physics-class-11-unit-14-chapter-03-examples-of-simple-harmonic-motion-l-3-6-zrgbjwbi2ie

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Equation of motion for SHM </primary>
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- $\frac{d^2 x}{d t^2} =-c x  = -\omega^2 x \quad(c>0)$

- where; $\omega$ = angular frequency

- **Solution**

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

- <span style="color:blue">Constants A and B are determined by two conditions </span>

- (i) x(0), v(0)

- (ii) $x(t_1), x(t_2)$ 
 
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<footer style = "font-size: 18px"> $\rightarrow$ Examples of Simple Harmonic Motion $\rightarrow$ <span style = "color:blue"> Equation of motion for SHM </span> $\rightarrow$ Spring-Mass System $\rightarrow$ Spring-Mass System </footer>

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Spring-Mass System </primary>
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- **Physically :** A spring mass system (Spring follow Hooke's law) performs SHM.

- k = Spring Constant

- m = mass of the black

- g = gravitational acceleration

- spring is stretches by l

- What happens if we pull / push the mass by a distance y and release it.

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<footer style = "font-size: 18px">Examples of Simple Harmonic Motion $\rightarrow$ Equation of motion for SHM $\rightarrow$ <span style = "color:blue"> Spring-Mass System </span> $\rightarrow$ Spring-Mass System $\rightarrow$ Example of Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Spring-Mass System </primary>
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- $F_{spring} = -k(l-y)$

- $F_m$ = mg

- Displacement is downwards and the force is upwards.

- Net force = $k(l-y)-mg$ =  -ky upwards

- or F= -ky downwards

- Now F = mg

- $m \frac{d^2 y}{dt^2}=-ky \text{or}   \frac{d^2 g}{d t^2}=-\frac{k}{m} g$

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

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<footer style = "font-size: 18px">Equation of motion for SHM $\rightarrow$ Spring-Mass System $\rightarrow$ <span style = "color:blue"> Spring-Mass System </span> $\rightarrow$ Example of Simple Harmonic Motion $\rightarrow$ Example of Simple Harmonic Motion </footer>

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Example of Simple Harmonic Motion </primary>
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- We are using examples to show you how different systems perform simple harmonic motion.

- In this system, take a string which is pulled so that it has a tension T.

- $\frac{x}{l} \ll 1$

- $F_{\text {net }} =T \sin \theta+T \sin \theta=2 T \sin \theta$

- Since $\quad x \ll l$

- $\sin \theta=\tan \theta \simeq\theta \simeq\frac{x}{l}$

- $F_{\text {net }}=-2 T \frac{x}{l}$

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Example of Simple Harmonic Motion </primary>
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- **Equation of motion**: $ m \ddot{x}=-\frac{2 T x}{l}$

- $\ddot{x}=-\left(\frac{2 T}{l m}\right) x$

- Angular frequency $\omega=\sqrt{\frac{2 T}{l m}}$

- Time period $2 \pi \sqrt{\frac{\mathrm{lm}}{2 T}}$

- $y(t)=A \cos \sqrt{\frac{2 T}{l m}} t+B \sin \sqrt{\frac{2 T}{l m}} t$

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Example of Simple Harmonic Motion </primary>
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- A wooden block floating in liquid.

- Uniform surface area: A

- m = mass of the block

- $\rho$ = Density of liquid

- l = Depth to which the block is submerged

- g = Gravitational acceleration

- Archimedes principle: A $\rho$ gl = mg

- A $\rho$ l = m

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Example of Simple Harmonic Motion </primary>
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- Mass of the liquid = (Al)$\rho$

- Equation of motion = $\frac{md^2y}{d} =  -2A$\rho$y

- Al$\rho \frac{d^2y}{dt^2}$ = - 2A$\rho$gy

- $\frac{d^2y}{dt^2}= - \frac{2g}{l}y$

- $\omega^2 = \frac{2g}{l}$

- Time = $2 \pi \sqrt{\frac{l}{2 g}}$

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Simple Pendulum </primary>
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- **Simple pendulum m:**

- Time period for oscilation of a simple pendulum.

- Motion of the pendulum is periodic.

- **Question:** Is the motion simple harmonic motion?

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Simple Pendulum </primary>
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- Force in the x direction:

- $F_\perp$ to string = mg $sin\theta$

- for $\theta \ll 1 \quad,  mg sin\theta \cong m g \theta$

- $F_\perp = -\frac{m g x}{l}$

- $m \frac{d^2x}{dt^2} = -\frac{mg}{l}x$

- $\frac{d^2x}{dt^2} = - \frac{g}{l}x$

- $\omega^2=\frac{g}{l} \quad \omega=\sqrt{\frac{g}{l}}$

- Time period $T=2 \pi \sqrt{\frac{l}{g}}$

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Simple Pendulum </primary>
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- Find the length of a pendulum that has time period of 1 second.

- $1=2 \pi \sqrt{\frac{l}{9}} $

- $ l=\frac{g}{4 \pi^2} \simeq \frac{1}{4} m$ = 25 cm

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- So for we have looked at Point masses performing simple harmonic motion.

