot be a solution.

- $x = C t^n, \ddot{x} = n(n - 1) C t^{n-1}$

- $(x \propto t^n)$ is NOT a solution

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-7.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-8.jpg)

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $\ddot{x}+\omega_0^2 x = \frac{F}{m} \cos \omega t$

- $x(t)=C \cos \omega t$, $\ddot{x}=-\omega^2 C \cos \omega t$

- $-\omega^2 C cos \omega t+\omega_0^2 C cos \omega t = \frac{F}{m}\cos \omega t$

- $C=\frac{F}{m(\omega_0^2-\omega^2)} \quad[\omega_0 \neq \omega]$

- $x(t)=\frac{F}{m(\omega_0^2-\omega^2)} \cos \omega t$

- General solution = $A \cos \omega_0 t+ B \sin \omega_0 t + \frac{F}{m(\omega_0^2-\omega^2} \cos \omega t$

- $x_{homogeneous}(t) + x_{particular}(t)$ = $x_{h}(t) + x_{p}(t)$

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-7.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-8.jpg)

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $x_h = A \cos \omega_0 t + B \sin \omega_0 t$

- $\ddot{x}_h + \omega_0^2 x_h = 0$,  $x_p=\frac{F}{m(\omega_0^2-\omega^2)}\cos \omega_0 t$

- $\ddot{x}_p+\omega_0^2 x_p=\frac{F}{m} \cos \omega t$

- $x=x_h+x_p$, $\ddot{x}=\ddot{x}_h+\ddot{x}_p$

- $\omega_0^2 x=\omega_0^2 x+\omega_0^2 x_p$

- $\ddot{x}+\omega_0^2 x=(\ddot{x}_h+\omega_0^2 x_h)+(\ddot{x}_p+\omega_0^2 x_p) =\frac{F}{m} cos \omega t$

- $x(t) = A cos \omega_0 t + B \sin \omega_0 t + \frac{F}{m(\omega_0^2-\omega^2)} \cos \omega t$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $x(t = 0) = 0, \dot{x}(t = 0) = 0$

- If there is no force  x(t) = 0

- F  =  F cos $\omega$t

- x(t) = A cos $\omega_0$t + B sin $\omega_0$t + $\frac{F}{m(\omega_0^2-\omega^2)}$ cos $\omega$t

- $x(t=0)=0 \Rightarrow A + \frac{F}{m(\omega_0^2-\omega^2)} = 0$

- $A = -\frac{F}{m(\omega_0^2-\omega^2)}$

- $\dot{x}(t = 0) = 0 $

- $\omega_0   A \sin \omega_0 t|_{t = 0}$ + $\omega_0 B \cos \omega_0 t  $+

- $\frac{-\omega F}{m(\omega_0^2-\omega^2)} \mid \sin \omega t|_{t = 0}$ = 0   $\Rightarrow B = 0$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $x(t)=\frac{F}{m(\omega_0^2-\omega^2)}[\cos \omega t-\cos \omega_0 t]$

- $\cos \omega t-\cos \omega_0 t=-2 \sin (\frac{\omega+\omega_0}{2} t) \sin (\frac{\omega_1-\omega_0}{2} t)$

- $x(t)=\frac{-2 F}{m(\omega_0^2-\omega^2)} \sin (\frac{\omega+\omega_0}{2} t) \sin (\frac{\omega-\omega_0}{2} t)$

- $\quad =\frac{2 F}{m(\omega_0^2-\omega^2)} \sin (\frac{\omega+\omega_0}{2} t) \sin (\frac{\omega_0-\omega}{2} t)$

- $\omega_0 > \omega$

- $x(t)= A \sin \Omega t \sin (\Delta \omega) t$

- $A=\frac{2 F}{m(\omega_0^2-\omega^2)}, \Omega=\frac{\omega+\omega_0}{2}, \Delta \omega=\frac{\omega_0-\omega}{2}$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- $x(t)=\frac{2 F}{m(\omega_0^2-\omega^2)} \sin \Omega t \sin (\Delta\omega) t$

