Simple Harmonic Motion is the periodic motion of an object in which the restoring force is proportional to the displacement. This is usually represented in a sinusoidal graph. This video gives a wide overview of Simple Harmonic Motion that we will be going over more of later in this.
Important Definitions:
Periodic Motion- A motion that is repeated over a certain amount of time. The restoring force is not proportional to the displacement.
Restoring force- The force that pushes the object back towards equilibrium.
Simple harmonic motion- Periodic motion in which the restoring force is proportional to the displacement (as the displacement increases the force needed to return the object to equilibrium increases). They are represented by a sin or cos function.
Equilibrium position- A position where there is no velocity and no displacement.
Amplitude (A)- The maximum displacement from the equilibrium.
Period (T)- The time it takes to complete one cycle.
Frequency (f)- How many cycles in one second
f = 1/T and the units are s^-1 or Hz
Angular frequency- ω- The number of rotations in one second.
Restoring force- The force that pushes the object back towards equilibrium.
Simple harmonic motion- Periodic motion in which the restoring force is proportional to the displacement (as the displacement increases the force needed to return the object to equilibrium increases). They are represented by a sin or cos function.
Equilibrium position- A position where there is no velocity and no displacement.
Amplitude (A)- The maximum displacement from the equilibrium.
Period (T)- The time it takes to complete one cycle.
Frequency (f)- How many cycles in one second
f = 1/T and the units are s^-1 or Hz
Angular frequency- ω- The number of rotations in one second.
Energy Transfers:
During Simple Harmonic Motion, there is a constant energy transfer as the object oscillates and loses potential energy and gains kinetic energy. When the object is at the maximum displacement it has all potential energy and all of the energy is potential because there is no movement and it is the furthest stretch that it can be. When it is passing through equilibrium it has the most kinetic energy since it is moving and there is no stretch of the spring.
This chart shows the energy transfer as an object goes through simple harmonic motion. As we can see the equilibrium is at 0 and that's where the kinetic energy is at the most and while the object is at its amplitude the potential energy is as its maximum.
http://www.unistudyguides.com/wiki/Oscillations
Variables that Effect Period:
The Period is the amount of time an object takes to make one cycle. For example when you pull a pendulum back and release the time it takes to get back to the point that you let it go from is called the period. Mathematically the period is 1/Frequency = 2π / ω. We learned about two different types of simple harmonic motion: A spring and a Pendulum. They both have different formulas and variables.
https://ncalculators.com/mechanical/simple-harmonic-motion-time-period-calculator.htm
This shows that the period of a spring is effected by the mass which will increase the period of the spring and that if the spring constant increases then the period will decrease. This makes sense because if you place a lot of mass on a spring it seems to be more "bouncy" while if you get a tighter spring then it doesn't bounce as much.
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https://socratic.org/questions/a-pendulum-swings-back-and-forth-with-a-period-of-0-5-s-what-is-the-length-of-th
This shows that a longer pendulum arm will lead to increased period while a stronger gravitational force will lead to a lesser period. This is the equation as the spring one, just wit different variables substituted in to fit te situation.
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Initial vs. Gravitational Mass:
The difference between gravitational mass and initial mass is not that they are different values. Mass is a universal constant of matter that does not depend on gravity. Inertial mass is based off of an objects ability to resist change and can be measure by placing an object into simple harmonic motion using a spring with a known spring constant and finding out the period, once you know that you can calculate for the mass. Gravitational mass is the same value as inertial mass it is just represented by how a gravitational field acts on an object. A simple scale can find gravitational mass.
Periodic Motion or Simple Harmonic Motion?
Not all periodic motion is simple harmonic motion. Simple harmonic motion requires the restoring force to be proportional to the displacement and is represented by a sin or cos graph. Periodic motion is just a motion that is repeated over a certain period of time.
https://www.slideshare.net/chelimin/h2-physics-2011-topic-15-oscillations-lecture-1
These are graphs of real life examples that while they are periodic motion are not simple harmonic motion. They repeat but the graph is not a sin or cos graph and the restoring force is not proportional to the displacement.
Determining Frequency and Wavelength from a Graph:
Finding amplitude from a graph is quite simple. It is the distance from the peak of the curve to the midline of the function, in most cases 0. To find the period of the function which is just the time it takes for it to complete one cycle all you have to do is measure the distance from peak to peak or where the function starts to repeat. After you find that you can plug that into Frequency = 1/T to find the frequency. The wavelength formula is Wave Velocity/Frequency.
http://quintic.com/education/case_studies/Simple_Harmonic_Motion.html
http://quintic.com/education/case_studies/Simple_Harmonic_Motion.html
Connecting Different Representations:
Amplitude: This is the distance from the midline to the peak of a velocity time graph.
Period: The time between peaks in a distance time graph
The maximum velocity is when the object is moving through the equilibrium and is where the object has the most kinetic energy.
The maximum and minimum of the function are the points where the function is stopped and has the most potential energy.
The reason that the acceleration is flipped from the distance time graph is because it is representing the restoring force that has to be opposite of the displacement which is also the net force of the object. Therefore, the restoring force is directed in the opposite direction of the displacement.
https://www.askiitians.com/iit-study-material/iit-jee-physics/oscillation/graphical-representation-of-shm/
Period: The time between peaks in a distance time graph
The maximum velocity is when the object is moving through the equilibrium and is where the object has the most kinetic energy.
The maximum and minimum of the function are the points where the function is stopped and has the most potential energy.
The reason that the acceleration is flipped from the distance time graph is because it is representing the restoring force that has to be opposite of the displacement which is also the net force of the object. Therefore, the restoring force is directed in the opposite direction of the displacement.
https://www.askiitians.com/iit-study-material/iit-jee-physics/oscillation/graphical-representation-of-shm/
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