1. Harmonic Oscillations:
Definition and Characteristics:
Harmonic oscillations refer to repetitive, back-and-forth movements or vibrations that follow a sinusoidal (or harmonic) pattern. Characteristics include a restoring force proportional to the displacement from equilibrium, resulting in periodic motion.
2. Damped Harmonic Motion:
Derivation of the differential equation:
The differential equation governing damped harmonic motion is derived from Newton's second law, including a damping term. It is typically of the form
\[ m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = 0 \]
where \( m \) is mass, \( c \) is damping coefficient, \( k \) is the spring constant, and \( x \) is displacement.
Solution of the differential equation:
The solution involves exponential decay and oscillatory terms. It's expressed as
\[ x(t) = Ae^{-\frac{c}{2m}t}\cos(\omega_dt + \phi) \]
where \( A \) is amplitude, \( \omega_d \) is damped angular frequency, and \( \phi \) is the phase constant.
Overdamped, Critically damped, and Underdamped cases:
- Overdamped: When damping is high, the system returns to equilibrium slowly without oscillation.
- Critically damped: A special case with the fastest return to equilibrium without oscillating.
- Underdamped: Damping is low, leading to oscillations while returning to equilibrium.
Quality factor expression:
The quality factor, \( Q \), is defined as the ratio of the angular frequency of oscillation to the rate of energy loss per cycle.
\[ Q = \frac{\omega_0}{\omega_d} \]
where \( \omega_0 \) is the natural angular frequency.
3. Forced Oscillations:
Differential equation derivation:
For forced oscillations, the differential equation is
\[ m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F_0\cos(\omega t) \]
where \( F_0 \) is the amplitude of the external force, \( \omega \) is its angular frequency.
Expressions for amplitude and phase of forced oscillations:
The amplitude and phase of forced oscillations are given by
\[ A = \frac{F_0}{m\sqrt{(\omega_0^2 - \omega^2)^2 + (\frac{c\omega}{m})^2}} \]
and
\[ \phi = \arctan\left(\frac{\frac{c\omega}{m}}{\omega_0^2 - \omega^2}\right) \].
4. Amplitude Resonance:
Expression for Resonant frequency:
Resonant frequency occurs when the driving frequency equals the natural frequency, \( \omega = \omega_0 \).
Quality factor and Sharpness of Resonance:
Quality factor \( Q \) is related to the sharpness of resonance:
\[ Q = \frac{\omega_0}{\Delta\omega} \], where \( \Delta\omega \) is the width of the resonance curve.
5. Electrical Analogy of Mechanical Oscillators:
Analogies between mechanical and electrical systems:
Mass (\(m\)) corresponds to inductance (\(L\)), damping (\(c\)) to resistance (\(R\)), and spring constant (\(k\)) to the reciprocal of capacitance (\(1/C\)).
6. Wave Motion:
Derivation of one-dimensional wave equation:
Derived from the principles of wave motion, the one-dimensional wave equation is
\[ \frac{\partial^2u}{\partial t^2} = v^2 \frac{\partial^2u}{\partial x^2} \],
where \( u \) is displacement and \( v \) is wave velocity.
Solution of one-dimensional wave equation:
Solutions involve functions representing traveling waves, e.g., \( u(x, t) = f(x - vt) \).
Three-dimensional wave equation and its solution:
The three-dimensional wave equation is
\[ \frac{\partial^2u}{\partial t^2} = v^2 \nabla^2u \].
Solutions depend on the specific form of the source and boundary conditions.
7. Distinction Between Transverse and Longitudinal Waves:
Explanation of transverse and longitudinal wave characteristics:
Transverse waves have vibrations perpendicular to the direction of wave propagation, while longitudinal waves have vibrations parallel to the direction of wave propagation.
8. Transverse Vibration in a Stretched String:
Description of transverse vibrations in a stretched string:
Transverse vibrations involve oscillations perpendicular to the length of the string, governed by the wave equation.
9. Statement of Laws of Vibration:
Basic principles governing vibrational motion:
These laws include the principle of superposition, which states that the net displacement at any point and time caused by several independent vibrations is the vector sum of the individual displacements. Additionally, the laws cover resonance, frequency, and amplitude relationships in vibrating systems.
0 Comments