understand and use the terms displacement, amplitude, period, frequency, angular frequency and phase difference in the context of oscillations, and express the period in terms of both frequency and angular frequency

Simple Harmonic Oscillations

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement from an equilibrium position and acts in the opposite direction. It can be described mathematically by sinusoidal functions.

Key Terms and Their Definitions

• Displacement (\$x\$): The instantaneous distance of the oscillating object from its equilibrium position. Positive or negative sign indicates direction.
• Amplitude (\$A\$): The maximum magnitude of displacement from equilibrium. It is the peak value of \$|x|\$.
• Period (\$T\$): The time taken for one complete cycle of oscillation. Measured in seconds (s).
• Frequency (\$f\$): The number of complete cycles per unit time. \$f = \dfrac{1}{T}\$, measured in hertz (Hz).
• Angular Frequency (\$\omega\$): The rate of change of the phase of the sinusoid, expressed in radians per second. \$\omega = 2\pi f = \dfrac{2\pi}{T}\$.
• Phase Difference (\$\phi\$): The angular offset between two sinusoidal motions of the same frequency. It indicates how much one motion leads or lags another.

Mathematical Description of SHM

The displacement of an object undergoing SHM can be written as:

\$x(t) = A \cos(\omega t + \phi)\$

or equivalently, using a sine function:

\$x(t) = A \sin(\omega t + \phi)\$

Both forms describe the same motion; the choice of cosine or sine simply changes the initial phase.

Relationships Between Period, Frequency, and Angular Frequency

The three quantities are interrelated:

\$T = \frac{1}{f}, \qquad \omega = 2\pi f, \qquad T = \frac{2\pi}{\omega}\$

These equations allow conversion between any two of the quantities.

Summary Table

Example: Mass–Spring System

For a mass \$m\$ attached to a spring with force constant \$k\$, the angular frequency is given by:

\$\omega = \sqrt{\frac{k}{m}}\$

Consequently, the period and frequency become:

\$T = 2\pi\sqrt{\frac{m}{k}}, \qquad f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\$

Phase Difference in Practice

If two pendulums of identical length are set into motion, and one starts \$30^\circ\$ (or \$\pi/6\$ rad) ahead of the other, the phase difference is \$\phi = \pi/6\$ rad. Their displacements can be written as:

\$x_1(t) = A \cos(\omega t)\$

\$x_2(t) = A \cos(\omega t + \phi)\$

Key Points to Remember

1. Amplitude is the maximum displacement; it does not affect period or frequency.
2. Period and frequency are reciprocal: \$T = 1/f\$.
3. Angular frequency incorporates the factor \$2\pi\$ to relate linear cycles to radians.
4. Phase difference determines how two SHM motions are offset in time.
5. All SHM equations can be expressed using either \$f\$ or \$\omega\$, depending on convenience.