{Use of Tech} Spring oscillations A spring hangs from the ce...

Question

{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t−10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>
c. At what times is the velocity of the mass zero?

c. At what times is the velocity of the mass zero?

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says a buoy is floating in the sea, bobbing up and down with the waves. The height H and feet of the buoy above the sea level after T seconds is given by the function H of T is equal to 3 multiplied by cosine of T minus 3 multiplied by sin of T, or T greater than or equal to 0. Determine the times at which the velocity of the buoy is 0. We give 4 possible choices as our answers. For choice A, we have T is equal to 1 divided by 4, multiplied by N, multiplied by pi seconds, where N is any non-negative integer. For choice B, we have T is equal to 1 divided by 2, multiplied by N multiplied by pi seconds, where N is any non-negative integer. For choice C, we have T is equal to n multiplied by pi plus pi divided by 4 seconds, where N is in a positive integer, and for choice D we have T is equal to N multiplied by i minus pi divided by 4 seconds, where N is any positive integer. Now we're asking the question to determine the times at which the velocity of the buoy is 0. So the first thing we need to do is determine our velocity as a function of time. So recall to determine our velocity, we need to take the time derivative of our position equation. That means that our velocity is a function of time, which we'll label as VFT. It's going to be equal to the derivative with respect to T of H of T. So this is going to be equal to the derivative with respect to T. Of the quantity of 3 multiplied by cosine of T. Minus 3 multiplied by the sine of T. So taking the derivatives, this is going to be equal to -3 multiplied by sin of T minus 3 multiplied by cosine of T. So recall that the derivative with respect to T of cosine of T is equal to minus sine of T, and the derivative with respect to T of sin of T is equal to cosine of T. So now that we have the velocity as a function of time, we just need to set this equation equal to 0. And now we need to solve 4 T. So the first step we can do is divide both sides of our equation by -3, and this will give us sign of T. Plus cosine of T. is equal to 0. So now what we want to do is multiply both sides by 1 divided by the square root of 2. So this will give us 1 divided by the square root of 2 multiplied by the sine of T, plus 1 divided by the square root of 2 multiplied by cosine of T is equal to 0. Now, the reason why we wanted to multiply by 1 divided by the square root of 2 is because 1 divided by the square root of 2 is equal to the sine of pi divided by 4, which is also equal to the cosine of pi divided by 4. So that means we can rewrite this equation as sine of T, multiplied by cosine of pi divided by 4. Plus, cosine of T multiplied by sine of pi divided by 4 is equal to 0. And we can actually simplify this expression using our angle addition formulas. So this is going to be equal to the sign. Of the quantity of T plus pi divided by 4 in quantity is equal to 0. Now, our sine function is 0 whenever its argument is equal to an integer multiple of pi. So that means that T plus pi divided by 4 has to be equal to an integer multiple pi, which will write as N multiplied by pi. Son here is an integer. So now we just need to solve for tea. We'll have tea. is equal to N multiplied by pi minus pi divided by 4. Now, we want our time T to be greater than or equal to 0, so we want T to be positive. So this is only true when N is greater than 0. So N here must be a positive integer. So, this is actually the answer that we were looking for. And if we come back up to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

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