1. How Motion Changes During an Oscillation
When an object oscillates, its position, speed, and the force acting on it keep changing smoothly in a repeating manner. Understanding displacement, velocity, and acceleration together helps build a clear picture of how the motion evolves at every moment.
I find it helpful to imagine a pendulum or a mass on a spring: sometimes it moves fast, sometimes slow, sometimes stops for a split second. All of this is captured by these three quantities.
2. Displacement in Oscillation
Definition: Displacement is the distance of the object from its mean (equilibrium) position at any instant, along with direction.
In oscillations, the object keeps moving to the left or right (or up and down), and at every moment its displacement tells you exactly where it is.
2.1. How Displacement Changes Over Time
At the extreme positions, displacement is maximum. At the centre (mean position), displacement becomes zero.
This back-and-forth change makes displacement vary like a sine or cosine wave.
2.2. Mathematical Form
The simplest way to express displacement in oscillation is:
\( x(t) = A \sin(\omega t + \phi) \)
Here:
- \( A \) is the amplitude
- \( \omega \) is the angular frequency
- \( \phi \) is the phase constant
This formula gives the position at any instant.
3. Velocity in Oscillation
Definition: Velocity in oscillation is the rate at which displacement changes with time.
Sometimes the object moves fast (near the centre), and sometimes it slows down (near the extremes). Velocity captures this changing speed and direction.
3.1. How Velocity Behaves
Velocity is maximum at the mean position because the restoring force has accelerated the object for some time.
Velocity becomes zero at the extreme positions because the object momentarily stops before reversing direction.
So the velocity graph looks like a cosine curve if displacement is a sine curve.
3.2. Mathematical Expression for Velocity
Velocity is the derivative of displacement:
\( v(t) = \dfrac{dx}{dt} = A \omega \cos(\omega t + \phi) \)
This expression shows how velocity changes smoothly during the motion.
4. Acceleration in Oscillation
Definition: Acceleration in oscillation is the rate at which velocity changes with time.
Acceleration is directly tied to the restoring force, so it tells us a lot about how the oscillating system pulls the object back toward the mean position.
4.1. How Acceleration Changes
Acceleration is maximum at the extremes because the restoring force is strongest there.
Acceleration is zero at the mean position because the restoring force momentarily becomes zero there.
Interestingly, acceleration always points toward the mean position.
4.2. Mathematical Expression for Acceleration
Acceleration is the derivative of velocity (or the second derivative of displacement):
\( a(t) = \dfrac{dv}{dt} = -A \omega^2 \sin(\omega t + \phi) \)
The negative sign shows that acceleration is always directed opposite to displacement — this is what makes the motion oscillatory.
5. How Displacement, Velocity and Acceleration Relate to Each Other
All three quantities follow a repeating pattern, but they reach their maximum and zero values at different times:
- When displacement is maximum, velocity is zero and acceleration is maximum.
- When displacement is zero, velocity is maximum and acceleration is zero.
- Velocity and displacement are out of phase by 90°, and acceleration is 180° out of phase with displacement.
Once you visualize this shifting behaviour, the whole oscillatory motion becomes very intuitive.
6. A Simple Way to Picture the Three Together
I like to imagine a mass on a spring:
- At the extreme top: displacement is maximum upward, velocity is zero, acceleration is maximum downward.
- At the centre: displacement is zero, velocity is maximum, acceleration is zero.
- At the extreme bottom: displacement is maximum downward, velocity is zero, acceleration is maximum upward.
These three quantities dance together in a repeating pattern — one reaches its peak while the other falls to zero, creating the smooth, continuous feel of oscillatory motion.