- <span style="color:blue">Amplitude :</span> $\frac{2 F}{m (\omega_0^2-\omega^2)}$ $\rightarrow$ very large if $\omega_0$ $\rightarrow$ $\omega_1$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- Time-dependence

- Product $\sin \Omega t \sin \Delta \omega t$

- $\Omega \gg \Delta \omega$

- $x(t)=\frac{2 F}{m(\omega_0^2-\omega^2)} \sin (\frac{\omega_0+\omega_1}{2} t)  \sin (\frac{\omega_0-\omega_1}{2} t)$

- Cannot substitute $\omega = \omega_0$ dencity in the formula above becomes we divided by $(\omega_0^2-\omega^2)$ assuming $\omega_0 \neq \omega$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Free Oscillator </primary>
<hr>
<main>

- Therefore for $\omega_0=\omega$, we take the limit $\omega_0 \rightarrow \omega$

- $\omega_0 \rightarrow \omega, \sin (\omega_0-\omega_t) \rightarrow \frac{(\omega_0-\omega)}{2} t$

- (Similar to $\sin \theta \rightarrow \theta$ for $\theta \rightarrow 0$)

- $\frac{\omega_0+\omega}{2} \rightarrow \omega_0$

- $x(t)=\frac{2 F}{m(\omega_0 +\omega)(\omega_0-\omega)}$

- $\sin \omega_0 t \frac{(\omega_0-\omega)}{2} t=\frac{F}{2 m \omega_0} t \sin \omega_0 t$

- Where  $\omega_0 = \omega$

- $x(t)=\frac{F}{2 m \omega_0} t \cdot \sin \omega_0 t$

- Amplitude will keep on increasing the time: the phenomena of resonance.

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Free Oscillator $\rightarrow$ Free Oscillator $\rightarrow$ <span style = "color:blue"> Free Oscillator </span> $\rightarrow$ Damped Harmonic Oscillator $\rightarrow$ Damped Harmonic Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Damped Harmonic Oscillator </primary>
<hr>
<main>

- Harmonic Oscillator experience constant frictional force.

- $m \ddot{x} = -kx + F$ (direction opposite to $\dot{x}$)

- $=-k x+F(-\hat{\dot{x}})$

- Energy does not get conserved due to friction = change in the energy of the Oscillator.

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Free Oscillator $\rightarrow$ Free Oscillator $\rightarrow$ <span style = "color:blue"> Damped Harmonic Oscillator </span> $\rightarrow$ Damped Harmonic Oscillator $\rightarrow$ Damped Harmonic Oscillator </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Damped Harmonic Oscillator </primary>
<hr>
<main>

- Spring-Mass System

- Energy = $\frac{1}{2} k A_1^2$

- **Change in energy**

- $=\frac{1}{2} k(A^2-A_1^2) \quad(A_1<A)$

- $= \frac{1}{2} k(A + A_1)(A-A_1)$

- **Work done against the friction**

- $= F(A + A_1)$

- $\Rightarrow (A - A_1) = (\frac{2 F}{k})$ independent of A

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Free Oscillator $\rightarrow$ Damped Harmonic Oscillator $\rightarrow$ <span style = "color:blue"> Damped Harmonic Oscillator </span> $\rightarrow$ Damped Harmonic Oscillator $\rightarrow$ Damped Harmonic Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Damped Harmonic Oscillator </primary>
<hr>
<main>

- In one cycle the decreases in amplitude  $=2(\frac{2 F}{k})=(\frac{4 F}{k})$

- Frictional force $\propto \dot{x}$ ( Stokes's law) $=-b \dot{x}$

- $m\ddot{x}=-k x-b \dot{x}$

- $m \ddot{x}+b \dot{x}+k x=0$

- $\ddot{x}+\frac{b}{m}\dot{x}+\omega_0^2 x=0$

- $\ddot{x}+\gamma \dot{x}+\omega_0^2 x$ = 0

- Equation of motion for a damped harmomic Oscilattor

- $\gamma$ = damping coefficient

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Damped Harmonic Oscillator $\rightarrow$ Damped Harmonic Oscillator $\rightarrow$ <span style = "color:blue"> Damped Harmonic Oscillator </span> $\rightarrow$ Damped Harmonic Oscilattor $\rightarrow$ Damped Harmonic Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Damped Harmonic Oscilattor </primary>
<hr>
<main>

