Simple Harmonic Motion
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$(\omega)$ :
PYQ-2023-Oscillations-Q4, PYQ-2023-Oscillations-Q7
$$\omega=\frac{2 \pi}{T}=2 \pi f$$
Time period $(\mathrm{T}) :$
PYQ-2023-Waves-And-Sound-Q8, PYQ-2023-Oscillations-Q8, PYQ-2023-Oscillations-Q10
$$\mathrm{T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$$
Speed :
$$v=\omega \sqrt{A^{2}-x^{2}} $$
Acceleration :
$$ a=-\omega^{2} x$$
Kinetic Energy (KE):
PYQ-2023-Motion-In-Two-Dimensions-Q3, PYQ-2023-Work-Power-And-Energy-Q1, PYQ-2023-Work-Power-And-Energy-Q3, PYQ-2023-Oscillations-Q9, PYQ-2023-Kinetic-Theory-Of-Gases-Q5, PYQ-2023-Kinetic-Theory-Of-Gases-Q6
$$\text{KE} = \frac{1}{2} m v^{2}=\frac{1}{2} m \omega^{2}\left(A^{2}-x^{2}\right)=\frac{1}{2} k\left(A^{2}-x^{2}\right)$$
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:
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Series Combination : $$1 / k_{eq}=1 / k_{1}+1 / k_{2}$$
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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
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Angular Frequency: $$\omega^{\prime} \approx \sqrt{\mathrm{k} / \mathrm{m}},=\omega_{0}$$
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Amplitude: $$A=A_{0} e^{\frac{-b t}{2 m}}$$
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Energy: $$E(t)=\frac{1}{2} K A^{2} e^{-b t / m}$$
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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: $$A=\frac{F_{0}}{\omega_{d} b}$$
