Equations and graphs of simple harmonic motion Knowledge-Based Study

Study notes

1. Defining Simple Harmonic Motion (SHM)

   Simple Harmonic Motion is a special type of periodic oscillatory motion where the restoring force acting on an object is directly proportional to its displacement from equilibrium and directed toward that equilibrium position.

   According to Hooke's Law and Newton's second law, this condition can be expressed as:

   $$F = -kx \implies m a = -kx \implies a = -\left(\frac{k}{m}\right)x$$

   Because the acceleration is proportional to the negative of displacement, the resulting motion is sinusoidal over time.

   Displacement Equation

   Choosing the initial time such that the object starts from maximum positive displacement at $t = 0$, the position $x(t)$ as a function of time is given by:

   $$x(t) = A \cos(\omega t)$$

   Where:

   • $x(t)$: Instantaneous displacement at time $t$ (measured in meters, $\text{m}$).
   • $A$: Amplitude, which is the maximum magnitude of displacement from the equilibrium position ($\text{m}$).
   • $\omega$: Angular frequency (measured in radians per second, $\text{rad/s}$), related to frequency $f$ and period $T$ by $\omega = 2\pi f = \frac{2\pi}{T}$.

   Velocity as a Function of Time

   Velocity is the time rate of change of displacement. Differentiating displacement with respect to time gives:

   $$v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t)$$

   The maximum speed occurs when the oscillating mass passes through the equilibrium position ($x = 0$):

   $$v_{\max} = A\omega$$

   Acceleration as a Function of Time

   Acceleration is the time rate of change of velocity. Differentiating velocity with respect to time gives:

   $$a(t) = \frac{dv}{dt} = -A\omega^2 \cos(\omega t)$$

   The maximum magnitude of acceleration occurs at the extreme displacement positions ($x = \pm A$):

   $$a_{\max} = A\omega^2$$

   The Defining Relationship of SHM

   By substituting $x(t) = A \cos(\omega t)$ into the acceleration formula, we obtain the fundamental equation that characterizes all simple harmonic motion:

   $$a(t) = -\omega^2 x(t)$$

   Interpreting the Kinematic Graphs

   When plotted against time, displacement, velocity, and acceleration each form sinusoidal curves with the same period $T$, but shifted relative to one another in phase.

   • Displacement vs. Time: Follows a cosine curve when starting at maximum displacement ($x = +A$ at $t = 0$).
   • Velocity vs. Time: Follows a negative sine curve. When displacement is at an extreme ($x = \pm A$), velocity is zero. When displacement is zero (equilibrium), velocity reaches its maximum positive or negative value.
   • Acceleration vs. Time: Follows an inverted cosine curve. Acceleration is always directed opposite to displacement.

   Phase Differences

   The relative shifts between the waveforms can be described in terms of phase angles:

   • Velocity leads displacement by a phase angle of $\frac{\pi}{2}\text{ rad}$ (or $90^\circ$, corresponding to a quarter of a period, $\frac{T}{4}$).
   • Acceleration leads velocity by $\frac{\pi}{2}\text{ rad}$ ($90^\circ$).
   • Acceleration is completely out of phase with displacement by $\pi\text{ rad}$ (or $180^\circ$, corresponding to half a period, $\frac{T}{2}$).

   Summary of Critical Values

   Understanding where these kinematic quantities reach their extremes is essential for analyzing oscillatory systems:

   • At the Equilibrium Position ($x = 0$):
     • Displacement: $x = 0$
     • Restoring Force: $F = 0$
     • Acceleration: $a = 0$
     • Speed: Maximum ($|v| = v_{\max} = A\omega$)
   • At Maximum Displacement ($x = +A$ or $x = -A$):
     • Displacement: Maximum magnitude ($|x| = A$)
     • Restoring Force: Maximum magnitude ($|F| = kA$)
     • Acceleration: Maximum magnitude ($|a| = a_{\max} = A\omega^2$)
     • Speed: Zero ($v = 0$)