Understanding Simple Harmonic Motion (SHM)

In simple harmonic motion, or SHM, displacement, velocity, and acceleration are all linked together. Let’s make this simple!

What is Displacement?

Displacement is how far something has moved from its starting position. In SHM, we can use a basic equation to show it:

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

Here’s what the letters mean:

• $x(t)$ is the position at time $t$.
• $A$ is the amplitude, which is the highest point it can reach.
• $\omega$ is the angular frequency, which tells us how fast it's moving.
• $\phi$ is the phase constant, helping adjust the position in time.

What is Velocity?

Velocity is how quickly displacement changes over time. We can find it by working with the displacement equation:

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

This means that velocity is at its highest when the sine part equals 1. This happens when the object is at its middle point, or equilibrium, where $x = 0$.

What is Acceleration?

Acceleration tells us how fast velocity changes. To find it, we differentiate the velocity equation:

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

From this, we see a key idea: acceleration always points back to the middle point (equilibrium), and it changes in the opposite direction of displacement.

Key Points to Remember

1. Velocity is highest when the object is at the middle point ($x = 0$), which means $v = \pm A \omega$.
2. Acceleration is highest at the maximum points ($x = \pm A$), leading to $a = \mp A \omega^2$.
3. The plus and minus signs remind us that velocity and acceleration are out of sync by 90 degrees. When velocity hits zero at the ends, acceleration is at its peak.

In SHM, knowing how displacement, velocity, and acceleration relate to each other helps us understand how energy moves in systems that swing back and forth!