The Equation of SHM:
The equation of motion for a particle in SHM is given by:
* x(t) = A * sin(ωt + φ)
where:
* x(t) is the displacement from the mean position at time t
* A is the amplitude (maximum displacement)
* ω is the angular frequency
* φ is the phase constant
Acceleration in SHM:
To find the acceleration, we differentiate the displacement equation twice with respect to time:
1. Velocity: v(t) = dx/dt = Aω * cos(ωt + φ)
2. Acceleration: a(t) = dv/dt = -Aω² * sin(ωt + φ)
Relationship between Acceleration and Displacement:
Notice that the acceleration equation has the same sine function as the displacement equation. This means:
* a(t) = -ω² * x(t)
Key Point: The negative sign indicates that the acceleration is always directed opposite to the displacement. This is what makes the motion "harmonic" – the restoring force always pulls the particle back towards the equilibrium position.
Inverse Proportionality:
The equation a(t) = -ω² * x(t) shows that the acceleration is proportional to the displacement. However, since there's a negative sign, it implies an inverse relationship. This means:
* As the displacement increases, the magnitude of the acceleration increases, but in the opposite direction.
* As the displacement decreases, the magnitude of the acceleration decreases.
In summary, the acceleration of a particle in SHM is inversely proportional to its displacement from the mean position. This relationship is fundamental to understanding the oscillatory nature of simple harmonic motion.
