๐ Understanding Simple Harmonic Motion (SHM) Graphs
Simple Harmonic Motion (SHM) is a specific type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. This results in oscillations around a central equilibrium position. We can represent these oscillations graphically by plotting displacement, velocity, and acceleration as functions of time.
๐ Displacement vs. Time
Displacement in SHM refers to the object's position relative to its equilibrium point at any given time. The graph of displacement versus time is a sinusoidal curve, either a sine or cosine function, depending on the initial conditions.
- ๐ Definition: The distance and direction of the object from its equilibrium position.
- ๐ Graph Shape: A sine or cosine wave. The amplitude (A) represents the maximum displacement, and the period (T) represents the time for one complete oscillation.
- ๐งฎ Equation: $x(t) = A \cos(\omega t + \phi)$, where $x(t)$ is the displacement at time $t$, $A$ is the amplitude, $\omega$ is the angular frequency, and $\phi$ is the phase constant.
๐ Velocity vs. Time
Velocity in SHM is the rate of change of displacement with respect to time. The graph of velocity versus time is also a sinusoidal curve, but it's 90 degrees out of phase with the displacement graph.
- ๐ Definition: The rate of change of displacement; how fast and in what direction the object is moving.
- ๐ Graph Shape: A sine or cosine wave, phase-shifted by $\frac{\pi}{2}$ relative to the displacement graph. When displacement is at its maximum (positive or negative), velocity is zero, and vice versa.
- ๐งฎ Equation: $v(t) = -A\omega \sin(\omega t + \phi)$, where $v(t)$ is the velocity at time $t$.
๐ Acceleration vs. Time
Acceleration in SHM is the rate of change of velocity with respect to time. The graph of acceleration versus time is also a sinusoidal curve, and it's 180 degrees out of phase with the displacement graph.
- ๐ Definition: The rate of change of velocity; how quickly the velocity is changing.
- ๐ Graph Shape: A sine or cosine wave, phase-shifted by $\pi$ relative to the displacement graph. When displacement is at its maximum, acceleration is also at its maximum but in the opposite direction.
- ๐งฎ Equation: $a(t) = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$, where $a(t)$ is the acceleration at time $t$.
๐ Comparing Displacement, Velocity, and Acceleration
Let's use a table to compare these three crucial aspects of SHM:
| Feature |
Displacement (x) |
Velocity (v) |
Acceleration (a) |
| Definition |
Position relative to equilibrium |
Rate of change of displacement |
Rate of change of velocity |
| Graph Shape |
Sine or Cosine wave |
Sine or Cosine wave (90ยฐ phase shift) |
Sine or Cosine wave (180ยฐ phase shift) |
| Equation |
$x(t) = A \cos(\omega t + \phi)$ |
$v(t) = -A\omega \sin(\omega t + \phi)$ |
$a(t) = -A\omega^2 \cos(\omega t + \phi)$ |
| Phase Relationship |
Reference |
$\frac{\pi}{2}$ out of phase with displacement |
$\pi$ out of phase with displacement |
| Maximum Value |
A (Amplitude) |
$A\omega$ |
$A\omega^2$ |
๐ Key Takeaways
- ๐ Phase Shifts: Velocity leads displacement by 90 degrees, and acceleration leads displacement by 180 degrees.
- ๐ Sinusoidal Nature: All three graphs (displacement, velocity, acceleration) are sinusoidal, but with different amplitudes and phase shifts.
- ๐ Equilibrium Point: At the equilibrium point, displacement is zero, velocity is maximum, and acceleration is zero.
- extremos Extrema Points: At the maximum displacement, velocity is zero, and acceleration is maximum (in the opposite direction).