⇒ The question we want to answer is this: how do the displacement, velocity, and acceleration of a simple harmonic oscillator vary with time?
⇒ This image gives us some insight
⇒ The following graph shows how the displacement of the mass varies with time if it is released from rest with an amplitude A
⇒ The graph has the shape of a cosine function, which can be written as: x = Acosθ
⇒ But the value of θ is 2π after one complete cycle so, at the end of the cycle x = Acos(2π)
⇒ However, we know that the oscillation is a function of t. The function that fits the equation is:
⇒ Once we have an equation that connects displacement with time, we can also produce equations that link velocity with time, and then also acceleration with time
⇒ We derive this assuming x = A when t = 0
⇒ Mathematicians will see that the velocity equation is the derivative of the displacement equation, and that the acceleration equation is the derivative of the velocity equation
⇒ Since the maximum value of a sine or cosine function is 1, we can write the maximum values for x, v, and a as follows:
⇒ We also write down one further useful equation now, which allows us to calculate the velocity, v, of an oscillating particle at any displacement, x:
⇒ This will be proved later when we consider the energy of an oscillating system
⇒ This shows graphically the relationship between x, v, and a. These graphs are related to each other
⇒ The graph of velocity, v, against time, t, links to the gradient of the displacement-time graph (x-t) graph because v = Δx⁄Δt
⇒ The graph of acceleration, a, against time, t, (c) links to the gradient of the velocity-time (v-t) graph (b) because a = Δv⁄Δt