- What happens if we deal with extended bodies or rigid bodies.

- Is it simple harmonic motion?

- When dealing with extended bodies or rigid bodies, we use to torque equation.

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- A uniform rod of mass m and length l is pivoted at one end. It is hanging vertically in equalibrium. If displaced by an angle $\theta$ from the vertical, write is equation of motion when it is released.

- Find if for $\theta$<< 1, it performs SHM and find its time period.

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

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### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- Torque = $\frac{mgl}{2} sin \theta$

- I $ \alpha$ = -$\frac{mgl}{2} sin \theta$, $\alpha = \frac{d^2\theta}{dt^2}$

- Equation of motion

- I $\frac{d^2\theta}{dt^2} = - \frac{mgl}{2} sin\theta$

- If $\theta$<< 1,  $sin \theta \simeq \theta$

- $\frac{d^2\theta}{dt^2} = - \frac{mgl}{2I} \theta$

- $\omega^2 = \frac{mgl}{2I}$, $I= \frac{ml^2}{3}$

- $\omega^2=\frac{m g l}{2 \times \frac{ml}{3}}=\left(\frac{3 g}{2 l}\right)$

- $T=\frac{2 \pi}{\omega} = 2 \pi \sqrt{\frac{2 l}{3 g}}$

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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> $\rightarrow$ Example $\rightarrow$ Solution </footer>

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Example </primary>
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- A uniform disc of mass an and radium R is pioted at a point on its p 
- What is the frequency of its oscilations when it is d by a small angle $\theta$ from the vertical and related?

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Solution </primary>
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- <span style="color:blue">Torque</span> = mg r $sin \theta \simeq$ mgr$\theta$

- <span style="color:blue">Equation of motion:</span> I$\alpha$ = - mgr$\theta$

- I $ \frac{d^2\theta}{dt^2} = - mgr \theta$

- I = $\frac{mr^2}{2} + mr^2 = \frac{3mr^2}{2}$

- $\frac{3mr^2}{2} \frac{d^2\theta}{dt^2} = -mgr\theta$

- $\frac{d^2\theta}{dt^2} = - \frac{2g}{3r} \theta$

- $\omega^2 = \frac{2g}{3r}, T= 2\pi \sqrt\frac{3r}{2g}$

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

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Problem </primary>
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- Shown below is a collection of positive and negative changes ( positive ions and electrions) is slab geometry. The positive charges are fixed while negative charge are mobile. If the slab of negative charge is displaced as shown, its starts oscillation. find the frequency of oscillations.

- These is a restoring force due to the electric field.

- Let, n = number density of the charges.

- Volume = Ax

- Number of charges = nAx

- Charge = - neAx, e: electronic charge.

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<footer style = "font-size: 18px">Solution $\rightarrow$ Plasma Oscillations $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Problem $\rightarrow$ Example </footer>

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- Charge / area =$\sigma = \frac{neAx}{A} = nex$

- $E = \frac{\sigma}{\epsilon} = \frac{nex}{\epsilon}$

- $F=+(n A l) e \times \frac{nex}{\epsilon} = \frac{n^2Ale^2}{\epsilon} $

- $ m_e \ddot{x}=-\frac{n e^2 x}{\epsilon}$.

- $\ddot{x}=-\frac{n e^2 }{m_e \epsilon} x$

- Plasma frequency: $\omega_p^2 = \frac{n e^2 }{m_e \epsilon}$

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<footer style = "font-size: 18px">Plasma Oscillations $\rightarrow$ Problem $\rightarrow$ <span style = "color:blue"> Problem </span> $\rightarrow$ Example $\rightarrow$ Example </footer>

### <Success style="font-size:25px"> Examples Of Simple Harmonic Motion L-3 </Success>
### <primary style="font-size:24px"> Example </primary>
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- $I \ddot{\theta}=I \alpha=\tau$

- $\tau=-m g \frac{a}{\sqrt{2}} \sin \alpha \simeq-\frac{m g a}{\sqrt{2} } \theta \quad (\theta - small )$

- $I \ddot{\theta}=\frac{-m g^a}{\sqrt{2}} \theta$

- $\omega^2=\left(\frac{m g a}{\sqrt{2} I}\right)$

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

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### <primary style="font-size:24px"> Example </primary>
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- $I =I_{CM}+I_{\text{about CM}}$

- $I =\frac{m a^2}{2}+\frac{m a^2}{6} =\frac{2 m a^2}{3}$

- $\omega^2=\frac{m g^2}{\sqrt{2} \cdot \frac{2 p^2}{3}}=\left(\frac{3 g}{2 \sqrt{2} a}\right)$

- $\omega=\left[\frac{3g}{2\sqrt{2} x}\right]^{1 / 2}$

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

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### <primary style="font-size:24px"> Conclude </primary>
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- $\frac{d^2 y}{d t^2}=-\omega^2 y \Rightarrow$ SHM

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