A 0.500-kg mass on a spring has velocity as a function of time given by vx(t) = -(3.60 cm/s) sin[(4.71 rad/s)t - (pi/2)]. What are (a) the period; (b) the amplitude; (c) the maximum acceleration of the mass; (d) the force constant of the spring?

Video transcript

Everyone in this problem. A frictionless trolley of mass 14.2 kg attached to an ideal spring has a velocity function. V sub X of t is equal to negative 8.5 centimeters per second and sign 5.2 radiance per second, T plus pi over three. And were asked to determine the amplitude and peak acceleration of the trolley. Alright, so let's start with part one. The amplitude. Now we're given a velocity function. Let's recall the general form of the velocity function. It's going to be V. X. Of T. Is equal to negative a omega sign omega, T plus phi. Where phi is the phase angle omega is the angular frequency and a is the amplitude we're looking for. Okay, so we want to compare this general equation to the equation. Were given. Okay, but we see that the amplitude A we're looking for. Okay, is multiplied by this angular frequency omega. So when we see it in our version of the function, okay, we have the number 8.5 is going to be eight times omega. Okay, so we can't we don't know which portion of that is the amplitude. A without first finding the angular frequency omega. Okay, so let's start there. Let's find omega and omega appears in the equation again. Okay, it's going to be the coefficient in front of the T. Inside of the sine function. Okay, so if we compare this to our equation, we see that omega Is going to be equal to 5.2 radiance per second. So now that we know omega, let's get back to finding the amplitude. Okay, so we can compare our general equation. Okay we have this negative a omega. That's gonna be the coefficient in front of the sign. Outside of that sign that coefficient is negative omega. And if we compare with the equation we're given we have negative 8. centimeters per second. Okay, so we know that negative a omega is equal to negative 8.5 centimeters per second. Okay and again we're looking for the amplitude a we know omega because we just found it above So we can isolate for the amplitude. A okay, so getting rid of those negatives, we have that the amplitude a times omega which is 5.2 radiance per second is equal to 8.5 cm/s. And it's a good idea when you're taking values from an equation like this to carry over the units. Okay, so in this case we were given centimeters per second and you'll see that that gives us our amplitude in centimeters. Okay, if you don't carry over the units from from the equation, you're not going to know whether your amplitude is given in centimeters meters millimeters. So always a good idea to carry over the units from that equation so you can follow through with them All right, so if we divide our amplitude a it's going to be 8.5 divided by 5.2. The unit of per second will cancel. We're gonna be left with the unit of cm and we get 1.6346cm. Okay. And so that is our amplitude a that we were looking for Now. If we look at our answer choices we see that answer. Choice B is the only one that has the amplitude of 1.63 cm that we found. Okay. So if this was on a test, if you were in um A set period of time you could go ahead, answer this B and move on. But we're gonna look at part two. Okay to see how you would go about solving this peak acceleration problem and just to double check that that is the right answer. Alright, so part two, we're asked to find the peak acceleration. Okay, when you hear peak acceleration you can think of maximum acceleration. Well, let's recall that the maximum acceleration a max is given by a omega squared. So the amplitude times the angular frequency squared? Well, we just found the amplitude and in order to find the amplitude we also had to find the angular frequency omega. So we have both of these values. We can plug them in and solve So we get 1.634, 6 cm times the angular frequency 5.2 radiance per second. All squared. Ok. And working this out on our calculators, we get a maximum acceleration of 44.2 in our unit centimeter per second squared. Again. Following those units from that equation that we took them from our accelerations in centimeters per second squared, not meters per second squared. Okay, Alright, so that's our maximum acceleration. So we go back up to our answer traces, may we see that we found an amplitude of 1. centimeters and a peak acceleration of 44.2 centimeters per second squared. So we have answer choice B. Thanks everyone for watching. I hope this video helped see you in the next one.