Where:
$A$A = amplitude, $\omega$ω= angular frequency, $\phi$ϕ= phase constant

Energy in SHM

Time Average Values:

$$\langle K \rangle = \langle U \rangle = \frac{1}{4}m\omega^2A^2$$⟨K⟩=⟨U⟩=41​mω2A2 $$\langle E \rangle = \frac{1}{2}m\omega^2A^2 = \text{constant}$$⟨E⟩=21​mω2A2=constant

⭐ Damped Oscillations

Equation of motion:$$m\ddot{x} + b\dot{x} + kx = 0$$mx¨+bx˙+kx=0

Where $b$b = damping constant

Types of damping

Solution (underdamped):$$x(t) = Ae^{-\gamma t}\cos(\omega’ t + \phi)$$x(t)=Ae−γtcos(ω′t+ϕ)

Where $\gamma = \frac{b}{2m}$γ=2mb​, $\omega’ = \sqrt{\omega^2 – \gamma^2}$ω′=ω2−γ2​

⭐ Forced Oscillations

Equation:$$m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t)$$mx¨+bx˙+kx=F0​cos(ωt)

Two states:

• Transient → short-lived
• Steady → long time motion

Amplitude:

$$A(\omega) = \frac{F_0}{\sqrt{(k – m\omega^2)^2 + (b\omega)^2}}$$A(ω)=(k−mω2)2+(bω)2​F0​​

⭐ Resonance

Maximum amplitude occurs at:$$\omega = \omega_0 = \sqrt{\frac{k}{m}}$$ω=ω0​=mk​​

Quality Factor:

$$Q = \frac{\omega_0}{2\gamma}$$Q=2γω0​​

Higher $Q$Q → sharper resonance → more energy stored

⭐ Bar Pendulum

Time period:$$T = 2\pi\sqrt{\frac{I}{mgd}}$$T=2πmgdI​​

Where $I$I = moment of inertia, $d$d = distance between pivot & center of mass

⭐ Kater’s Pendulum

Precision pendulum to determine acceleration due to gravity ggg:$$g = 4\pi^2\frac{L}{T^2}$$g=4π2T2L​

Where $L$L is equivalent length.

⚡ 2️⃣ Special Theory of Relativity

⭐ Michelson–Morley Experiment

• Aim: Detect motion of Earth through aether
• Result: No fringe shift → aether does not exist
• Conclusion → Speed of light is constant in all inertial frames

⭐ Einstein’s Postulates

1️⃣ Laws of physics same in all inertial frames
2️⃣ Speed of light $c$c is constant and maximum, independent of motion of source or observer

⭐ Lorentz Transformations

Connecting space-time coordinates of two inertial observers:$$x’ = \gamma(x – vt)$$x′=γ(x−vt) $$t’ = \gamma\left(t – \frac{vx}{c^2}\right)$$t′=γ(t−c2vx​)

Where:$$\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}$$γ=1−c2v2​​1​

🕒 Time Dilation

$$\Delta t = \gamma \Delta t_0$$Δt=γΔt0​

📏 Length Contraction

$$L = \frac{L_0}{\gamma}$$L=γL0​​

🔹 Relativistic Velocity Transformation

$$u’ = \frac{u – v}{1 – \frac{uv}{c^2}}$$u′=1−c2uv​u−v​

🔹 Relativistic Mass

$$m = \gamma m_0$$m=γm0​

🌟 Mass–Energy Equivalence

$$E = mc^2 = \gamma m_0c^2$$E=mc2=γm0​c2

When at rest:$$E_0 = m_0c^2$$E0​=m0​c2

🔹 Relativistic Doppler Effect

If source & observer moving closer:$$\nu = \nu_0 \sqrt{\frac{1 + \beta}{1 – \beta}}$$ν=ν0​1−β1+β​​

Where $\beta = v/c$β=v/c

🔹 Relativistic Momentum & Energy

$$p = \gamma m_0 v$$p=γm0​v $$E^2 = p^2c^2 + m_0^2c^4$$E2=p2c2+m02​c4

📌 Summary Table

⭐ Applications of Relativity

• Particle accelerators
• GPS satellite clock corrections
• Nuclear reactions $E = mc^2$E=mc2
• Space travel physics

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