- **Damped harmonic oscillator**

- $\ddot{x}+\gamma \dot{x}+\omega_0 x=0$

- <span style="color:blue">Case (i): $\gamma >> \omega_0$</span>

- If $\gamma >> \omega_0$ $\gamma$ and $\omega_0$ have the same dimension $sec^{-1}$.

- Approximately:

- $\ddot{x}+\gamma \dot{x}=0 $

- $\frac{d^2 x}{d t^2}+ \gamma \frac{d x}{d t}=0$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Damped Harmonic Oscillator $\rightarrow$ Damped Harmonic Oscillator $\rightarrow$ <span style = "color:blue"> Damped Harmonic Oscilattor </span> $\rightarrow$ Damped Harmonic Oscilattor $\rightarrow$ Heavily Damped Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Damped Harmonic Oscilattor </primary>
<hr>
<main>

- $\frac{d^2 x}{d t^2}+\gamma \frac{d x}{d t}=0$

- Let $\frac{d x}{d t} = z(t), \frac{d^2 x}{d t^2}=\frac{d z}{d t}$

- $\frac{d z}{d t}+\gamma z=0 \Rightarrow \quad z=C e^{-\gamma t}$

- $\frac{d x}{d t}=C e^{-\gamma t}=x(t)=C_1 e^{-\gamma t}$

- <span style="color:blue">Initial condition</span>

- $x(t)|_{t=0}=A$

- $x(t) = Ae^{-\gamma t}$

- $x(t) \propto e^{-\gamma t}$

</div>

</div>

</main>
<hr>
<footer style = "font-size: 18px">Damped Harmonic Oscillator $\rightarrow$ Damped Harmonic Oscilattor $\rightarrow$ <span style = "color:blue"> Damped Harmonic Oscilattor </span> $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ Heavily Damped Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Heavily Damped Oscilattor </primary>
<hr>
<main>

- Case (II): $\omega_0 \gg \gamma$

- $\ddot{x}+\gamma \dot{x}+\omega_0^2 x(t)=0 \Rightarrow \ddot{x}+\omega_0^2 x(t)=0$

- x = A cos $\omega_0$ t + B sin $\omega_0$ t

- x(t) = A cos $\omega_0$ t (B = 0)

- We have $x(t=1)=A$, and $\dot{x}(t=0)=0$

- $x(t)=A \cos \omega_0 t$

- Correspondence to x(t=0)= A and $\dot{x}(t=0) = 0$

- If there is damping, the amplitude decrease slowly .

- $\gamma \neq 0 \Rightarrow x(t)= A(t) cos \omega_0 t$

- We want to find A(t). Also, Find how much evergy is lost due to friction.

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-21.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Damped Harmonic Oscilattor $\rightarrow$ Damped Harmonic Oscilattor $\rightarrow$ <span style = "color:blue"> Heavily Damped Oscilattor </span> $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ Heavily Damped Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Heavily Damped Oscilattor </primary>
<hr>
<main>

- $F(t)=-b \dot{x} =-b \frac{d}{d t}[A(t) cos \omega_0 t] = -b(\frac{d A}{d t}) \cos\omega_0 t + b \omega_0 A(t) \sin \omega_0 t$

- Energy lost due to $F(t)=F \cdot \dot{x} /$ time = Loss in time

- Energy lost per unit time = $F\dot{x} = -b \dot{x}^2 = -b\{\frac{d}{d t}[A(t) cos\omega_0 t]\}^2$

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-23.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-24.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Damped Harmonic Oscilattor $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ <span style = "color:blue"> Heavily Damped Oscilattor </span> $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ Heavily Damped Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Heavily Damped Oscilattor </primary>
<hr>
<main>

- Since, $\gamma \ll \omega_0$, anticipate $\frac{d A}{d t}$ is neglegible in comparison with $\frac{d}{d t} (\cos \omega_0 t) =\omega_0 \sin \omega_0$t

- $\gamma \ll \quad \omega_0 \frac{1}{\gamma} \gg \omega_0^{-1}$

- Time scale involved with amplitude decrease

- time scale of Oscilattor

- **Physically:**<ins></ins> We see many-many oscillations before observing any significant decrease in amplitude

- $\Rightarrow$ That in why we can talk about amplitude A(t) as a function of time.

