Period, frequency and amplitude Knowledge-Based Study

Study notes

1. Oscillatory Motion and Equilibrium

   An oscillation is a repetitive back-and-forth motion around a central stable point known as the equilibrium position. When an object is displaced from this position, an internal restoring force acts to bring it back toward equilibrium.

   A complete repetition of the motion, from an initial position back to the exact same position moving in the original direction, is called one complete cycle or oscillation.

   Simple Harmonic Motion (SHM) is a special and fundamental type of periodic motion in which the restoring force is directly proportional to the displacement and directed toward the equilibrium position.

   Defining Amplitude

   Amplitude ($A$) is the maximum magnitude of displacement of an oscillating object from its central equilibrium position.

   • SI Unit: meter (m), though centimeters (cm) or millimeters (mm) are frequently used in practical settings.
   • Key Characteristic: Amplitude is always a positive scalar quantity representing the peak distance from equilibrium, not the peak-to-peak distance. The total range of motion between opposite turning points is equal to $2A$.
   • Physical Significance: In an ideal oscillator without damping, the amplitude remains constant. The mechanical energy of the system is directly proportional to the square of the amplitude ($E \propto A^2$).

   Period and Frequency

   The time characteristics of an oscillation are quantified using period and frequency.

   Period ($T$)

   The period ($T$) is the time required for an oscillating object to complete one full cycle of motion.

   • SI Unit: second (s).
   • For an object completing $N$ oscillations in total elapsed time $t$, the period is given by:

   $$T = \frac{t}{N}$$

   Frequency ($f$)

   The frequency ($f$) represents the number of complete oscillations occurring per unit time.

   • SI Unit: hertz (Hz), where $1\text{ Hz} = 1\text{ s}^{-1}$.
   • Using the same count of oscillations $N$ over time $t$:

   $$f = \frac{N}{t}$$

   The Reciprocal Relationship

   Because period and frequency quantify inverse aspects of the same periodic cycle, they are inversely related:

   $$T = \frac{1}{f} \quad \text{and} \quad f = \frac{1}{T}$$

   Angular Frequency ($\omega$)

   In Simple Harmonic Motion, it is often mathematically convenient to express how rapidly the phase of the oscillation changes in radians per second. This quantity is called angular frequency ($\omega$):

   $$\omega = 2\pi f = \frac{2\pi}{T}$$

   Reading a Displacement-Time Graph

   When displacement $x$ is plotted against time $t$, the resulting curve is sinusoidal:

   • Identifying Amplitude ($A$): Measure the vertical distance from the center line ($x = 0$) to any crest or trough.
   • Identifying Period ($T$): Measure the horizontal time interval between two consecutive identical points, such as from crest to crest, from trough to trough, or across one full sine wave cycle.
   • Calculating Frequency ($f$): Once period $T$ is read from the time axis, calculate frequency using $f = \frac{1}{T}$.