The amplitude of an oscillator decreases to 36.8% of its initial value in 10.0 s. What is the value of the time constant?

Video transcript

Welcome back, everybody. We are making observations about a forth shaped metallic plate oscillating between the north and south poles of a magnet. We are told that after 8.6 seconds. So a time of 8.6 seconds that the amplitude is reduced to 42.8% of the initial amplitude. So the amplitude at that time is 0.4 to eight of our initial amplitude. And we are tasked with finding what is the time constant for the fork shaped metallic plate? We know that the amplitude at a given time is just going to be equal to the initial amplitude times E to the negative time divided by two times our desired time constant. Now, what I'm gonna go ahead and do is I'm gonna go ahead and plug in this value right here. What we get is 20. times, our initial amplitude is equal to our initial amplitude times E to the negative T divided by two times our time constant. And if you'll see I can divide by our initial amplitude on both sides. And that cancels out. Now using a property of natural logs, what I'm able to do is I'm able to say that the natural log of 0.428. So the natural log of 0.428 is equal to negative T over to T. Now, what I'm gonna do is I'm actually gonna go ahead and take this or take the equation and drag this down here because we are going to manipulate it a little bit to get our desired time content alone. What I'm gonna do is I'm going to multiply both sides of this equation by two T over the natural log of 20.428. I'll do the same thing to this side or to the left side two T times the natural log 0.428. And you'll see that on the left side here, these terms cancel out and on the right side, the two T terms cancel out what this leaves us with is two times our time constant is equal to negative T divided by the natural log of 20.428. Now, the last thing to do to isolate our time constant is divide both sides by two. You'll see the twos cancel out on the left side and we have that our time constant is equal to negative T over two times the natural log of 20.428. Let's go ahead and plug in our values here. We have that our time constant is equal to negative 8.6 divided by the natural log two times the natural log of 20.428. And when you plug into your calculator, we get a time constant of five point oh seven seconds, which corresponds to our final answer. Choice of C Thank you all so much for watching. Hope this for your help. We'll see you all in the next one.