Displacement, Velocity, and Acceleration in Simple Harmonic Motion

Introduction

Key Concepts

1. Simple Harmonic Motion (SHM)

Simple Harmonic Motion refers to a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that displacement. Mathematically, SHM can be described by the equation: $$ F = -kx $$ where \( F \) is the restoring force, \( k \) is the force constant, and \( x \) is the displacement from the equilibrium position.

2. Displacement in SHM

Displacement in SHM refers to the distance and direction of the oscillating object from its equilibrium position at any given time. It is a vector quantity and can be expressed as: $$ x(t) = A \cos(\omega t + \phi) $$ where:

• A is the amplitude (maximum displacement)
• \(\omega\) is the angular frequency
• t is the time
• \(\phi\) is the phase constant

The cosine function indicates that the motion is sinusoidal, characteristic of SHM.

3. Velocity in SHM

Velocity in SHM is the rate of change of displacement with respect to time. It describes how quickly the object moves through its oscillatory path. The velocity as a function of time is given by: $$ v(t) = \frac{dx(t)}{dt} = -A \omega \sin(\omega t + \phi) $$ Key points about velocity in SHM:

• Velocity is maximum when displacement is zero.
• Velocity is zero at maximum displacement.
• The motion is sinusoidal and 90 degrees out of phase with displacement.

4. Acceleration in SHM

Acceleration in SHM is the rate of change of velocity with respect to time. It indicates how quickly the velocity of the object is changing during its oscillatory motion. The acceleration as a function of time is: $$ a(t) = \frac{dv(t)}{dt} = -A \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t) $$ Key characteristics of acceleration in SHM:

• Acceleration is directly proportional to displacement but opposite in direction.
• The object experiences maximum acceleration at maximum displacement.
• The motion is sinusoidal and 180 degrees out of phase with displacement.

5. Angular Frequency and Period

Angular frequency (\(\omega\)) is a measure of how quickly the oscillations occur and is related to the period (\(T\)) and frequency (\(f\)) of the motion: $$ \omega = 2\pi f = \frac{2\pi}{T} $$ The period is the time taken for one complete cycle of oscillation, while frequency is the number of cycles per second. These quantities are inversely related.

6. Energy in SHM

In Simple Harmonic Motion, energy oscillates between kinetic and potential forms:

• Kinetic Energy (KE): Maximum when displacement is zero.
• Potential Energy (PE): Maximum at maximum displacement.

The total mechanical energy (\(E\)) in SHM remains constant and 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.

7. Phase Constant

The phase constant (\(\phi\)) determines the initial conditions of the oscillation, such as the starting position and direction of motion. It shifts the cosine wave horizontally and is crucial for matching SHM equations to specific physical scenarios.

8. Equations of SHM

The primary equations governing SHM are:

• Displacement: \( x(t) = A \cos(\omega t + \phi) \)
• Velocity: \( v(t) = -A \omega \sin(\omega t + \phi) \)
• Acceleration: \( a(t) = -A \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t) \)

These equations describe the harmonic oscillatory motion in terms of position, velocity, and acceleration over time.

9. Damped and Driven Harmonic Motion

While SHM assumes no energy loss, real-world oscillations often experience damping (energy loss) or are subject to external forces (driven SHM):

• Damped SHM: Incorporates a damping force proportional to velocity, reducing amplitude over time.
• Driven SHM: Involves an external periodic force, allowing sustained oscillations.

These variations lead to more complex behaviors, essential for understanding systems like pendulums and springs in real environments.

10. Applications of SHM

SHM principles are applied in various fields:

• Designing springs and suspension systems.
• Understanding molecular vibrations.
• Analyzing electrical circuits with inductors and capacitors.
• Studying wave phenomena and sound vibrations.

These applications demonstrate the versatility and importance of SHM in both theoretical and practical contexts.

Comparison Table

Aspect	Displacement	Velocity	Acceleration
Definition	Position relative to equilibrium	Rate of change of displacement	Rate of change of velocity
Equation	$x(t) = A \cos(\omega t + \phi)$	$v(t) = -A \omega \sin(\omega t + \phi)$	$a(t) = -\omega^2 x(t)$
Maximum Value	Amplitude (\(A\))	$A \omega$	$A \omega^2$
Phase Relationship	Reference	90° out of phase with displacement	180° out of phase with displacement
Significance	Defines the oscillatory path	Describes motion speed	Indicates force direction

Summary and Key Takeaways