Question

Consider the movement of the mass and spring with friction

A * [e ^ - (b / 2m) t] * cos (sqrt (k / m - (b / 2m) ^ 2)) t + φ)

With A = 2m, k = 3N / m, m = 5Kg and φ = 0, we could change to the value of "b" from zero until the oscillations disappear for the first time and the function does not change to negative values. In this case, the friction now has a very noticeable effect: it lengthens the period of the oscillation so much that no oscillations will be seen (T -> infinity). The theory indicates when that should happen. Clear for b from the equation and record the predicted value for b = .... (Kg / s) (at least two decimal places to the right of the point) The question is complete, the exercise is clear for b, I don't get how to do it. The answer is 7.746

Answer 1

The equation provided to us is

Now, the values provided are

So, substituting the values in (1), we get

Now, we must have the value inside cosine to be positive or 0 for the oscillation to occur.

The oscillation will just disappear right after the term inside cosine is 0.

So, to get the value of b when the oscillation just disappears

Now, as t can be any value,

So,

For b < 7.746, the oscillation occurs as shown

For the critical value of b = 7.745966692, the oscillation occurs as shown (I didn't use b = 7.746 here as the graphing tool I used takes 7.746 to be bigger than 7.745966692 and so no oscillation can be seen)

But, when b > 7.746, the oscillation disappears as the term inside cosine becomes imaginary.

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