Have you ever seen a slinky go down a set of stairs? It's pretty cool. If you haven't seen it yet, go and find a slinky and go to the top of a flight of stairs. Now, put the slinky on the very edge of the top stair and grab the end that isn't touching the floor. Pull the end up and outward so that one end of the slinky is on the floor and the other hangs over the edge of the stair. Drop the end of the slinky that you were holding. The slinky should descend the stairs independently, transferring its energy from step to step.
Fig. 1 - The slinky is a great example of a simple harmonic oscillator.
The slinky can do this because it is a simple harmonic oscillator. The energy in the simple harmonic motion of the slinky allows it to go from step to step. This article will teach us about energy in simple harmonic motion: its definition, formula and derivative, and equilibrium position. Then, we will dive into some examples of energy in simple harmonic motion to cap it off.
Energy in Simple Harmonic Motion Definition
Just like kinetic and potential energy, there is also energy that comes from simple harmonic motion. Additionally, I need to explain an essential distinction right off the bat. I will use the words "simple harmonic motion" and "simple harmonic oscillator" interchangeably throughout this article. However, they are not interchangeable. A simple harmonic oscillator is an object that undergoes simple harmonic motion. The oscillator is the object, and the motion refers to actual oscillatory motion. But to understand both concepts, we must first define restoring forces.
Restoring Forces
The universe likes to be in equilibrium; it wants to be balanced. The universe has laws to correct the imbalance whenever something disrupts that balance. One of these laws involves restoring forces. When an object is in equilibrium, everything is great. To quote the Lego Movie, "Everything is awesome!" But once that object moves from equilibrium, the balance has been destroyed, and the universe gets angry.
A restoring force is exerted opposite to an object's displacement from an equilibrium position.
A restoring force is a force that acts on that object to put it back to its equilibrium position: to put it back into balance. That is the reason the restoring force always acts opposite to an object's displacement; it is trying to put the object back in its rightful place. That "rightful place" is the equilibrium position.
The equilibrium position is the location where no net force is acting on a system or object.
For energy in simple harmonic motion, the equilibrium position is where there is no energy when the object is stationary because there are no net forces on the object or system.
Fig. 2 - Point B is the equilibrium: the place where the displacement \(x=0\) and the net forces acting on the object \(F_\text{net} = 0\).
Energy in Simple Harmonic Motion and its Derivation
Before we get into the energy of simple harmonic oscillators, we need to talk about simple harmonic motion, a type of periodic motion.
Simple harmonic motion results when the magnitude of a restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position.
The derived equation for simple harmonic motion is given by:
$$ma_x = -k\Delta x$$
where \(m\) is the mass, \(a_x\) is the acceleration in the horizontal direction, \(k\) is the harmonic oscillator constant, and \(\Delta x\) is the displacement.
Energy in Simple Harmonic Motion Formula
The mechanical energy of simple harmonic motion is the sum of its potential and kinetic energies:
$$E_\text{tot} = U + K$$
where \(E_\text{tot}\) is the total mechanical energy, \(U\) is the potential energy, and \(K\) is the kinetic energy.
Fig. 3 - This image shows how potential and kinetic energy have an inverse relationship, that is, as one decreases the other increases. But, due to conservation of energy, their combination cannot go over the total energy of the system.
As you can see from the equation and graph above, potential energy and kinetic energy have quite a rocky relationship. The more potential energy there is, the less kinetic energy there has to be to keep the total energy balanced. Simple harmonic motion is nothing more than a balancing act.
Conservation of energy governs this balance. For energy to be conserved, the total energy of an isolated system with simple harmonic motion must be constant. Therefore, to keep the balance, the potential energy is at a maximum when the kinetic energy is at a minimum. Furthermore, the kinetic energy is at a maximum when the potential energy is at a minimum. As one goes up, the other must go down, or the energy will not remain constant: quite the rocky relationship.
An object's minimum kinetic energy possible in simple harmonic motion is zero.
We can think of harmonic motion in terms of potential energy. Potential energy is energy inherent in an object based on its position relative to another object. Therefore, if my two objects were a ball and the earth, the higher I move the ball from the earth's surface, the more potential energy it gains. This is similar to energy stored in simple harmonic oscillators. As the object is displaced from its equilibrium more, the maximum potential energy it can experience is increased, so its total energy is ameliorated.
$$E_\text{total} = \frac{1}{2}\\k A^2$$
is the equation we use to describe the energy for a spring object system with \(E_\text{total}\) as the total energy of the system, \(k\) as the spring constant, and \(A\) as the amplitude.
Fig. 4 - A spring is a perfect example of a simple harmonic oscillator that exhibits simple harmonic motion.
Energy in Simple Harmonic Motion Example
Now armed with all this knowledge on simple harmonic motion, let's do some examples.
Energy in Simple Harmonic Motion Derivation
First, we'll derive the equation for the simple harmonic motion of a spring.
