Try some values and see.Calpalned said:Homework Statement
My textbook states that for simple harmonic motion, the sinusoidal graph of the x (displacement) as a function of time can be created using the "Equation of motion".
Homework Equations
The equation of motion ##A \cos (\omega t + \phi)##
The Attempt at a Solution
I know that ##A## stretches the cosine graph vertically, but that the frequency is unaffected. How do ##\omega## and ##\phi## affect the graph?
Thank you so much! Is my understanding below valid?SammyS said:Try some values and see.
or ...
What do you know about shifting and stretching/shrinking of graphs in general?
What is the period of y = cos(x) ?
How does the graph of y = f(x + k) compare with the graph of y = f(x) ?
etc.
Correct.Calpalned said:Thank you so much! Is my understanding below valid?
##A## stretches/compresses the graph vertically
##\omega## affects the period. The larger ##\omega## is, the shorter the period.
##\phi## is the horizontal shift and it is negative. That is, a positive value of ##\phi## will shift the graph to the left.
Simple harmonic motion is a type of periodic motion where an object oscillates back and forth around an equilibrium position with a constant amplitude and frequency. It is characterized by a sinusoidal wave pattern on a graph.
The equation for simple harmonic motion is x = A sin(ωt + φ), where x is the displacement of the object from its equilibrium position, A is the amplitude, ω is the angular frequency, and φ is the phase constant.
To analyze a graph of simple harmonic motion, you can look at the amplitude, period, and frequency of the wave. The amplitude is the maximum displacement from the equilibrium position, the period is the time it takes for one complete cycle of oscillation, and the frequency is the number of cycles per unit time.
Some real-life examples of simple harmonic motion include a pendulum swinging back and forth, a mass on a spring oscillating up and down, and a guitar string vibrating.
In simple harmonic motion, the total mechanical energy (kinetic + potential) of the system remains constant. This is due to the fact that the force acting on the object is directly proportional to its displacement from equilibrium, resulting in a conservation of energy as the object oscillates back and forth.