S.H.M.:

$$\mathrm{F}=-\mathrm{kx}$$

General equation of S.H.M. is $$x=A \sin (\omega t+\phi)$$

$(\omega t+\phi)$ is phase of the motion and $\phi$ is initial phase of the motion.

Angular Frequency$(\omega)$ :

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

Time period $(\mathrm{T}) :$

$$\mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$$

Speed :

$$v=\omega \sqrt{A^{2}-x^{2}} $$

Acceleration :

$$ a=-\omega^{2} x$$

Kinetic Energy (KE):

$$\text{KE} = \frac{1}{2} m v^{2}=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)=\frac{1}{2} k\left(A^{2}-x^{2}\right)$$

Potential Energy (PE) :

$$\text{PE} =\frac{1}{2} \mathrm{Kx}{ }^{2}$$

Total Mechanical Energy (TME):

$$ \text{TE} =K . E .+P . E .=\frac{1}{2} k\left(A^{2}-x^{2}\right)+\frac{1}{2} K x^{2}=\frac{1}{2} K A^{2} \text{(which is constant)}$$

Spring-Mass System

(1)

$$\Rightarrow \quad T=2 \pi \sqrt{\frac{m}{k}}$$

(2)

$$T=2 \pi \sqrt{\frac{\mu}{K}}$$

where: $\mu=\frac{m_1 m_2}{\left(m_1+m_2\right)}$ known as reduced mass

Combination Of Springs:

• Series Combination : $$1 / k_{eq}=1 / k_{1}+1 / k_{2}$$

• Parallel combination : $$ k_{eq}=k_1+k_2$$

Simple Pendulum:

In accelerating Reference Frame: $$T=2 \pi \sqrt{\frac{\ell}{g}}=2 \pi \sqrt{\frac{\ell}{g_{\text {eff. }}}}$$

$g_{\text {eff }}$ is net acceleration due to pseudo force and gravitational force.

Compound Pendulum / Physical Pendulum:

Time period (T):

$$T=2 \pi \sqrt{\frac{\mathrm{I}}{\mathrm{mg} \ell}}$$

where, $\mathrm{I}=\mathrm{I}_{\mathrm{CM}}+\mathrm{m} \ell^{2} ; \ell$ is distance between point of suspension and centre of mass.

Torsional Pendulum

Time period $$(T): \quad T=2 \pi \sqrt{\frac{I}{C}}$$

where, $C=$ Torsional constant

Superposition of SHM:

Superposition of SHM’s along the same direction

$$x_{1}=A_{1} \sin \omega t$$

$$x_{2}=A_{2} \sin (\omega t+\theta)$$

If equation of resultant SHM is taken as: $$\mathrm{x}=\mathrm{A} \sin (\omega \mathrm{t}+\phi)$$

$$A=\sqrt{A_{1}^{2}+A_{2}^{2}+2 A_{1} A_{2} \cos \theta}$$

$$ \tan \phi=\frac{A_{2} \sin \theta}{A_{1}+A_{2} \cos \theta}$$

Damped Oscillation

- Damping force

$$\vec{\mathrm{F}}=-\mathrm{b} \vec{\mathrm{v}}$$

- equation of motion is

$$\frac{\mathrm{mdv}}{\mathrm{dt}}=-\mathrm{kx}-\mathrm{bv}$$

• Over Damping: $$b^{2}-4 m K>0$$
• Critical Damping: $$b^{2}-4 m K=0$$
• Under Damping: $$b^{2}-4 m K<0$$
• For small damping the solution is of the form.

$$x=\left(A_{0} e^{-b t / 2 m}\right) \sin \left[\omega^{1} t+\delta\right]$$

Where $\omega^{\prime}=\sqrt{\left(\frac{k}{m}\right)-\left(\frac{b}{2 m}\right)^{2}}$

For small b

• Angular Frequency: $$\omega^{\prime} \approx \sqrt{\mathrm{k} / \mathrm{m}},=\omega_{0}$$

• Amplitude: $$A=A_{0} e^{\frac{-b t}{2 m}}$$

• Energy: $$E(t)=\frac{1}{2} K A^{2} e^{-b t / m}$$

• Quality factor or $Q$ value, $$Q=2 \pi \frac{E}{|\Delta E|}=\frac{\omega^{\prime}}{2 \omega_{Y}}$$

where $, \omega^{\prime}=\sqrt{\frac{k}{m} \cdot \frac{b^{2}}{4 m^{2}}} \quad, \omega_{Y}=\frac{b}{2 m}$

Forced Oscillations And Resonance

External Force $$F(t)=F_{0} \cos \omega_{d} t$$

$$x(t)=A \cos \left(\omega_{d} t+\phi\right)$$

$$A=\frac{F_{0}}{\sqrt{\left(m^{2}\left(\omega^{2}-\omega_{d}^{2}\right)^{2}+\omega_{d}^{2} b^{2}\right)}}$$

$$\tan \phi=\frac{-v_{0}}{\omega_{d} x_{0}}$$

(a) Small Damping: $$A=\frac{F_{0}}{m\left(\omega^{2}-\omega_{d}^{2}\right)}$$

(b) Driving Frequency Close to Natural Frequency: $$A=\frac{F_{0}}{\omega_{d} b}$$