Vibration, fundamentally an oscillatory motion, is described by various mathematical formulas depending on the specific physical system undergoing the oscillation. The most common type of vibration is Simple Harmonic Motion (SHM), which serves as the foundation for understanding many oscillatory phenomena.
Understanding Simple Harmonic Motion (SHM)
Simple Harmonic Motion is a repetitive back-and-forth movement through an equilibrium position, where the restoring force is directly proportional to the displacement and acts in the opposite direction.
Key Formulas for SHM:
The displacement of an object in SHM can be described by:
$$x(t) = A \cos(\omega t + \phi)$$
Where:
- $x(t)$ is the displacement from the equilibrium position at time $t$.
- $A$ is the amplitude, the maximum displacement from equilibrium.
- $\omega$ (omega) is the angular frequency, representing the rate of oscillation in radians per second.
- $t$ is the time.
- $\phi$ (phi) is the phase constant, determining the initial position of the oscillator at $t=0$.
From this, other important quantities are derived:
- Angular Frequency ($\omega$):
$$\omega = 2\pi f = \frac{2\pi}{T}$$ - Frequency ($f$): The number of oscillations per unit time.
$$f = \frac{1}{T} = \frac{\omega}{2\pi}$$ - Period ($T$): The time taken for one complete oscillation.
$$T = \frac{1}{f} = \frac{2\pi}{\omega}$$
Velocity and Acceleration in SHM:
The velocity ($v(t)$) and acceleration ($a(t)$) of an object in SHM are the first and second derivatives of the displacement equation, respectively:
- Velocity:
$$v(t) = -A\omega \sin(\omega t + \phi)$$
The maximum speed is $v_{max} = A\omega$. - Acceleration:
$$a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$$
The maximum acceleration is $a_{max} = A\omega^2$.
Formulas for Specific Vibrating Systems
Different physical systems exhibit SHM under certain conditions, each with its own specific formula for angular frequency, frequency, and period.
1. Mass-Spring System
A mass ($m$) attached to an ideal spring with spring constant ($k$) undergoes SHM when displaced and released.
- Angular Frequency ($\omega$):
$$\omega = \sqrt{\frac{k}{m}}$$ - Frequency ($f$):
$$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$ - Period ($T$):
$$T = 2\pi\sqrt{\frac{m}{k}}$$
Practical Insight: These formulas are crucial in designing shock absorbers, understanding vehicle suspension, and even modeling molecular vibrations. Stiffer springs (larger $k$) or smaller masses (smaller $m$) lead to higher frequencies and shorter periods.
2. Simple Pendulum
For small angles of displacement ($\theta < 15^\circ$), a simple pendulum of length ($L$) with a point mass ($m$) approximates SHM.
- Angular Frequency ($\omega$):
$$\omega = \sqrt{\frac{g}{L}}$$ - Frequency ($f$):
$$f = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$$ - Period ($T$):
$$T = 2\pi\sqrt{\frac{L}{g}}$$
Note: The period and frequency of a simple pendulum, for small angles, are independent of the mass of the bob and the amplitude of oscillation. They depend only on the length of the string and the acceleration due to gravity ($g$).
3. Vibrating String
The frequency of vibration of a string, such as those found on musical instruments, depends on its physical properties and the manner in which it vibrates. The formula for the frequency of a vibrating string is given by:
$$f = \frac{p}{2L} \sqrt{\frac{F}{\mu}}$$
Where:
- $f$ is the frequency of vibration in Hertz (Hz).
- $p$ is the harmonic number (also called the number of segments or antinodes).
- For the fundamental frequency (first harmonic), $p=1$.
- For the first overtone (second harmonic), $p=2$, and so on.
- $L$ is the length of the vibrating portion of the string in meters (m).
- $F$ is the tension in the string in Newtons (N).
- $\mu$ (mu) is the linear mass density of the string, which is its mass per unit length, in kilograms per meter (kg/m).
Example: A guitar string's pitch (frequency) can be changed by:
- Pressing down on frets: This changes the effective length ($L$) of the string. A shorter length increases the frequency.
- Tuning pegs: These adjust the tension ($F$) in the string. Higher tension increases the frequency.
- Using different gauge strings: Thicker strings have a higher linear mass density ($\mu$). A higher linear mass density decreases the frequency.
Dimensional Formula for Linear Mass Density ($\mu$):
The linear mass density ($\mu$) is defined as mass per unit length.
- The dimension of mass is
[M]. - The dimension of length is
[L].
Therefore, the dimensional formula for linear mass density ($\mu$) is[M L⁻¹].
Summary of Key Vibration Formulas
| System | Displacement ($x(t)$) | Angular Frequency ($\omega$) | Frequency ($f$) | Period ($T$) |
|---|---|---|---|---|
| General SHM | $A \cos(\omega t + \phi)$ | $2\pi f = \frac{2\pi}{T}$ | $\frac{1}{T} = \frac{\omega}{2\pi}$ | $\frac{1}{f} = \frac{2\pi}{\omega}$ |
| Mass-Spring | $A \cos(\sqrt{\frac{k}{m}} t + \phi)$ | $\sqrt{\frac{k}{m}}$ | $\frac{1}{2\pi}\sqrt{\frac{k}{m}}$ | $2\pi\sqrt{\frac{m}{k}}$ |
| Simple Pendulum | $A \cos(\sqrt{\frac{g}{L}} t + \phi)$ | $\sqrt{\frac{g}{L}}$ | $\frac{1}{2\pi}\sqrt{\frac{g}{L}}$ | $2\pi\sqrt{\frac{L}{g}}$ |
| Vibrating String | (Complex waveform) | (N/A for single $\omega$) | $\frac{p}{2L}\sqrt{\frac{F}{\mu}}$ | N/A (frequency is primary focus) |
These formulas provide the mathematical framework for analyzing and predicting the behavior of oscillating systems, from macroscopic movements to microscopic vibrations within materials.