Explain how to go from the equation
$$E_\text{tot} = U + K$$
to our equation for the total energy of a spring-object system
$$E_\text{total} = \frac{1}{2}\\k A^2\mathrm{.}$$
Solution: Knowing that the total energy of a simple harmonic oscillator equals its potential plus its kinetic energy gives us a hint for solving this problem. The potential energy in a spring is given by the formula
$$U=\frac{1}{2}\\kx^2$$
because this is equivalent to the total work done on the system. The work done on a system is equal to the energy transferred into or out of the system, which is precisely what this formula gives.
The kinetic energy formula is
$$K=\frac{1}{2}\\mv^2\mathrm{,}$$
but oscillatory motion is written in terms of angular velocity. Therefore, we have to rewrite the velocity like this:
$$v^2=\omega ^2 (A^2 - x^2)\mathrm{.}$$
Our new equation would then become
$$K= \frac{1}{2}\\m\omega ^2 (A^2 - x^2)\mathrm{.}$$
Knowing that
$$\frac{k}{m} = \omega ^2$$
allows us to make the substitution \(k\) for
$$m\omega ^2$$
so that we get
$$K=\frac{1}{2}\\k(A^2 - x^2)$$
for the final kinetic energy equation.
We then add these two equations together to get the total energy of the spring system:
$$\frac{1}{2}\\ kx^2 + \frac{1}{2}\\k (A^2 - x^2)\mathrm{.}$$
Using the distributive property gives
$$\frac{1}{2}\\ kx^2 + \frac{1}{2}\\kA^2 - \frac{1}{2}\\x^2\mathrm{,}$$
which cancels out the two \(\frac{1}{2}\\kx^2\) terms and leaves us with a final total energy for the spring-mass system of
$$E_\text{total}=\frac{1}{2}\\kA^2\mathrm{.}$$
Example 2
Now, we'll use that equation to find the energy of a spring-object system.
What is the energy in a spring-object system whose spring has a constant \(k\) of \(5\,\mathrm{\frac{N}{m}\\}\) and an amplitude of \(3\,\mathrm{m}\)?
Recall our simple harmonic motion energy equation for a spring-object system:
$$E_\text{total} = \frac{1}{2}\\k A^2\mathrm{.}$$
According to the problem, we have a \(k\) value of
$$k = 5\,\mathrm{\frac{N}{m}\\}$$
and an amplitude of
$$A= 3\,\mathrm{m}\mathrm{.}$$
Replacing these values in our equation yields
$$E_\text{total} = \frac{1}{2}\\ (5.00\,\mathrm{\frac{N}{m}\\}) (3.00\,\mathrm{m})^2\mathrm{,}$$
which gives us a total energy of
$$E_\text{total}= 22.5\,\mathrm{N\,m}\mathrm{.}$$
Newton meters \((\mathrm{N\,m})\) are the same thing as joules \((\mathrm{J})\).
Finally, we've reached the bottom of a long staircase; here is the essential knowledge for the energy of simple harmonic oscillators.
Energy of Simple Harmonic Oscillators - Key takeaways
- A simple harmonic oscillator is an object that undergoes simple harmonic motion. The oscillator is the object, and the motion refers to actual oscillatory motion.
- A restoring force is exerted opposite to an object's displacement from an equilibrium position.
- The equilibrium position is the location where no net force is acting on a system or object.
- Simple harmonic motion results when the magnitude of a restoring force exerted on an object is proportional to that object’s displacement from its equilibrium position.
- The derived equation for simple harmonic motion is
$$ma_x = -k\Delta x\mathrm{.}$$
The mechanical energy of simple harmonic motion is the sum of its potential and kinetic energies:
$$E_{tot} = U + K\mathrm{.}$$
For energy to be conserved, the total energy of an isolated system with simple harmonic motion must be constant. Therefore, to keep the balance, the potential energy is at a maximum when the kinetic energy is at a minimum. Furthermore, the kinetic energy is at a maximum when the potential energy is at a minimum.
An object's minimum kinetic energy possible in simple harmonic motion is zero.
As the object on a spring is displaced from its equilibrium more, the maximum potential energy it can experience is increased, so its total energy is ameliorated.
$$E_{total} = \frac{1}{2}\\k A^2$$ is the equation we use to describe the energy for an oscillating spring-mass system.
References
- Fig. 1 - Slinkies (https://pxhere.com/en/photo/1559395) by rawpixel.com (https://pxhere.com/en/photographer/795663) is licensed under Public Domain (https://creativecommons.org/publicdomain/zero/1.0/)
- Fig. 2 - Pendulum (https://physics.stackexchange.com/questions/170291/forces-acting-on-an-shm) by RogUE (https://physics.stackexchange.com/users/60846/rogue) is licensed by CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0/)
- Fig. 3 - Conservation of Energy in Simple Harmonic Motion, StudySmarter Originals
- Fig. 4 - Conservation of Energy in a Spring (https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/15-2-energy-in-simple-harmonic-motion/) by OpenStax CNX (https://openstax.org/books/university-physics-volume-1/pages/preface) is licensed by CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)
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