physics-class-11-unit-14-chapter-01-introduction-to-periodic-motion-l-1-6-2wzwdbova3e

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Oscillations and Waves </primary>
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- Oscillatory motion and wave motion.

- Oscillatory motion:- Periodic motion.

- Periodic motion: Is a motion that repeats itself.

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<footer style = "font-size: 18px"> $\rightarrow$ Introduction to Periodic Motion $\rightarrow$ <span style = "color:blue"> Oscillations and Waves </span> $\rightarrow$ Periodic Motion $\rightarrow$ Periodic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Periodic Motion </primary>
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- Periodic motion: Something is a motion that repeats itself.

- $x(t)=R \cos \theta(t)$

- $y(t)=R \sin \theta(t)$

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<footer style = "font-size: 18px">Introduction to Periodic Motion $\rightarrow$ Oscillations and Waves $\rightarrow$ <span style = "color:blue"> Periodic Motion </span> $\rightarrow$ Periodic Motion $\rightarrow$ Examples of Periodic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Circular Motion </primary>
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- Motion of particle in a circle.

- Specialize to motion when the speed of the particle is constant / uniform.

- Time period **$T=\frac{2 \pi R}{v}$**

- $f  =\frac{1}{T} =\frac{V}{2 \pi R}=\left(\frac{\omega}{2 \pi}\right)$

- Angular speed $=\frac{2 \pi}{T}=2 \pi \mathrm{f}$

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<footer style = "font-size: 18px">Examples of Periodic Motion $\rightarrow$ Examples of Periodic Motion $\rightarrow$ <span style = "color:blue"> Circular Motion </span> $\rightarrow$ Circular Motion $\rightarrow$ Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Circular Motion </primary>
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- $\theta=\omega t, \quad \omega=\frac{v}{R}$

- $ x(t)=R \cos \omega t$

- $y(t)=R \sin \omega t$

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<footer style = "font-size: 18px">Examples of Periodic Motion $\rightarrow$ Circular Motion $\rightarrow$ <span style = "color:blue"> Circular Motion </span> $\rightarrow$ Simple Harmonic Motion $\rightarrow$ Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Periodic Motion </primary>
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- x(t)  $\rightarrow$ Periodic.

- Changing periodically with time period T.

- x(t) is periodic.

- Velocity = $(\frac{dx}{dt})$ is also periodic.

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<footer style = "font-size: 18px">Simple Harmonic Motion $\rightarrow$ Simple Harmonic Motion $\rightarrow$ <span style = "color:blue"> Periodic Motion </span> $\rightarrow$ Periodic Motion $\rightarrow$ Periodic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Periodic Motion </primary>
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- x(t) is periodic.

- v(t) is periodic.

- a(t) is periodic.

- $\int_0^{t_1} a(t) d t=-2 v$

- x(t): displacement

- v(t) = $\frac{dx}{dt}$

- a(t)=$(\frac{dv}{dt}) = \frac{d^2x}{dt^2}$

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<footer style = "font-size: 18px">Simple Harmonic Motion $\rightarrow$ Periodic Motion $\rightarrow$ <span style = "color:blue"> Periodic Motion </span> $\rightarrow$ Periodic Motion $\rightarrow$ Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Periodic Motion </primary>
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- A ball dropped from height 'h' and bouncing without losing any energy.

- v(t)=$\frac{dy{(t)}}{dt}$

- a(t)=$\frac{dv(t)}{dt}=\frac{d^2y}{dt^2}$

- $ \int_{t_1}^{t_2} a(t) d t=2 v_0 =2 \sqrt{2 g h}$

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<footer style = "font-size: 18px">Periodic Motion $\rightarrow$ Periodic Motion $\rightarrow$ <span style = "color:blue"> Periodic Motion </span> $\rightarrow$ Simple Harmonic Motion $\rightarrow$ Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $x(t)=R \cos \omega t \quad y(t)=R \sin \omega t$

- $v(t) =\frac{d x(y)}{d t} =d R \sin w L$

- $ a(t) =\frac{d v}{d t} =-\omega^2 R \cos u t =-\omega^2 x(t)$

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<footer style = "font-size: 18px">Periodic Motion $\rightarrow$ Periodic Motion $\rightarrow$ <span style = "color:blue"> Simple Harmonic Motion </span> $\rightarrow$ Simple Harmonic Motion $\rightarrow$ Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- x(t)= $A cos \omega t + B sin \omega t$

- v(t) = $\frac{dx}{dt}= -\omega a\sin\omega t + \omega b\cos \omega t$

- a(t)= $\frac{dv}{dt}= -\omega^2 A\cos\omega t-\omega^2 B\ sin \omega t = -\omega^2x (t)$

| x- axis | y-axis  | 
| -------- | -------- |
|$ x(t)=R \cos \omega t$ |$y(t) =R \sin \omega t$   |
|$v(t) =\frac{d x}{d t}=-\omega R \sin \omega t$| $v(t) =\frac{d y}{d t} =\omega R \cos \omega t$     |
|$a(t)  =\frac{d v}{d t}=\frac{d^2 x}{d t^2}=-\omega^2 x$|$a(t)=\frac{d v}{d t}=\frac{d^2 y}{d t^2}=-\omega^2y$      |

