Introduction
Oscillations are a fundamental concept in physics and are pivotal for understanding various physical phenomena. In the context of NEET Physics, oscillations encompass the study of periodic motions where an object moves back and forth around an equilibrium position. This study note will break down the complex ideas of oscillations into digestible sections, ensuring a comprehensive understanding of the topic.
Simple Harmonic Motion (SHM)
Definition and Characteristics
Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
Key Characteristics:
- Restoring Force: $F = -kx$
- Displacement: $x(t) = A \cos(\omega t + \phi)$
- Velocity: $v(t) = -A \omega \sin(\omega t + \phi)$
- Acceleration: $a(t) = -A \omega^2 \cos(\omega t + \phi)$
Differential Equation of SHM
The differential equation governing SHM is derived from Newton's second law:
$$ F = ma \quad \Rightarrow \quad -kx = m \frac{d^2 x}{dt^2} $$
Rewriting, we get:
$$ \frac{d^2 x}{dt^2} + \left(\frac{k}{m}\right) x = 0 $$
Let $\omega^2 = \frac{k}{m}$, thus the equation simplifies to:
$$ \frac{d^2 x}{dt^2} + \omega^2 x = 0 $$
Energy in SHM
The total mechanical energy in SHM is conserved and is the sum of kinetic and potential energies.
- Potential Energy (U): $U = \frac{1}{2} k x^2$
- Kinetic Energy (K): $K = \frac{1}{2} m v^2 = \frac{1}{2} m (\omega A \sin(\omega t + \phi))^2$
- Total Energy (E): $E = U + K = \frac{1}{2} k A^2$
The total energy in SHM remains constant and is independent of time.
Damped Harmonic Motion
Definition
Damped harmonic motion occurs when a resistive force (such as friction or air resistance) acts on the oscillating system, causing the amplitude to decrease over time.
Equation of Motion
The differential equation for damped harmonic motion is:
$$ m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = 0 $$
where $b$ is the damping coefficient.
Types of Damping
- Underdamped: Oscillations with gradually decreasing amplitude.
- Critically Damped: The system returns to equilibrium as quickly as possible without oscillating.
- Overdamped: The system returns to equilibrium without oscillating, but slower than in the critically damped case.
For NEET, focus on understanding the qualitative differences between underdamped, critically damped, and overdamped systems.
Forced Oscillations and Resonance
Forced Oscillations
When an external periodic force drives the system, it is termed forced oscillation. The equation of motion is:
$$ m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \cos(\omega t) $$
Resonance
Resonance occurs when the driving frequency $\omega$ matches the natural frequency $\omega_0$ of the system, leading to maximum amplitude oscillations.
ExampleConsider a swing being pushed at regular intervals. If the pushes are timed with the natural frequency of the swing, the amplitude of the swing increases significantly.
Common MistakeA common misconception is that resonance always leads to infinite amplitude. In reality, damping always limits the amplitude.
Applications of Oscillations
Pendulums
Simple Pendulum
A simple pendulum consists of a mass $m$ attached to a string of length $L$.
- Period (T): $T = 2\pi \sqrt{\frac{L}{g}}$
Physical Pendulum
A physical pendulum is an extended object swinging about a pivot point.
- Period (T): $T = 2\pi \sqrt{\frac{I}{mgh}}$
where $I$ is the moment of inertia about the pivot point.
LC Circuits
An LC circuit consists of an inductor (L) and a capacitor (C) and exhibits electrical oscillations analogous to mechanical SHM.
- Angular Frequency: $\omega = \frac{1}{\sqrt{LC}}$
In LC circuits, energy oscillates between the electric field of the capacitor and the magnetic field of the inductor.
Summary
Oscillations are a critical concept in physics, with applications ranging from simple pendulums to electrical circuits. Understanding the principles of SHM, damped and forced oscillations, and resonance is essential for mastering this topic. Make sure to practice solving problems and conceptual questions to reinforce your understanding.
