physics-class-11-unit-14-chapter-05-damped-harmonic-oscillator-l-5-6-lcmz0al8ggi

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Damped Harmonic Oscillator </primary>
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- Damped Harmonic Oscillator

- $m\ddot x + b\dot x + kx = 0$

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

- $x(t)=A e^{-\gamma t / 2} \text { cos } \omega_0 t$

- $\gamma=\frac{b}{m}, \quad \omega_0^2=\frac{k}{m}$

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

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Forced Free Oscillator </primary>
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- $m \ddot{x} =-k x+F \cos \omega t$

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

- ${r}-x(t)=(A \cos \omega \cdot t+B \sin \omega \cdot t) $

- $ +\frac{F}{m\left(\omega_0^2-\omega^2\right)} \cos \omega t$

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

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Forced Damped Harmonic Oscillator </primary>
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* Swing in the park.

* Pendulum in clocks - you need a1 battery to keep it running.

- $m \ddot{x}=-b \dot{x}-k x+F \cos \omega t \quad\left(\omega \neq \omega_0\right)$

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

- Equation for forced damped harmonic oscillator.

- Recall from forced undamped oscillator.

* Motion was a combination of two diffrent frequences $\rightarrow \omega_0$  and $\omega$

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

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Undamped Oscillator </primary>
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- $\ddot x + \omega_0^2x = \frac{F}{m} \cos\omega t$

- $x(t)=\left(A \cos \omega_2 t+B \sin \omega t\right) +\frac{F}{m\left(\omega_1^2-\omega^2\right)} \cos \omega t$
 
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<footer style = "font-size: 18px">Forced Damped Harmonic Oscillator $\rightarrow$ Forced Damped Harmonic Oscillator $\rightarrow$ <span style = "color:blue"> Undamped Oscillator </span> $\rightarrow$ Undamped Oscillator $\rightarrow$ Steady State Solution </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Undamped Oscillator </primary>
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- $\ddot{x}+\gamma \dot{x}+\omega_0^2 x=\frac{F}{m} \cos at$

- The solution is a combination (superposition) of two different solution

- $x(t)=A e^{-\frac{\gamma}{2} t} \cos \omega_0 t+B e^{-\frac{\gamma}{2} t} \ sin \omega_0 t $

- $x_{p}$ = depends on the applied force

- As t become large, the solution

- A $e^{-\gamma / 2 t} \cos\omega_0 t$ + $Be^{-\gamma / 2 t} \sin\omega_0 t \rightarrow 0$

- We are left with steady state solution $\rightarrow$ $x_{\rho}(l)= x(t)$

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<footer style = "font-size: 18px">Forced Damped Harmonic Oscillator $\rightarrow$ Undamped Oscillator $\rightarrow$ <span style = "color:blue"> Undamped Oscillator </span> $\rightarrow$ Steady State Solution $\rightarrow$ Steady State Solution </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Steady State Solution </primary>
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- Steady State Solution for a Forced Damped Harmonic Oscillator.

- $A e^{-\frac{\gamma}{2} t} \cos\omega t$ + $B e^{-\frac{\gamma}{2} t} \sin\omega t \rightarrow 0$

- Transient solution have died down.

- We are left with only the steady state solution motion.

- $\ddot x + \gamma \dot x + \omega+0^2x = \frac{F}{m}\cos\omega t$

- Recall in undamped oscillator

- $\ddot x+\omega_0^2 x = \frac{F}{m} \cos \omega t$
 
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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-14-chapter-05-damped-harmonic-oscillator-l-5_6-lcmz0al8ggi-7.jpg)
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<footer style = "font-size: 18px">Undamped Oscillator $\rightarrow$ Undamped Oscillator $\rightarrow$ <span style = "color:blue"> Steady State Solution </span> $\rightarrow$ Steady State Solution $\rightarrow$ Steady State Solution </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Steady State Solution </primary>
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- We had oscillator a solution x(t) = A $\cos \omega t$

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

- We want to try the solution of the form x(t) = A $\cos \omega t$

- Substitute this is the equation of motion

- $\dot x(t) = - \omega A sin \omega t$

- $\ddot x (t) = - \omega A cos\omega t$

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

- LHS contains both $\cos\omega t$ and $\sin\omega t$.

- RHS contains only $\cos\omega t$.

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<footer style = "font-size: 18px">Undamped Oscillator $\rightarrow$ Steady State Solution $\rightarrow$ <span style = "color:blue"> Steady State Solution </span> $\rightarrow$ Steady State Solution $\rightarrow$ Steady State Solution </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Steady State Solution </primary>
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- x(t) = $A \cos\omega t + B \sin\omega t$

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

- $\ddot x (t) = - \omega^2 A\cos\omega t - \omega^2 B \sin\omega t$

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

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<footer style = "font-size: 18px">Steady State Solution $\rightarrow$ Steady State Solution $\rightarrow$ <span style = "color:blue"> Steady State Solution </span> $\rightarrow$ Steady State Solution $\rightarrow$ Steady State Solution </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Steady State Solution </primary>
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- $-\omega^2 A + \gamma\omega B + \omega_0^2 A=\frac{A}{m}$

- $-\omega^2 B - \gamma\omega A + \omega_0^2 B =0$

- Two equation and two unknowns.

