Physics – I (A & B) Organizer : Mechanics, Oscillations, Optics, Electromagnetism, Dielectrics, Magnetic Properties of Materials, Quantum Mechanics, and Statistical Mechanics

Key Concepts in Mechanics and SHM:

Newton's Laws of Motion are defined, including the laws of inertia, force proportionality to acceleration, and equal and opposite action/reaction. Simple Harmonic Motion (SHM) is a periodic motion where acceleration (or force) is directly proportional to displacement and directed towards a fixed point.

Total Energy of SHM is $E = \frac{1}{2} m \omega^2 a^2$.

Damped Harmonic Motion occurs when a body's amplitude continuously decays due to dissipative forces like friction.

The differential equation for damped motion is $m\frac{d^2x}{dt^2} = -kx - L\frac{dx}{dt}$.

Motion is classified as overdamped (dead beat) if $K > \omega$, critically damped if $K = \omega$, and damped oscillatory if $K < \omega$.

Relaxation Time ($\tau$) is the time taken for the amplitude to fall to $1/e$ times its initial amplitude. $\tau$ is also defined as the time for energy to fall to $1/e$ of its initial value, where $\tau = \frac{1}{2K}$.

Forced Vibration is produced when a body vibrates under an applied strong periodic force.

Velocity Resonance occurs at the natural frequency, $\omega$.

Amplitude Resonance occurs at a resonant frequency $q_{res} = \sqrt{\omega^2 - 2K^2}$.

Quality Factor (Q) is $2\pi$ times the ratio of average energy stored to the energy dissipated per cycle. For weak damping, $Q \approx \frac{\omega}{2K}$.

Key Concepts in Vector Calculus:

A vector field $\vec{A}$ is solenoidal if $\nabla \cdot \vec{A} = 0$.

A vector field $\vec{A}$ is irrotational if $\nabla \times \vec{A} = 0$.

Stokes' Theorem relates the surface integral of the curl of a vector field to the line integral around the boundary of the surface.

Gauss's Divergence Theorem relates the volume integral of the divergence of a vector field to the surface integral of the vector field.

For a conservative field, the line integral around a closed path is zero ($\oint \vec{E} \cdot d\vec{l} = 0$)