An object in SHM oscillates with a period of 4.0 s and an amplitude of 10 cm. How long does the object take to move from x = 0.0 cm to x = 6.0 cm?

Video transcript

Welcome back, everyone. We are making observations about a wooden block and we are told that it follows simple harmonic motion. We are told that it has a period of 1. seconds with a maximum displacement of 4.2 centimeters and or point oh 42 m. And we are tasked with finding what is the time it takes for the block to move. Let's see here from a from X equals 02 X equals 3.5 centimeters. So our difference in time is going to be equal to the time when X is equal to 3.5 centimeters or point oh 35 m minus the time when X is zero. All right. Wonderful. Well, what are we gonna use for this? Well, for simple harmonic motion, we can use the position equation that X F T is equal to the amplitude times the cosine of Omega T pull us our phase constant. What is Omega here, Omega is going to be two pi divided by our period, which we have. So we can calculate that real quick. It's gonna be two pi divided by 1.7 giving us 3.696 radiance per second. But what about our phase constant here? Well, what we can do is we can actually plug in values of T to this equation to find our face constant. And one of our points of interest is X equals zero anyways. So let's plug in X equals zero, X equals zero. Or rather when T is equal to zero, we get that X is equal to zero. So that solves this part of our delta T. And what do we get to follow from that? We have zero is equal to using this equation right here, 00.42 times the cosine of zero plus our phase constant. Now, if I isolate the phase constant term, what I get is that the face constant is equal to the inverse cosine of zero, which can either be positive or negative pi over two. Now here's the rule about face constant after a time of zero. Does it move towards a positive or negative amplitude? Right? If it moves towards a positive amplitude, then what we get is a negative phase constant. But if we have that, we move towards a negative amplitude initially, then we get a positive phase constant. We are moving towards a positive amplitude right out of the gate. So we will need a negative phase constant. Therefore, our phase constant is equal to pi over two great. So now that we have our face constant, what we can go ahead and do is we can go ahead and calculate our time at which X is equal to 0.35. So X is equal to 0.35. What is our time? Let's plug everything in. We have that. Let's see here, we have 0.0 actually, before plugging in values. Let's simplify this a little bit. We have that X of T is equal to a cosine of Omega T now minus pi over two. But this is just the same thing as our amplitude times the sign of Omega T which, which makes simplifying a lot easier. So now let's plug in all of our values with this equation right here, we have 0.35, equal 2.42 times the sign of 3.696 times T, we'll going to divide both sides by 0.42, divided by 0.42. And then I will take the inverse sign of both sides. And lastly, then we just need to divide by the 3.696 term, 3.696 term. Now let me go ahead and scroll down just a little bit more here. What this gives us is that our time is equal to the inverse sign of 0.35, divided by 0.42, all divided by 3.696, which when you plug into our calculator, we get 0.27 seconds. So now to find our delta T. Remember delta T scrolling back up here, we just have our time at X equals 0.35 minus our time at X equals zero. So what this is going to be equal to is delta T is 00.27 minus zero, which just gives us 00.27 seconds, which as you can see corresponds to our final answer. Choice of D. Thank you all so much for watching. I hope this video helped. We will see you all in the next one.