- $B=\left(\frac{\gamma \omega}{\omega_0^2-\omega^2}\right) A$

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### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Steady State Solution </primary>
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- $A=\frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{F}{m}\right)$

- $B=\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)} A$
- $=\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{F}{m}\right)$

- <span style="color:blue">Steady - state solution</span>

- $x(t)=\left[\frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2} \text { as } \omega t\right.+\left.\frac{\gamma \omega}{\left(\omega^2-\omega y+\gamma^2 \omega\right.} \sin \omega t\right] \frac{F}{m}$

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### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Steady State Solution </primary>
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- $x(t)= \frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2+\gamma^2 \omega^2\right.}\left(\frac{F}{m}\right) \cos \omega t$

- $ +\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)+\gamma^2 \omega^2}\left(\frac{F}{m}\right) \sin \omega t$

- $x (t)$ is periodic with the same frequency.

- i.e, $\omega$ as the frequency of the applied force.

- It contains both $\cos\omega t$  and $\sin\omega t$ terms.

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<footer style = "font-size: 18px">Steady State Solution $\rightarrow$ Steady State Solution $\rightarrow$ <span style = "color:blue"> Steady State Solution </span> $\rightarrow$ Amplitude of Motion $\rightarrow$ Amplitude of Motion </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Amplitude of Motion </primary>
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- **Amplitude of Motion (The Phase of Motion)**

- $ x(t)= \frac{\left(\omega_0^2-\omega^2\right)}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{E}{m}\right) \cos \omega t$

- $+\frac{\gamma \omega}{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}\left(\frac{F}{m}\right) \sin \omega t$

- $\frac{\omega_0^2-\omega^2}{\left(\omega_0^2-\omega^2\right)^2+
 \gamma^2 \omega^2}$ =
 - $\frac{1}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}} \left (\frac{\omega_0^2-\omega^2}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2}}\right)$

- $\frac{\gamma\omega}{\left(\omega^2-\omega)^2+\gamma^2 \omega^2\right.}$=
- $ \frac{1}{\sqrt{(\omega_0^2-\omega^2)^2}+\gamma^2\omega^2}
\left( \frac{\gamma\omega}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2}}\right)$

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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-14-chapter-05-damped-harmonic-oscillator-l-5_6-lcmz0al8ggi-11.jpg)
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<footer style = "font-size: 18px">Steady State Solution $\rightarrow$ Steady State Solution $\rightarrow$ <span style = "color:blue"> Amplitude of Motion </span> $\rightarrow$ Amplitude of Motion $\rightarrow$ Amplitude </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Amplitude of Motion </primary>
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- $\left|c_1\right| \leqslant 1 \left|c_2\right| \leqslant 1 \quad c_1^2+c_2^2 = 1$

- $x(t)=\frac{F}{m \sqrt{\left(\omega_0^2-\omega_0\right)^2+\gamma^2 \omega^2}} \cos (\omega t-\phi)$

- $c_1 = \cos \phi=\frac{\omega_0^2-\omega^2}{\sqrt{\left(\omega_0^2-\omega^2\right)+\gamma^2 \omega^2}}$

- $c_2 = \sin \phi=\frac{\gamma \omega}{\sqrt{(\omega_0^2-\omega^2)^2+\gamma^2 a^2}}$

- $\tan \phi=\frac{\gamma \omega}{\left(\omega _0^2-\omega^2\right)}$

- <span style="color:blue">The steady state solution lags behind the applied force.</span>

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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-14-chapter-05-damped-harmonic-oscillator-l-5_6-lcmz0al8ggi-11.jpg)
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<footer style = "font-size: 18px">Steady State Solution $\rightarrow$ Amplitude of Motion $\rightarrow$ <span style = "color:blue"> Amplitude of Motion </span> $\rightarrow$ Amplitude $\rightarrow$ Resonance </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Amplitude </primary>
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- Amplitude  $\frac{F}{m \sqrt{\left(\omega_2^2-\omega^2\right)+\gamma^2 \omega^2}}$ , $\phi=-\frac{\gamma \omega}{\left(\omega_1^2-\omega^2\right)}$

- Same frequency on the applied force
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<footer style = "font-size: 18px">Amplitude of Motion $\rightarrow$ Amplitude of Motion $\rightarrow$ <span style = "color:blue"> Amplitude </span> $\rightarrow$ Resonance $\rightarrow$ Resonance </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Resonance </primary>
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- It there is reasonance?

