Mechanical Energy in Simple Harmonic Motion

Potential Energy in Simple Harmonic Motion

We consider a spring-mass system to find the potential energy expression in an example of simple harmonic motion. The spring force is a conservative force, so we can define the potential energy for it. As a quick reminder, a conservative force is any force that is independent of an object's path, such that it depends only on the displacement of the object. The spring's potential energy is the energy accumulated in the spring when it is compressed. To find the expression, we can determine the work done by the force solving an integral or, more easily, we can plot the spring force as a function of position and determine the area under the curve.

$$\triangle U=-\int_a^b{\overset\rightharpoonup F}_{\text{conservative}}\cdot\overset\rightharpoonup{\operatorname dr}$$

Fig 2 - Spring force as a function of position to illustrate how to determine the spring's potential energy by calculating the area below the curve.

From the above figure, we see that the area under the curve is a triangle. So, we can easily determine the expression of the spring's potential energy:

$$\begin{array}{rcl}U&=&\frac12(x)(kx),\\U&=&\frac12kx^2.\end{array},$$

where \(k\) is the spring constant that measures the stiffness of the spring in Newtons per meter \((\frac{\mathrm N}{\mathrm m})\) and \(x\) is the object's displacement from equilibrium in meters \((\mathrm m)\). This is a model based on experimental evidence of Hooke's Law.

We know the conservative force that acts on the spring-mass system, so we can solve the integral to find the potential energy stored in the spring.

$$U_s=-\int_0^x(-kx)dx=\frac12kx^2$$

Kinetic Energy in Simple Harmonic Motion

Now that we know energy is conserved in simple harmonic motion, and we have the expression for the potential energy, we can determine the expression for velocity, as we know the equation for the kinetic energy:

$$\begin{array}{rcl}K_i+U_i&=&K_f+U_f,\\\frac12mv_i^2+\frac12\omega^2mx_i^2&=&\frac12mv_f^2+\frac12\omega^2mx_f^2.\end{array}$$

Initially, we are at maximum displacement so, \(v_i=0,\;x_i=A,v_f=v,\;and\;x_f=x\). We substitute the values into the equation above and solve for the velocity:

$$\begin{array}{rcl}\frac12\omega^2mA^2&=&\frac12mv^2+\frac12\omega^2mx^2,\\v^2&=&\omega^2(A^2-x^2),\\v&=&\omega\sqrt{A^2-x^2}.\end{array}$$

Now that we know the expression for the velocity of the object undergoing simple harmonic motion, we can determine the equation for the kinetic energy of simple harmonic oscillators:

$$\begin{array}{rcl}K&=&\frac12mv^2,\\K&=&\frac12m\omega^2(A^2-x^2).\end{array}$$

Total Mechanical Energy in Simple Harmonic Motion

Now that we know the expressions for kinetic and potential energy in simple harmonic motion, we can determine the equation for the total mechanical energy of a simple harmonic oscillator.

$$\begin{array}{rcl}E&=&K+U,\\E&=&\frac12k(A^2-x^2)+\frac12kx^2,\\E&=&\frac12kA^2.\end{array}$$

Another way to visualize the total mechanical energy is to think in terms of the expressions for the position and velocity of a simple harmonic oscillator given by

$$\begin{array}{rcl}x&=&A\cos\left(\omega t+\phi\right),\\v&=&-A\omega\sin\left(\omega t+\phi\right).\end{array}$$

If we examine the energy of the system, we see that the potential energy oscillates as a cosine-squared function, while the kinetic energy oscillates as a sine-squared function.

$$\begin{array}{rcl}E&=&K+U,\\E&=&\frac12mv^2+\frac12kx^2,\\E&=&\frac12mA^2\omega^2\sin^2\left(\omega t+\phi\right)+\frac12kA^2\cos^2\left(\omega t+\phi\right),\\E&=&\frac12\cancel mA^2\left(\frac k{\cancel m}\right)\sin^2\left(\omega t+\phi\right)+\frac12kA^2\cos^2\left(\omega t+\phi\right),\\E&=&\frac12kA^2\sin^2\left(\omega t+\phi\right)+\frac12kA^2\cos^2\left(\omega t+\phi\right),\end{array}$$

where the trigonometric identity \(\cos^2\left(\theta\right)+\sin^2\left(\theta\right)=1\). The mechanical energy is given by

$$E=\frac12kA^2.$$

So, we see the total mechanical energy of the simple harmonic oscillator is the sum of the kinetic energy of the mass and the potential energy stored in the spring. The energy of the system is proportional to the square of the amplitude, such that the amplitude affects the total energy. Also, we see that the total energy in this system is constant over time.