physics-class-11-unit-14-chapter-02-introduction-to-simple-harmonic-motion-l-2-6-m7yj7e6s6pg

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- x(t) = R cos $\omega$ t

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

- v(t) = – $\omega$ A sin $\omega$ t

- a(t) = – $\omega^2 \text{A cos} \omega$ t = – $\omega^2 \text{x}$

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

- $\frac{d^2 x}{d t^2}=-c x$

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<footer style = "font-size: 18px"> $\rightarrow$ Introduction to 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 Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- Equation for simple harmonic motion, $\frac{d^2 x}{d t^2}=-c x \quad c \equiv \omega^2$.

- Solution: $x(t)=A \cos \sqrt{c} t+B \sin \sqrt{c} t$.

- Force = ma

- If force on a particle $\propto$ - displacement Minus sign $\Rightarrow$ force is in the direction opposite to the displacement.

- The motion of particle is going to be simple harmonic motion.

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- Visulalize SHM (Simple harmonic motion)

- A particle moving uniformly in a circle with constant angular speed or constant speed.

- Phasor diagrams

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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> $\rightarrow$ Problems $\rightarrow$ Problems </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Problems </primary>
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- $\left(\frac{d^2x}{d t^2}=-k x\right) \frac{d^2 y}{d x^2}=-k y \quad(k>0)$

- $y(x) = A \cos \sqrt{k} x + B \sin \sqrt{k} x$

- A and B are some unknown constants if we worth to detriment these constants we need further information.

- A & B $\rightarrow$ two constants

- (1) Displacement and velocity at t = 0

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- (2) Displacement at two different times

- $\frac{d^2 x}{d t^2}=-\omega^2 x$

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

- $x(t = 0) = X_0$ and $\frac{d x}{d t}=\left.v(t)\right|_{t = 0} = V_0$

-  x(t = 0) = A cos 0 + B sin 0 = A = $x_0$

- $v(t)=\frac{d x}{d t}=$

- $-\omega A \sin \omega t+\left.\omega B \cos \omega t\right|_{t=0}  =\omega B = v_0 \Rightarrow B=\left(\frac{v_0}{\omega}\right)$

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- $x(t)=\quad X_0 \cos \omega t + \frac{V_0}{\omega} \sin \omega t$

- when $x_0$ is displacement at time t = 0 and $v_0$ is velocity at time t = 0.

- **Example:** Find the solution of

- $\frac{d^2 x}{d t^2} = -25 x \text { with }$

- x(t = 0) = 3m and v(t = 0) = –2 m$s^{-1}$

- x(t) = 3 cos 5t - $\frac{2}{5} sin 5t$

- = 3 cos 5t - 0.4 sin 5t

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Simple Harmonic Motion </primary>
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- $\frac{d^2 x}{d t^2} = -c x$

- Assume solution of the form $e^{\lambda t}$

- $x(t) = e^{\lambda t}$

- $\frac{dx}{dt} = \lambda e^{\lambda t}$

- $\frac{d^2x}{dt^2} = \lambda^2 e^{\lambda t}$

- Substitute this in the equation = $\lambda^2 e^{\lambda t} = -c e^{\lambda t}  \Rightarrow \lambda = \pm i \sqrt{c}$

- Solutions are $e^{i \sqrt{c} t},  e^{-i \sqrt{c} t}$ = $ x(t) = A_1 e^{i \sqrt{c} t} + B_1 e^{-i \sqrt{c} t}$

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- $x(t) = A_1 e^{i \sqrt{c} t} + B_1 e^{-i \sqrt{c} t}$

- $e^{i \sqrt{c} t} = \cos \sqrt{c} t + i \sin \sqrt{c} t$

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

- $A = A_1 + B_1$

- $B = i\left(A_1 - B_1\right)$

- If c < 0 Force $\propto$ displacement

- $\frac{d^2 x}{d t^2}  = c x (c > 0)$

- $x(t)=e^{\lambda t} \lambda^2 = c \Rightarrow \lambda = \pm \sqrt{c}$

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Summary </primary>
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- F $\propto$ - displacement

- Equation of motion : $\frac{d^2 x}{d t^2}  = -c x$

- General solution : $x(t) - A \cos \sqrt{e} k + B \sin \sqrt{c} t$

- SIMPLE HARMONIC MOTION

- A and B are determined by some condition (two condition) given

- If F $\propto$ displacement $\frac{d^2 x}{d t^2} = c x  (c > 0)$

- The particle tends to move away when ankled.