- Does the amplitude go to a very large value for center applied frequency

- $A=\frac{F}{m \sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2 \omega^2}}$

- $\frac{d A}{d \omega}=0$

- $\frac{d A}{d \omega}=-\frac{F}{m} \quad \frac{1}{2} \frac{1}{\left.\left(\omega _ 0 ^2-\omega^2\right)^2+\gamma^2 \omega^2\right)^{3 / 2}} \times \frac{\left(2\left(\omega_0^2-\omega^2\right) \times-2 \omega+2 \gamma^2 \omega\right)}{1} \\ =  0$

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<footer style = "font-size: 18px">Amplitude of Motion $\rightarrow$ Amplitude $\rightarrow$ <span style = "color:blue"> Resonance </span> $\rightarrow$ Resonance $\rightarrow$ Resonance </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Resonance </primary>
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- $2 \left(\omega_0^2-\omega^2\right)(- 2\omega) + 2\gamma^2\omega=0$

- $- 2\left(\omega_0^2-\omega^2\right)= -\gamma^2$

- or $\omega_0^2-\omega^2= \frac{\gamma^2}{2}$

- $\omega^2=\omega_0^2-\frac{\gamma^2}{2}$

- $\omega=\omega_0\left(1-\frac{\gamma^2}{2\omega_0^2}\right)^{\frac{1}{2}}$

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<footer style = "font-size: 18px">Amplitude $\rightarrow$ Resonance $\rightarrow$ <span style = "color:blue"> Resonance </span> $\rightarrow$ Resonance $\rightarrow$ Power </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Resonance </primary>
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- For $\gamma << \omega_0$

- $\omega=\omega_0\left(1-\frac{\gamma^2}{4\omega_0^2}\right)$

- $=\omega_0-\frac{\gamma^2}{t\omega_0} \simeq\omega_0$

- $A \left(\omega\right)$ = $\frac{c}{\sqrt{\left(\omega_0^2-\omega^2\right)^2+\gamma^2\omega^2}}$

- At frequency $\omega=\omega_0$, the amplitude of steady state motion becomes large.

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<footer style = "font-size: 18px">Resonance $\rightarrow$ Resonance $\rightarrow$ <span style = "color:blue"> Resonance </span> $\rightarrow$ Power $\rightarrow$ Average power </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Power </primary>
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- Power supplied by the force = Fv

- $x(t) =  A \cos\left(\omega t -\phi\right)$

- $v(t) = -\omega A \sin \left(\omega t - \phi\right)$

- Power = Fv = $- \omega A F \cos\omega t . sin\left(\omega t -\phi\right)$

- = $- \omega A F \left(\cos \omega t \sin\omega t \cos\phi - cos^2 \omega t \sin\phi\right)$

- Power = $-\omega AF \left(\cos\omega t \sin\omega t \cos\phi - cos^2 \omega t \sin\phi\right)$

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<footer style = "font-size: 18px">Resonance $\rightarrow$ Resonance $\rightarrow$ <span style = "color:blue"> Power </span> $\rightarrow$ Average power $\rightarrow$ Average power </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Average power </primary>
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- $\frac{\text{Average power}}{Cycle} = \frac{1}{T} \int_0^T \operatorname{Power}(t) d t$

- = $-\omega A F\left[\frac{1}{T} \int_0^T \cos \omega t \sin \omega t d t \cdot \cos \phi-\frac{1}{T} \int_0^T \cos ^2 \omega t \quad \sin \phi d t\right]$

- $ T =\frac{2\pi}{\omega}$

- $\int_0^T \cos \omega t \sin \omega t d t= 0 $
 
 - $\frac{1}{T} \int \cos ^2 \omega t d t=\frac{1}{2}$

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<footer style = "font-size: 18px">Resonance $\rightarrow$ Power $\rightarrow$ <span style = "color:blue"> Average power </span> $\rightarrow$ Average power $\rightarrow$ Average Power </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Average power </primary>
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- Average power = $\frac{\omega AF}{2} \sin\phi$

- $\sin \phi$ = Power factor

- This will be maximum value $sin\phi$ is maximum.

- Average power=  $\frac{\omega F^2}{m \sqrt{\left(\omega^2-\omega_0^2\right)+x^2 \gamma^2}} \sin \phi$

- $\left.\operatorname{sin} \phi\right|_{\text {maximum }}=1 \Rightarrow \phi=\frac{\pi}{2} \Rightarrow \cos \phi=0$

- $\frac{\omega_0^2-\omega^2}{m \sqrt{\left(\omega^2-\omega_0^2\right)+x^2 \gamma^2}} = 0 \rightarrow \omega_0^2 = \omega^2$

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<footer style = "font-size: 18px">Power $\rightarrow$ Average power $\rightarrow$ <span style = "color:blue"> Average power </span> $\rightarrow$ Average Power $\rightarrow$ Conclude </footer>

### <Success style="font-size:25px"> Damped Harmonic Oscillator L-5 </Success>
### <primary style="font-size:24px"> Conclude </primary>
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* Resonance in forced undamped harmonic oscillation

* Damping into the system

- Friction

- Velocity Dependent Damped

- $x(t)=  A e^{-\gamma / 2 t} \cos t$

- $+B e^{-\gamma / 2} \sin \omega t$

* Forced damped harmonic oscillation

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