Your lab instructor has asked you to measure a spring constant using a dynamic method—letting it oscillate—rather than a static method of stretching it. You and your lab partner suspend the spring from a hook, hang different masses on the lower end, and start them oscillating. One of you uses a meter stick to measure the amplitude, the other uses a stopwatch to time 10 oscillations. Your data are as follows:

Use the best-fit line of an appropriate graph to determine the spring constant.

Video transcript

Hey, everyone in this problem. A mass M is attached to the lower end of a light spring with a spring constant K suspended from a clamp. The system is made to oscillate. A motion detector measures the amplitude and time needed for the mass to complete five oscillations. The procedure is repeated with different masses and the results are shown in the table. We're asked to calculate the spring constant using a suitable graph representation. Now, the table were given, OK. It has four rows in the first column. It shows the mass in kilograms. The second column has the time and seconds and the third column has the amplitude in meters. We're given four answer choices. Option, a one newton per meter, option B 1.5 newtons per meter, option C 2.5 newtons per meter and option D four newtons per meter. OK. So we're giving some information about mass about time about amplitude. We wanna find the spring constant and we wanna do this in a graphical way. Now, let's recall that we can write the period of oscillation T is equal to two pi multiplied by the square root of M divided by K, no K is the spring content. That's what we're interested in. So let's square both sides to get rid of that square root. We're gonna have that T squared is equal to four pi squared multiplied by M divided by K. I'm gonna rewrite this. OK. It's gonna be the same thing just rearranged. So we have T squared is equal to four pi squared divided by K. OK? And then multiplied by now, if we imagine T squared being our Y value and the mass being our X value, then this is the equation of a line. OK. And this line in particular has a slope given by four pi squared divided by K. So we plot T squared as a function of the mass M, then four pi squared divided by K will be the slope. And we know there's a few different ways to find slope. All right. So let's start by calculating these periods or yeah, these periods T and T squared. So we're gonna add this to our table as extra columns. So we're gonna have T the period in seconds and then we're gonna have T squared, which is gonna be in second squid. Now, we're told that the time given is to complete five oscillations, we know that the period is the time it takes to complete one oscillation. And so the period is gonna be the total time for five oscillations divided by five. OK. So for each of these, for the first row, we have a time of 14 seconds divided by five, gives us a period of 2.8 seconds. The second row has a total time of 12.6 seconds divided by five oscillations gives us a period of 2.52 seconds. Ok. Next row, 10.9 seconds divided by five, gives us a period of 2.18 seconds. And finally a time of 8.89 seconds giving us a period of 1. seconds. So we found the period simply by taking the total time to complete five oscillations, dividing it by five to find the time for one oscillation. And now to find T squared, the period squared, we're just gonna square these values that we bet. So 2.8 squared gives us 7.84, 2.52 squared, gives us 6.3504, 2.18 squared, gives us 4.7524. And finally, 1.778 squared gives us 3.16, 1284. All right. So again, what are we doing? This is a little bit different way that we're approaching this problem. We want T squared as like our Y value. OK, we want the mass M as our X value. So if we look at our table, now this first column with the mass M is gonna be like our X value. And this T squared column at the end is gonna be like our Y value. OK. So we have X and Y now we have regular table of values. Now we're gonna graph those or can imagine we're graphing those and the slope is gonna be four pi squared divided by K. Now, how do you find the soap? Right. Well, let's scroll down, give yourself some room to do this. OK. We know that the slope is equal to the rise divided by the run, which can also be written as the change in the Y values divided by the change in the X values. Now, the change in the Y values, OK. We've said that T squared is kind of functioning as our Y variable. And so this is actually gonna be the change in T squared divided by the change in M OK. M is functioning as our X variable. OK? We can do this, OK? We can pick, we can find the change. We're just gonna take the first period, subtract the, this last period. OK. So we have 7. seconds squared minus 3.161284 seconds squared. OK. So this is a change in the Y variable or the Y direction of our graph. And when we change that much in the Y direction, what happens in the OK. So what happens to the mass? Well, we go from a mass of 0.5 kg. OK. That corresponds with our period squared of 7.84 seconds squared down to a mass of 0.2 kg when we have a period squared of 3.161284. And so our change in mass is going to be 0.5 kg minus 0.2 kg. All right. Now, we know that the slope is equal to four pi squared divided by K. And this is gonna be equal to the right hand side here. And we're gonna simplify, this is gonna be 4.6787, 16 seconds squared divided by 0. kg. And so isolating for K, OK. K is going to be equal to four part squared multiplied by 0.3 kg divided by mhm 4. 2nd squared. And what we get is a value of 2.53, we have kilograms per second squared. Recall that a Newton is equal to a kilogram meter per second squared. And so kilogram per second squared is actually equivalent to a Newton per meter. All right. And so we found that spring constant K by using this graphical interpretation. OK. Saying that the period squared was our Y variable, the mass M was functioning as our X variable and K was related to the slope of that line. So if we compare our answer to the answer choices, OK, we need to round to two significant digits and our answer corresponds with option C 2.5 newtons per meter. Thanks everyone for watching. I hope this video helped see you in the next one.