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

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Spring-mass System </primary>
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- $m \frac{d^2 x}{d t^2} = -k x$

- $\frac{d^2 x}{d t^2} = -\frac{k}{m} x$

- Of the form $ \frac{d^2 x}{d t^2} = -\omega^2 x$

- $\omega^2=\frac{k}{m} \quad \frac{d^2 x}{d t^2}=-\omega^2 x$

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

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

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<footer style = "font-size: 18px">Summary $\rightarrow$ Spring-mass System $\rightarrow$ <span style = "color:blue"> Spring-mass System </span> $\rightarrow$ Spring-mass System $\rightarrow$ Spring-mass System </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
### <primary style="font-size:24px"> Spring-mass System </primary>
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- x(t) = A cos $\omega$t + B sin $\omega$t

- $\omega=\sqrt{\frac{k}{m}}$

- x(t) = $x_0$ cos $\omega$t

- $x(0) = A = x_0$

- $v(0) = -\omega A \sin \omega t + \omega B cos \omega t|_{t=0} = 0$

- $\Rightarrow B = 0$

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<footer style = "font-size: 18px">Spring-mass System $\rightarrow$ Spring-mass System $\rightarrow$ <span style = "color:blue"> Spring-mass System </span> $\rightarrow$ Spring-mass System $\rightarrow$ Examples </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- A mass of 2 kg is attached to a spring of spring constant k = 500 $Nm^{-1}$. What will be the frequency of oscillation if the mass is displaced from its equilibrium position and released.

- k = 500 $Nm^{-1}$, m = 2 kg

- $\omega=\sqrt{\frac{k}{m}} = \sqrt{\frac{500}{2}} = \sqrt{250} \quad 5 \sqrt{10}$ rad/s

- $f = \frac{\omega}{2 \pi}=\frac{5 \sqrt{10}}{2 \pi} = \frac{2.5 \sqrt{10}}{\pi}$ Hz or $\left(s^{-1}\right)$

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<footer style = "font-size: 18px">Spring-mass System $\rightarrow$ Spring-mass System $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Examples $\rightarrow$ Phase </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- (2) A mass of 5 kg is attached to a spring of spring costant 400 $Nm^{-1}$. When it is polled from its equilibrium position by 0.5 m and released, on a frictionless horizontal table, What well be its displacement as a function of time.

- k = 400 $Nm^{-1}$, m = 5 kg

- $\omega=\sqrt{\frac{k}{m}}=\sqrt{\frac{400}{5}} = \sqrt{80} ~rad/s = 4 \sqrt{5} ~rad/s$

- $x(t)= A \cos 4 \sqrt{5} t + B \sin 4 \sqrt{5} t$

- x(t = 0) = 0.5,  v(t = 0) = 0, A = 0.5 m  , x(0) = A + 0 = 0.5

- $v =\left.\frac{d x}{d t}\right|_{t = 0}$ + $\omega$

- x(t) = 0.5 cos 4$\sqrt{5}t $

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<footer style = "font-size: 18px">Spring-mass System $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Examples </span> $\rightarrow$ Phase $\rightarrow$ Phase </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- $\ddot{x}=-\omega^2 x$

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

- A and B are constants

- The solution can also be written in the from

- $x(t)=\bar{A} \cos (\omega t+\phi)$ ---(1)

- or $x(t)=\bar{A} \cos (\omega t-\phi)$, or $=\bar{A} \sin (\omega t+\phi)$

- From (1) $\quad \dot{x}(t)=-\bar{A} \omega \sin (\omega t+\phi)$

- $\dot{x}(t)=-\omega^2 \bar{A} cos(\omega t+\phi)=-\omega^2 x$

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<footer style = "font-size: 18px">Examples $\rightarrow$ Examples $\rightarrow$ <span style = "color:blue"> Phase </span> $\rightarrow$ Phase $\rightarrow$ Phase </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- x(t) = A cos $\omega$ t + B sin $\omega$ t $=\sqrt{A^2+B^2}[\frac{A}{\sqrt{A^2+B^2}} \cos \omega t+\frac{B}{\sqrt{A^2+B^2}} \sin \omega t]$

