Interestingly, there have been several studies using cadavers to determine the moments of inertia of human body parts, information that is important in biomechanics. In one study, the center of mass of a 5.0 kg lower leg was found to be 18 cm from the knee. When the leg was allowed to pivot at the knee and swing freely as a pendulum, the oscillation frequency was 1.6 Hz. What was the moment of inertia of the lower leg about the knee joint?

Video transcript

Hey, everyone in this problem, we're told that moment of inertia is an integral part of designing objects in systems that undergo rotations. The length of a 12 kg bar in a crane model is 1.3 m. The center of mass at the bar is located 25 centimeters from the rotational axis. The bar oscillates about the rotational axis like a pendulum at a frequency of two Hertz. We're asked to find the moment of inertia of the bar about the rotational axis. We're given four answer choices. Option A 0.968 kg meters squared. Option B 0.782 kg meters squared, option C 0.186 kg meters squared and option D 1.15 kg meters squared. So we're looking for a moment of inertia here. We're given information about length about mass, about the distance from the center of mass and about frequency. So if you try to calculate the moment of inertia here with your typical formula from your table, you would have a hard time because we aren't given information about what direction the rotation is occurring and things like that. OK. So we would have a hard time picking which formula to use from our moment of inertia table. However, recall that we have a formula that links the period of rotation to the moment of inertia. So we have the period T is equal to two pi multiplied by the square root of I divided by M G D. Now, you might be saying why are we using this equation with period? We don't have information about the period. And you're right. However, we have information about the frequency and we know that the frequency in the period are inversely related. So the period is going to actually be one divided by the frequency. So we have one divided by the frequency is equal to two pi multiplied by the square root of I divided by M G D. Yeah. All right. So let's write out the variables that we know. We know that the frequency given is two Hertz. Hey, we're just gonna work left to right in this equation. The moment of inertia eye is what we're trying to find. And it's gonna be the mass and we're told that this mass or, or bar has a mass of 12 kg. OK? G gravitational acceleration which we know is 9. m per second squared and D D is gonna be the distance from the rotational axis to the center of gravity. OK. And we're told that the center of mass is 25 centimeters from the rotational axis. And so D is gonna be 25 centimeters. We want to convert this to meters. So we're gonna multiply and we know that in one m there are centimeters. So we're gonna multiply by one m divided by 100 centimeters. The unit of centimeter cancels, we're effectively dividing by 100 and we get 0. m. So we have everything we need, we have all of these variables that we need in order to solve our eye. So before I plug them in, I'm gonna rearrange this equation. Um So it's a little bit less messy, isolate for I and then we'll substitute in all of these values. So let's divide both sides by two pi first, we get one divided by two pi F is equal to the square root of I divided by M G D. And we can square both sides. Now to get rid of that square root, we have one divided by two pi F, all squared is equal to I divided by M G D. And now we're going to multiply both sides by M G D. OK, to isolate I one last step. So we get I is equal to. Now in the numerator on the left hand side, we have one squared, that's just gonna be one. So when we multiply M G D, we're gonna get M G D in the numerator. And then in the denominator, we're still gonna have our two pi F squared. All right. So we've isolated for I now it's substituting in all of those values that we have. So we had that the mass was 12 kg multiplied by the acceleration due to gravity 9.8 m per second squared multiplied by the distance from the rotational axis to the center of mass which is 0.25 m. All of this is gonna be divided by two pi multiplied by the frequency two Hertz all squid. If we simplify in the numerator, this is gonna be 29.4 in the unit, we have kilogram meter per second squared multiplied by meter. And so we get kilogram meter squared per second squared and this is divided by that denominator. OK? Two pi multiplied by two Hertz. That's gonna give us four Pi Hertz. All of that squared is gonna give us 16 pi squared Hertz squared. Can I ever remember that A Hertz is going to be one divided by second in terms of the unit. OK. So the unit of per second squared will divide out and we're gonna be left with the unit of kilogram meters second, sorry meters squared, kilogram meter squared. All right. And when we do this division, you can work this out on your calculator and you're gonna get 0.186 kg meters squared. OK? And that is the moment of inertia eye for that bar that we were looking for. If we compare this to our answer traces. You see that this corresponds with answer choice. C thanks everyone for watching. I hope this video helped see you in the next one.