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-23.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-24.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Heavily Damped Oscilattor $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ <span style = "color:blue"> Heavily Damped Oscilattor </span> $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ Heavily Damped Oscilattor </footer>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Heavily Damped Oscilattor </primary>
<hr>
<main>

- Energy loss = -b $\dot{x}^2$

- $\dot{x}=\frac{d}{d t}[A(t) cos \omega_0 t]$

- $=\frac{d A}{d t} cos \omega_0 t-\omega_0 A(t) \sin \omega_0 t$

- Neglect this terms because $\frac{1}{A}(\frac{d A}{d t})<<<\omega_0$

- $\dot{x}=-\omega_0 A(t) \sin \omega_0 t$

-  Energy loss = -b $\dot{x}^2$ = $-\gamma m \omega_0^2 A^2(t) \sin ^2 \omega_0 t$

- More meaningfull is to talk about every loss /cycle

- $-\gamma m \omega_0^2 A^2(t) \int_0^T \sin ^2 \omega_0 t ~dt$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Heavily Damped Oscilattor </primary>
<hr>
<main>

- $\frac{d E}{d t}=\frac{ \text{ Energy loss / cycle }}{T}$

- $=-\gamma m \omega_0^2 A^2(t) \frac{1}{T} \int_0^T \sin^2 \omega_0 t dt$

- $=-\gamma \frac{1}{2} m \omega_0^2 A^2(t)$

- $E(t) = \frac{1}{2} k A^2(t)$

- In case $\gamma \ll \omega_0 \quad \frac{d E}{d t}=-\gamma E(t)$

- $E(t)=E_0 e^{-\gamma t}$

- $\frac{d E}{d t}=-\frac{1}{2} \gamma k  A^2(t)$
</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Heavily Damped Oscilattor </primary>
<hr>
<main>

- $\frac{d E}{d t}=\frac{d}{d t}(\frac{1}{2} k A^2)= k A \frac{d A}{d t}$

- $kA \frac{d A}{d t}=-\frac{1}{2} \gamma kA^2$

- $\frac{d A}{d t}=-\frac{\gamma}{2} A$ or $A(t)=e^{-\gamma / 2 t} A_0$

- $x(t) = A(t) \cos \omega_0 t$

- $\omega_0 \gg \gamma = A_0 e^{-\gamma/2 t} \cos \omega_0 t$

</div>

</div>

### <Success style="font-size:25px"> Force Oscillation Of Undamped Damped Oscillatormotion L-4 </Success>
### <primary style="font-size:24px"> Conclude </primary>
<hr>
<main>

- (i) Forced undamped Oscilattor

- (ii) Resonance

- (iii) Highly doped oscillator

- (iv) Lightly doped oscillator

- Constant forces on an oscillator

- $\rightarrow$ Constant dampang forces on and oscillator

</div>

![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-28.jpg)
![image](https://temp-public-img-folder.s3.ap-south-1.amazonaws.com/sathee.prutor.images/subject-images/iitpal/image/physics-class-11-unit-14-chapter-04-force-oscillation-of-undamped-damped-oscillatormotion-l-4_6-irxoxk-b20e-29.jpg)

</div>

</main>
<hr>
<footer style = "font-size: 18px">Heavily Damped Oscilattor $\rightarrow$ Heavily Damped Oscilattor $\rightarrow$ <span style = "color:blue"> Conclude </span> $\rightarrow$ Thank You $\rightarrow$  </footer>