SOLUTION: Find the trigonometric model for a simple harmonic motion that has initial displacement of 0 at t=0, and amplitude of 5 inches, and a frequency 4/3 cycles per second

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Question 1197680: Find the trigonometric model for a simple harmonic motion that has initial displacement of 0 at t=0, and amplitude of 5 inches, and a frequency 4/3 cycles per second
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
if you are graphing this using the sine function, i would think the equation would be:
y = 5 * sin(4/3 * x).
that would give you an amplitude of 5 inches with a displacement of 0 at x = 0 (x take the place of t in the graph).
there would be no horizontal or vertical shift.
the graph would look like this:



if you used the cosine function, then a horizontal shift would be required.
i think the equation would then be:
y = 5 * cos(4/3 * (x - 3pi/8))
that would give you an amplitude of 5 inches with a displacement of 0 at x = 0 (x take the place of t in the graph).
there would be no vertical shift, although a horizontal shift of 3pi/8 would be required.
the graph would look like this:



the graphs show that you would complete 4 full cycles in what would normall be 3 full cycles.
that's because the frequency is 4/3 * the normal frequency, which would be 1.

the graphs, themselves, would be identical, except for the equations used.

the initial displacement is 0 in both graphs, and the amplitube is 5 in both graphs.



Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find the trigonometric model for a simple harmonic motion that has initial displacement of 0
at t=0, and amplitude of 5 inches, and a frequency 4/3 cycles per second
~~~~~~~~~~~~~~~~~~ 

The frequency is  4/3 per second.  It means that the period is 3/4 of a second.


The problem says NOTHING about the phase shift.


It only says that the initial displacement is 0 (zero).


There are TWO harmonic (sinusoidal) models satisfying the imposed conditions.


One model is  y = 5%2Asin%28%28%282%2Api%29%2F%28%283%2F4%29%29%29%2At%29%29 = 5%2Asin%28%28%288%2Api%29%2F3%29%2At%29%29.


Another model is y = -5%2Asin%28%28%288%2Api%29%2F3%29%2At%29%29.


Both models have zero displacements at t= 0.

Solved.

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The lesson to learn from my solution is that there are TWO different models
(different geometrically and physically) satisfying the imposed conditions.

These models are shifted half-period one from the other.

@Theo missed one of them.


It is, of course, the major intention of this problem for the student to find BOTH harmonics.


I would like to add something at the end.

        It is good if you have formal knowledge of a subject.

        But in many cases it is not enough to solve problems correctly.

        In addition to formal knowledge, you should have adequate physical and geometrical intuition,
        which is the same as common sense.

        The problems like this one help you to develop such intuition and common sense.
        It is the major benefit of solving such problems.