Wave & Optics - Simple Harmonic Motion

Classification of Motion

• Motion Types Overview:

  • Periodic Motion: Motion that repeats itself at regular intervals of time.

    • Oscillatory Motion: To-and-fro motion about a fixed mean position.

      • Examples: Pendulum, vibrating string.

    • Non-Oscillatory Motion: Repeating motion that does not involve to-and-fro movement around a central mean position.

      • Examples: Hands of a clock, motion of the Sun / Earth's orbit.

  • Non-Periodic Motion: Motion that does not repeat itself at regular intervals.

    • Example: Projectile / Projective motion.

Fundamental Mechanics of Simple Harmonic Motion

• Definition and Characteristics:

  • Simple Harmonic Motion (SHM) is a specific type of oscillatory motion characterized by repeated back-and-forth ("to and fro") movement around a mean position.

  • In SHM, the magnitude of acceleration is directly proportional to the displacement from the mean position:         $a \propto x$         $a = k x$         where $a$ is acceleration, $x$ is displacement, and $k$ is a constant of proportionality.

• Restoring Force and Hooke's Law:

  • By Newton's second law of motion:         $F = m a$

  • Substituting the proportionality relation for acceleration directed towards the mean position yields the restoring force equation:         $F = -k x$

  • This force relation ($F = -k x$) directly models systems such as an oscillating spring, where the force acts in the direction opposite to displacement to restore the body to its equilibrium position.

Derivation of the Standard Equation of SHM

• Geometric Representation via Circular Motion:

  • Simple harmonic motion can be analyzed as the projection of uniform circular motion along a reference line or axis.

  • Consider a right triangle formed by the radius/amplitude vector $A$, the horizontal displacement $x$, and an angular position $\theta$:         $\cos(\theta) = \frac{x}{A}$

  • Solving for displacement $x$ gives:         $x = A \cos(\theta)$ $\quad \text{--- (1)}$

• Incorporating Angular Velocity:

  • Angular velocity $\omega$ is defined as the rate of change of angular displacement over time $t$:         $\omega = \frac{\theta}{t}$

  • Rearranging for the angle $\theta$:         $\theta = \omega t$ $\quad \text{--- (2)}$

• Standard SHM Displacement Equation:

  • Substituting equation (2) into equation (1):         $x = A \cos(\omega t)$

  • This represents the standard equation of Simple Harmonic Motion, where:

    • $x$ is the displacement at time $t$

    • $A$ is the maximum displacement or amplitude

    • $\omega$ is the angular velocity

    • $t$ is the elapsed time