- Where: $\frac{A}{\sqrt{A^2 + B^2}} \leqslant 1 ,\quad \frac{B}{\sqrt{A^2 + B^2}} \leqslant 1$

- $\frac{B}{\sqrt{A^2+B^2}}=\sqrt{1-(\frac{A}{\sqrt{A^2+B^2}})^2}$

- $\cos \phi=\frac{A}{\sqrt{A^2+B^2}}, \quad \sin \phi=\frac{B}{\sqrt{A^2+B^2}}$

- $x(t)=\sqrt{A^2+B^2}[\cos \omega t \cos \phi+\sin \omega t \sin \phi] = \sqrt{A^2+B^2} \cos (\omega t -\phi)$

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<footer style = "font-size: 18px">Examples $\rightarrow$ Phase $\rightarrow$ <span style = "color:blue"> Phase </span> $\rightarrow$ Phase $\rightarrow$ Two Springs Attached to a Mass </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- $x(t)=\bar{A} \cos (\omega t-\phi)$

- when $\bar{A}=\sqrt{A^2 + B^2}$

- $\cos \phi =\frac{A}{\sqrt{A^2+B^2}}=\frac{A}{\bar{A}}$

- $\sin \phi=\frac{B}{\bar{A}} $

- $\tan \phi=\frac{B}{A}$

- $cos \phi=\frac{A}{\bar{A}} \quad \sin \phi=-\frac{B}{\bar{A}}$

- $x(t)={\bar{A}} cos(\omega t+\phi)$

- $\phi$ initial phase of the motion.

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<footer style = "font-size: 18px">Phase $\rightarrow$ Phase $\rightarrow$ <span style = "color:blue"> Phase </span> $\rightarrow$ Two Springs Attached to a Mass $\rightarrow$ Two Springs Attached to a Mass </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- $x(t)=\bar{A} \cos (\omega t+\phi)$

- $x(0)=\bar{A} \cos \phi$

- $\dot{x_0}(0)=-\omega \bar{A} \sin (\omega t+\phi)|_{t=0} =-\omega \bar{A} \sin \phi$

- Two springs attached to a mass

- Two identical springs and attach a mass 'm' to then in the following two configuration.

- Find the frequency of oscillation of mass m in the two cases.

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<footer style = "font-size: 18px">Phase $\rightarrow$ Phase $\rightarrow$ <span style = "color:blue"> Two Springs Attached to a Mass </span> $\rightarrow$ Two Springs Attached to a Mass $\rightarrow$ Two Springs Attached to a Mass </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- Springs are massless.

- Stretch in the second spring is (x - y).

- Since the spring is massless the net force on it must be zero.

- $\Rightarrow k y = k(x - y)-\left(y-\frac{x}{2}\right)$

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<footer style = "font-size: 18px">Phase $\rightarrow$ Two Springs Attached to a Mass $\rightarrow$ <span style = "color:blue"> Two Springs Attached to a Mass </span> $\rightarrow$ Two Springs Attached to a Mass $\rightarrow$ Two Springs Attached to a Mass </footer>

### <Success style="font-size:25px"> Introduction To Simple Harmonic Motion L-2 </Success>
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- Force on the mass = $\frac{k x}{2}$

- $m \ddot{x}=-\frac{k x}{2}$

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

- $\omega=\frac{1}{\sqrt{2}} \sqrt{\frac{k}{m}}$

-  Find = 2 kx

- $m \ddot{x}=-2 k x \quad \ddot{x}=-\frac{2 k}{m} x$

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

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<footer style = "font-size: 18px">Two Springs Attached to a Mass $\rightarrow$ Two Springs Attached to a Mass $\rightarrow$ <span style = "color:blue"> Two Springs Attached to a Mass </span> $\rightarrow$ Two Springs Attached to a Mass $\rightarrow$ Thank You </footer>