**Chapter
25 : Simple Harmonic Motion**

**Introduction**

Simple Harmonic Motion (SHM) is a fundamental type of vibratory motion found in various natural phenomena and engineered systems. It is characterized by a sinusoidal oscillation, where the restoring force acting on an object is directly proportional to its displacement from the equilibrium position and directed opposite to that displacement. In this chapter, we explore the principles of SHM, deriving equations for displacement, velocity, acceleration, time period, and frequency, and discussing free, forced, and resonant vibrations.

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**Principles of Simple
Harmonic Motion**

**Equation of Motion**

The general equation of motion for a particle undergoing simple harmonic motion is given by:

*x(t)=Acos(Ï‰t+**Ï•**)* Where:

- x(t) is the displacement at time t,
- A is the amplitude (maximum displacement),
- Ï‰ is the angular frequency,
- Ï• is the phase angle.

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**Derivation of
Displacement, Velocity, and Acceleration**

**Displacement (x(t)**

*x(t)=Acos(Ï‰t+**Ï•**) *

**Velocity (v(t)**

*v(t)=−AÏ‰sin(Ï‰t+**Ï•**)*

**Acceleration (a(t))**

*a(t)=−AÏ‰ ^{2}cos(Ï‰t+*

*Ï•*

*)*

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**Time Period and
Frequency**

**Time Period (T)**

The time period is the time taken for one complete oscillation. It is related to the angular frequency (Ï‰) by:

T=2Ï€Ï‰

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**Frequency (f)**

The frequency is the number of oscillations per unit time. It is the reciprocal of the time period:

f=1/T

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**Simple Harmonic
Motion of a Cantilever**

A cantilever is a common example of a system undergoing simple harmonic motion. When a cantilever beam is displaced from its equilibrium position and then released, it oscillates back and forth. The displacement of the cantilever tip at any time tt can be described by the SHM equation.

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**Types of Vibrations**

**Free Vibrations**

Free vibrations occur when a system oscillates without any external force acting on it after an initial displacement. In the case of a cantilever, if it is displaced and then released, the subsequent motion is a free vibration.

**Forced Vibrations**

Forced vibrations occur when an external force is applied to a system at regular intervals, driving it to oscillate. In the context of a cantilever, applying a periodic force to the free end can result in forced vibrations.

**Resonant Vibrations**

Resonance occurs when the frequency of the externally applied force matches the natural frequency of the system. This leads to a significant increase in amplitude and can result in structural damage if not properly controlled. Understanding resonant frequencies is crucial in various engineering applications, such as preventing bridge collapse due to resonant vibrations induced by wind.

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**Conclusion**

Simple Harmonic Motion is a fundamental concept in physics, describing the repetitive oscillatory behavior observed in countless natural and engineered systems. The equations governing displacement, velocity, and acceleration provide a comprehensive understanding of the dynamics of simple harmonic motion. The principles of free, forced, and resonant vibrations further enhance our ability to analyze and design systems undergoing oscillatory motion. Whether it's the swinging of a pendulum, the vibrations of a guitar string, or the oscillations of a cantilever beam, simple harmonic motion plays a pivotal role in our understanding of dynamic systems.