The graph of Figure 18.15a, shown earlier, represents how the displacement of an oscillator varies during s.h.m. We have already mentioned that this is a sine curve. We can present the same information in the form of an equation. The relationship between the displacement x and the time t is as follows:

$x = {x_0}\sin \omega t$

where ${x_0}$ is the amplitude of the motion and $\omega $ is its frequency. Sometimes, the same motion is represented using a cosine function, rather than a sine function:

$x = {x_0}\cos \omega t$

The difference between these two equations is illustrated in Figure 18.19. The sine version starts at $x =0$; that is, the oscillating mass is at its equilibrium position when $t = 0$.
The cosine version starts at $x = {x_0}$, so that the mass is at its maximum displacement when $t = 0$.
Note that, in calculations using these equations, the quantity $\omega t$ is in radians. Make sure that your calculator is in radian mode for any calculation (see Worked example 2). The presence of the $\pi $ in the equation should remind you of this.

Acceleration and displacement

In s.h.m., an object’s acceleration depends on how far it is displaced from its equilibrium position and on the magnitude of the restoring force. The greater the displacement x, the greater the acceleration a. In fact, a is proportional to x. We can write the following equation to represent this:

$a =  - {\omega ^2}x$

where $a =the  acceleration$ of an object vibrating in s.h.m., $\omega $ is the angular frequency of the object, $x =displacement$ 

This equation shows that a is proportional to x; the constant of proportionality is ${\omega ^2}$. The minus sign shows that, when the object is displaced to the right, the direction of its acceleration is to the left.
The acceleration is always directed towards the equilibrium position, in the opposite direction to the displacement.
It should not be surprising that angular frequency $\omega $ appears in this equation. Imagine a mass hanging on a spring, so that it can vibrate up and down. If the spring is stiff, the force on the mass will be greater; it will be accelerated more for a given displacement and its frequency of oscillation will be higher.
The equation $a =  - {\omega ^2}x$ helps us to define simple harmonic motion. The acceleration a is directly proportional to displacement x; and the minus sign shows that it is in the opposite direction.
An object vibrates in simple harmonic motion if its acceleration is directly proportional to its displacement from its equilibrium position and is in the opposite direction to the displacement.
If a and x were in the same direction (no minus sign), the body’s acceleration would increase as it moved away from the fixed point and it would move away faster and faster, never to return.
Figure 18.20 shows the acceleration–displacement ($a – x$) graph for an oscillator executing s.h.m. Note the following:
- The graph is a straight line through the origin ($a \propto x$).
- It has a negative slope (the minus sign in the equation $a =  - {\omega ^2}x$). This means that the acceleration is always directed towards the equilibrium position.
- The magnitude of the gradient of the graph is ${\omega ^2}$.

- The gradient is independent of the amplitude of the motion. This means that the frequency f or the period T of the oscillator is independent of the amplitude and so a simple harmonic oscillator keeps steady time.
If you have studied calculus, you may be able to differentiate the equation for x twice with respect to time to obtain an equation for acceleration and thereby show that the defining equation $a =  - {\omega ^2}x$ is satisfied.

Equations for velocity

The velocity v of an oscillator varies as it moves back and forth. It has its greatest speed when it passes through the equilibrium position in the middle of the oscillation. If we take time $t = 0$ when the oscillator passes through the middle of the oscillation with its greatest speed ${v_0}$, then we can represent the changing velocity as an equation:

$x = {x_0}\,\cos \,\omega t$

We use the cosine function to represent the velocity since it has its maximum value when $t = 0$.
The equation $v = {v_0}\,\cos \,\omega t$ tells us how v depends on t. We can write another equation to show how the velocity depends on the oscillator’s displacement x:

$v =  \pm \omega \sqrt {{x_0}^2 - {x^2}} $

This equation can be used to deduce the speed of an oscillator at any point in an oscillation, including its maximum speed.

Maximum speed of an oscillator

If an oscillator is executing simple harmonic motion, it has maximum speed when it passes through its equilibrium position. This is when its displacement x is zero. The maximum speed ${v_0}$ of the oscillator depends on the frequency f of the motion and on the amplitude ${x_0}$. Substituting $x = 0$ in the equation:

$v = \omega \sqrt {{x_0}^2 - {x^2}} $

$x = 0$ when the speed is at a maximum:

${v_0} = \omega {x_0}$

According to this equation, for a given oscillation:

${v_0} \propto {x_0}$

A simple harmonic oscillator has a period that is independent of the amplitude. A greater amplitude means that the oscillator has to travel a greater distance in the same time–hence it has a greater speed.
The equation also shows that the maximum speed is proportional to the frequency. Increasing the frequency means a shorter period. A given distance is covered in a shorter time–hence it has a greater speed.
Have another look at Figure 18.15. The period of the motion is $0.40 s$ and the amplitude of the motion is $0.02 m$. The frequency f can be calculated as follows:

$\begin{array}{l}
f = \frac{1}{t}\\
 = \frac{1}{{0.40}}\\
 = 2.5\,Hz
\end{array}$

We can now use the equation ${v_0} = (2\pi f){x_0}$ to determine the maximum speed ${v_0}$: 

$\begin{array}{l}
{v_0} = (2\pi f){x_0} = (2\pi  \times 2.5) \times 2.0 \times {10^{ - 2}}\\
{v_0} \approx 0.31\,m\,{s^{ - 1}}
\end{array}$

This is how the values on Figure 18.15b were calculated.