Frequency & Period of SHM

Introduction

Key Concepts

Definition of Simple Harmonic Motion

Simple Harmonic Motion 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. Mathematically, it is expressed as: $$ F = -kx $$ where \( F \) is the restoring force, \( k \) is the force constant, and \( x \) is the displacement from the equilibrium position.

Displacement, Velocity, and Acceleration in SHM

In SHM, the displacement of the oscillating object varies sinusoidally with time. The position as a function of time can be described by: $$ x(t) = A \cos(\omega t + \phi) $$ where:

• A is the amplitude of motion.
• \(\omega\) is the angular frequency.
• \(\phi\) is the phase constant.

Frequency and Period Defined

Frequency (\( f \)) and period (\( T \)) are two fundamental characteristics of SHM.

• Frequency (f): The number of complete oscillations per unit time, measured in hertz (Hz).
• Period (T): The time taken to complete one full oscillation, measured in seconds (s).

Angular Frequency

Angular frequency (\( \omega \)) relates to frequency and period as follows: $$ \omega = 2\pi f = \frac{2\pi}{T} $$ It represents the rate of change of the phase of the oscillation and is measured in radians per second (rad/s).

Energy in SHM

In SHM, energy oscillates between kinetic and potential forms without any loss, assuming no damping. The total mechanical energy (\( E \)) is given by: $$ E = \frac{1}{2} k A^2 = \frac{1}{2} m \omega^2 A^2 $$ where \( m \) is the mass of the oscillating object.

Derivation of Period and Frequency

For a mass-spring system undergoing SHM, Hooke's Law (\( F = -kx \)) and Newton’s second law (\( F = ma \)) can be combined: $$ ma = -kx \Rightarrow m \frac{d^2x}{dt^2} = -kx $$ This differential equation leads to the solution: $$ x(t) = A \cos\left(\sqrt{\frac{k}{m}} t + \phi\right) $$ Comparing with the standard SHM equation, we identify: $$ \omega = \sqrt{\frac{k}{m}} \Rightarrow f = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $$ Thus, the period is: $$ T = \frac{1}{f} = 2\pi \sqrt{\frac{m}{k}} $$

Dependence of Period on Mass and Spring Constant

From the period equation: $$ T = 2\pi \sqrt{\frac{m}{k}} $$ we observe that:

• T increases with an increase in mass (\( m \)).
• T decreases with an increase in the spring constant (\( k \)).

Damped and Driven SHM

Real-world oscillatory systems often experience damping and may be subjected to external driving forces:

• Damped SHM: Includes a damping force (often proportional to velocity) that causes the amplitude to decrease over time.
• Driven SHM: Involves an external periodic force driving the system, which can lead to phenomena like resonance.

Applications of Frequency and Period in SHM

Understanding frequency and period is essential in various applications:

• Timekeeping: Oscillations in pendulums and quartz crystals serve as the basis for clocks and watches.
• Vibrational Analysis: Used in engineering to assess the stability of structures and machinery.
• Acoustics: The frequency of sound waves determines the pitch of the sound heard.
• Electrical Engineering: LC circuits exhibit oscillatory behavior with specific frequencies important for signal processing.

Graphical Representation of SHM

SHM can be graphically represented in different forms:

• Displacement vs. Time: A cosine or sine wave showing periodic motion.
• Velocity vs. Time: A sine or cosine wave out of phase with displacement.
• Acceleration vs. Time: A negative cosine or sine wave, indicating acceleration opposite to displacement.

Phase Difference in SHM

Phase difference indicates the relative displacement between two oscillating quantities. In SHM:

• The velocity is 90° out of phase with displacement.
• The acceleration is 180° out of phase with displacement.

Comparison Table

Summary and Key Takeaways