75. {Use of Tech} Oscillator displacements Suppose a mass on a...

Question

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:

s(t) = e⁻ᵗ sin t

a. Graph the position function. At what times does the oscillator pass through the position s = 0?

Accepted Answer

Welcome back, everyone. In this problem, consider a damped oscillator with a position functions of T equals E to the negative 0.5 T multiplied by the cosine of 2T. Determine the times at which the oscillator passes through the equilibrium position S equals 0. Now, for us to solve for these times, first we need to set the position equal to 0. So that means a of T. Which is e to the negative 0.5 T multiplied by the cosine of 2 T is going to be equal to 0. Now analyzing the product, the product to the negative 0.5 T multiplied by the cosine of 2T will equal 0 if either factor is 0. So if instead of saying no, I should change this to say if. S of T is equal to 0, then either E to the negative 0.5 T equals 0 or the cosine of 2 T equals 0. So let's try to solve in T for both. Now for our exponential factor, recall that the exponential function is never zero for any real T. Now that tells us then that E to the negative 0.5 T is gonna be greater than 0 for all T, so we can ignore this factor. Thus, only the cosine term can be zero, so we know that this cannot be zero. Now if only the cosine factor, OK. or only the cosine term can be zero serving for the cosine of 2 t equals 0. Then remember we know that the cosine of an angle theta will be equal to 0 when our angle is equal to 1/2 of pi plus a multiple of n pi, OK, where N is an integer. Here Theta would be equal to 20, OK, so what we're really saying then. Is that 2 t would be equal to a half of pi plus n pi and if that's the case, we can serve for t dividing both sides by 2, which meanst would be equal to 14th of pi plus a half of n pi, OK, where N is an integer. And no, since our time, OK, since our time T is gonna be greater than equal to 0, greater than or equal to 0, that means we will require 1/4 OK. Well, let me write it below here. Our expression a 4th of pi plus 1/2 of n pi to be greater than our equal to 0. And if that's the case, this means that 4th plus n divided by 2 must be greater than our equal to 0, which means That 1 + 2 n must be greater than or equal to 0, which implies that n must be greater than or equal to 0. So, in that case then, generally, the oscillator passes through the equilibrium position S equals 0, OK. At, well, let me write that the oscillator. Passes through. S equals 0, OK. At T equals 4th of pi plus 1/2 of N pi, OK. Where N is greater than or equals to 0, so it could be 0. 123, and so on. Thus this is going to be the value of tea. Thanks a lot for watching everyone. I hope this video helped.

75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:
s(t) = e⁻ᵗ sin t
a. Graph the position function. At what times does the oscillator pass through the position s = 0?

Key Concepts

Damped Harmonic Motion

Zeros of a Function

Graphing Transcendental Functions

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