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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-01-introduction-to-periodic-motion-l-1_6-2wzwdbova3e-16.jpg)

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<footer style = "font-size: 18px">Periodic Motion $\rightarrow$ Simple Harmonic Motion $\rightarrow$ <span style = "color:blue"> Simple Harmonic Motion </span> $\rightarrow$ Simple Harmonic Motion $\rightarrow$ Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $\dot{x}=\frac{d^2x}{dt^2}= -C_x (C> 0)$

- $\dot{x(t)}$ = $ A_1 cos \sqrt{c}\ t + B_1 sin \sqrt{c}$ t

- $\dot{x(t)}$ = $ A cos \sqrt{c}\ t + B_1 sin \sqrt{c}$ t

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

### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- A Function is given as:

-  f(t) = $\cos 2\pi t + cos 3\pi t$

- Plot the function and find its period.

- ${cos}2 \pi t \rightarrow$ has a period of T=1

- $\left.\operatorname{Cos}(2 \pi t) \quad\right|_{t=0}=1$

- Next it becomes 1 is at t=1.

- $\cos 2\pi= 1$

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### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $f(t)= \cos 2 \pi t+\cos 3 \pi t= \cos \left(\left(\frac{2 \pi+3 \pi}{2} t\right)+\frac{(3 \pi-2 \pi)}{2} t\right) cos 3\pi t+\cos \left(\frac{2 \pi+3 \pi}{2} t-\frac{(3 \pi-2 \pi)}{2} t\right)cos 2\pi t$

- $=\cos \left(\frac{5 \pi}{2} t+\frac{\pi}{2} t\right)+\cos \left(\frac{5 \pi}{2} t-\frac{\pi}{2} t\right)$

- = $f(t)=2 \cos \frac{5 \pi}{2} t \cos \frac{\pi}{2} t$

- Now,

- $ f(0)=2 \times 1 \times 1=2$

- $f(T)=2 \cos \frac{5 \pi T}{2} \cos \frac{\pi}{2} T\$

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### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $m=$ mass of the particle, $k=$ constant

- $V(x) =\frac{1}{2} m k x \quad x\geq 0$

- $\quad =-\frac{1}{2} m k x \quad  x\leqslant 0$

- V(x) =$\frac{1}{2} m k|x|$

- A particle with energy E is going to perform a periodic motion $E=\frac{1}{2} m v_0^2 \Rightarrow v_0$ is the speed when v(x)=0

- By Energy conservation $E=\frac{1}{2} m v^2+\frac{1}{2} m k|x|$

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### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $\frac{1}{2}  m v_0^2  =\frac{1}{2} m v^2+\frac{1}{2} m k|x|$

- $v^2  =v_0^2-k|x|$

- $ v(x)  =\sqrt{v_0^2-k|x|}$

- $d t  =\frac{d x}{V-(x)}=\frac{d x}{\sqrt{v_0^2-k|x|}}$

- $v=0 \Rightarrow \frac{1}{2} m v_0^2=\frac{1}{2} m k|x|$

- $|x|=\frac{v_0^2}{k}$

- $\frac{T}{2}=\int_{-v_0^2 / k}^{v_i^2 / k} d t=\int_{-v_i^2 / k}^0 \frac{d x}{\sqrt{v_0^2+k x}}+\int_0^{\frac{v_0^2}{k}} \frac{d x}{\sqrt{v_0^2-k x}}$

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### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $\frac{T}{2}=\int_{-\frac{v_0^2}{k}}^0 \frac{d x}{\sqrt{v_0^2+k x}}+\int_0^{v_0^2 / k} \frac{d x}{\sqrt{v_0^2-k x}}$

- y= - x

- So, x= - y $\Rightarrow$ dx= -dy

- $\frac{T}{2}=-\int_{+v_0^2 / k}^0 \frac{d y}{\sqrt{v_0^2-k y}}+\int_0^{v_0^2 / k} \frac{d x}{\sqrt{v_0^2-k x}}$

- = $+\int_0^{v_0^2 / k} \frac{d y}{\sqrt{v_0^2-k y}}+\int_0^{v_0^2 / k} \frac{d y}{\sqrt{v_0^2-k y}}$

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### <Success style="font-size:25px"> Introduction To Periodic Motion L-1 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $\frac{T}{2}=2 \cdot \int_0^{v_0^2 / k} \frac{d y}{\sqrt{v_0^2-k y}}$

- $y=\frac{v_0^2}{k} \sin ^2 \theta \quad d y=\frac{2 v_0^2}{k} \sin \theta \cos \theta\ d \theta$

- $\frac{T}{2}=2 \int_0^{\pi / 2} \frac{2 v_0^2 / k \sin \theta \cos \theta d \theta}{v_0 \cos \theta}$

- $=4 \frac{v_0}{k} \cdot \int_0^{\pi / 2} \sin \theta d \theta=\frac{4 v_0}{k}$

- $T=\frac{8 v_0}{k}$ and frequency =$\frac{k}{8 nv^2}$.

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