Simple Harmonic Motion

What is Hooke's Law and how is it related to simple harmonic motion?

If a mass is attached to a spring and then displaced from its initial resting position, it will oscillate about the initial position in simple harmonic motion.

Hooke's law states that the restoring force that is required to either extend or compress the spring from its initial resting position by a distance x, is proportional to the spring constant k, which is a characteristic of the spring's stiffness as shown below where F is the force, k is the constant, and x is the displacement. The negative sign on the formula below indicates that the force has a negative direction from the displacement. The period of an oscillating spring can also be found using the equation below where T is period, and m is the mass of the spring.

\[\text{Hooke's Law: } F[N] = -k[N/m] \cdot [m] \qquad \text{Newton's second law: } F[N] = m[kg] A[m/s^2] \qquad T [s] = 2 \pi \sqrt{\frac{m}{k}}\]

Hooke's law has the same form as Newton's second law, where mass is the reciprocal of the spring's stiffness and acceleration is the reciprocal of the negative displacement. Hence, the acceleration in simple harmonic motion is proportional to the displacement and has the opposite direction (see below; where x is the distance of a mass oscillating from its equilibrium position).

What are the equations for simple harmonic motion?

There are various equations used to describe a mass performing simple harmonic motion.

Simple harmonic motion period equation

The time period T, of an object performing simple harmonic motion, is the time it takes for a system to go through one full oscillation and return to its equilibrium position. One full oscillation is assumed to be completed when an object has moved from its initial position, reached the two maximum displacement points, and then returned to its initial position.

The time period can be found from the equation below, where ω is the rate of change of angular displacement with respect to time, T is the period, and f is the number of full oscillations completed in one second.

\[F [Hz] = \frac{1}{T} [s]\omega [rad/s] = \frac{2 \pi}{T[s]} = 2 \pi \cdot f\]

Simple harmonic motion acceleration and displacement equation

The maximum acceleration, a, of an object oscillating in simple harmonic motion is proportional to the displacement, x, and the angular frequency ⍵. The formula indicates that the acceleration has an opposite direction from the displacement indicated from the minus sign. It also shows that the acceleration reaches its maximum when the displacement is at the maximum amplitude, which is the furthest point from equilibrium.

\[\alpha_{max} [m/s^2] = -\omega ^2 [rad/s] \cdot [m]\]

The given equation is described below, where an acceleration vs position graph illustrates that acceleration is a function of displacement. The slope of the given graph is equal to the negative squared angular frequency as shown in the equation below. The maximum and minimum displacement are therefore + x₀ and -x₀, as respectively shown.

\[\text{slope} = \frac{a}{x} = -\omega^2[rad/s]\]

The position of an object in harmonic motion can be found using the equation below if the angular frequency and amplitude at a given time are known.

\[x(t) = x_0 \sin(\omega t)\]

This equation can be used when the object is oscillating from the initial equilibrium position. A sine graph can be used to describe this motion as shown in the figure below, which illustrates the example of a pendulum starting from the equilibrium position.

If an object is oscillating from its maximum displacement position where the amplitude is equal to either -x₀ or x₀, then the equation below can be used.

height="25" id="2135825" \(x(t) = x_0 \cos(\omega t)\)

An illustration of a pendulum example starting to oscillate at its maximum amplitude position can be described by a cosine graph and equation as shown below.

Simple harmonic motion speed equation

The speed of an object oscillating in simple harmonic motion at any given time can be found using the equation below where Vo is the maximum velocity, t is time, and ω is the angular frequency.

\(V(t) = V_{max} \cos (\omega t) \qquad V_{max} = \omega \cdot x_0\)

This equation can also be derived from the position equation by deriving in terms of time; remember, velocity is the derivative of position over time. Another equation is used to describe the behaviour of speed with respect to the displacement and frequency of the harmonic oscillator shown below, where Xo is the amplitude and X is the displacement.

\(V = \pm \omega \sqrt{x_0^2 - x^2}\)

Simple harmonic motion acceleration equation

The acceleration of an object oscillating in simple harmonic motion at any given time can be found using the equation below, where a_max is the maximum acceleration, t is time, and ω is the angular frequency. This equation can also be derived from the velocity equation by deriving in terms of time, as acceleration is the derivative of velocity over time.

\(a (t) = -a_{max} \cdot \cos (\omega t) \qquad a_{max} = \omega^2 \cdot x\)

What are phase shift and phase angle?

When the initial position of the oscillating mass m at the initial time is not equal to the amplitude, and the initial velocity is not zero, then the resulting cosine function representing the motion of the mass will appear slightly shifted to the right.

This is known as a phase shift and it can be measured in terms of phase angle φ measured in rad. When phase shift is present, the equations of simple harmonic motion that were introduced as a function of time can also be written as a function of the phase angle.

\(x(t) = x_0 \cdot \cos(\omega t + \phi) \quad u(t) = V_{max} \cdot \sin (\omega t+ \phi) \quad a(t) = -a_{max} \cdot \cos(\omega t + \phi)\)

The phase angle can be determined from the position of the mass m oscillating, or its graph. The phase shift can be described as an angle measured in radians using the equation below where ω is the angular frequency, t is the time, and \(\phi_0\) is the initial phase shift. The table below describes the phase shift in terms of angle and cycle.

\(\phi = \omega t + \phi_0\)