A 15-cm-long, 200 g rod is pivoted at one end. A 20 g ball of clay is stuck on the other end. What is the period if the rod and clay swing as a pendulum?

Video transcript

Hey, everyone in this problem, a decoration consists of a 22 centimeter long bar of mass 270 g. One end is used like a pivot while the opposite end is fitted with a spherical ball, which we're told can be considered a point mass with a mass of 48 g. So let's stop there for a minute. We're gonna draw out what we have and then we're gonna come back and figure out what this question is asking us to do. So we have a bar, OK? That is 22 centimeters long. One end, there is gonna be a pivot. So we're gonna put the pivot on the left hand end and the opposite end has a spherical ball, the spherical bob with a mass of 48 g. And so this is what we have going on and what the question is asking us to do asking if the bar bob combination oscillates like a pendulum, find the period of the motion. We're given four answer choices. Option A 1.23 seconds. Option B 0.13 seconds, option C 1.74 seconds and option D 0.82 seconds. So we have a pendulum. Let's recall that the period t that we're trying to find it is given by two pi multiplied by the square root of I divided by MGD where I is the moment of inertia, M is the mass G is the acceleration due to gravity and D is the distance from the pivot to the center of mass. OK. So we know the mass, we know the acceleration due to gravity. We need to find the moment of inertia. I in this distance D in order to calculate the period. OK. So let's calculate each of those first. In order to do those, we're gonna need these masses in standard units. So let's go ahead and convert those now. So we don't have to do it inside of our equation. OK? You can do either but sometimes it's just a little bit cleaner and less messy if we do the conversions first. So we have the mass of the bar which we'll call MB is equal to 270 g. We wanna convert to kilograms. So we're gonna multiply by 1 kg divided by 1000 g. And we know that there are 1000 g in every kilogram, the unit of gram will divide out. And so what we're doing is essentially dividing by 1000. And we get that this is equal to 0.27 kg for the mass of the Bob. OK. We're gonna call this MS for the mass of the spherical bob, this is 48 g. We're gonna do the same thing multiplying by 1 kg, divided by 1000 g, the unit of gram divides out and we're left with 0.048 kilograms. The last thing we wanna do is convert this length. We're given the length in centimeters. We're gonna need that length of the bar when we do our moment of inertia calculation. And so let's convert that length L into our standard unit as well. We have 22 centimeters. We're gonna multiply by 1 m divided by 100 centimeters. OK? Because we know that there are 100 centimeters in every 1 m. The unit of centimeters divides out what we're essentially doing is dividing by 100. Let get 0.22 m. So we have all of our values written out in standard notation. Let's go ahead and move to our calculations. And the first thing we're going to calculate is that moment of inertia eye. Now, looking at the moment of inertia, I, we have two things to consider. We have the moment of inertia of the bar plus the moment of inertia of the bob. OK. So we have the moment of inertia of the bar plus the moment of inertia of the spherical ball. Now, the moment of inertia of this bar, OK. This bar can be treated as like a thin rod and it's rotating about its end. And so if you look up in the table in your textbook or that your professor provided the moment of inertia here is gonna be given by the mass of the bar and B multiplied by the length squared divided by three. We're told that this spherical blob can be treated as a point mass. And so the moment of inertia there is gonna be the mass of the sphere MS multiplied by R squared. OK? Or the distance from that point mass to the axis of rotation or the pivot point. Now, we have our equation, we need to substitute in our values. So the mass of the bar is 0.27 kilograms multiplied by the length of the bar, 0.22 m all squared two divided by three. And then we're gonna add the mass of the sphere, 0.048 kilograms multiplied by the distance from that point mass that sphere to our pivot. Now the sphere is at one end of the bar, the pivot is at the other. So the distance between them is actually gonna be the length of the bar. OK. So R squared is actually just L squared. OK. So this is also gonna be 0.22 m squared. If we work this out on our calculators, we get the moment of inertia, I is gonna be equal to 0.0066 792. And we have kilogram meters squared. OK. So we found our moment of inertia. I remember the other thing we needed to calculate was D which is gonna be the distance from the center of mass to the pivot now, because the pivot is at the left end. If we use the left end as our reference to calculate the center of mass, then the distance D is just going to be the position of that center of mass. OK. That value of the center of mass. And so D just gonna be X, yeah, A that X position of the center of mass and recall that the center of mass is given by M one, X one plus M two, X two divided by M one plus M two. If we have two objects, OK? Where X one is the position mass, one center of mass and X two is the position of mass, two center of mass. OK? And our two masses here are the mass of the bar and the mass of the bob. All right. So for this first one, again, the first mass we have is the mass of the bar. So we have MB multiplied by the position of its center of mass. Yes, we have this bar, the center of mass is going to be in the middle. OK. So from the left end, that's at a distance of L over tip where L is the length of that bar moving to our next, we have M two, which is the mass of our spherical Bob MS and X two is going to be its position relative to that left end. OK. The sphere is at the end of the bar and so it is at a distance of L away. So we have MB multiplied by L divided by two plus MS multiplied by L. And all of this is divided by that sum of mass is MB plus MS. Now, we can substitute in our values again, we're gonna use those masses and that length that we converted to our standard unit, we have 0.27 kg multiplied by 0.22 m divided by two plus 0.0 or 8 kg, multiplied by 0.22 m. All of this divided by that sum of masses, 0.27 kg plus 0.048 kg. And this gives us a position for our center of mass of 0.1266 m. OK? And that is that value D we were looking for the distance from our center of mass to our pivot, which is at the left end of a bar. OK. So we have our moment of inertia. I, we have this distance D let's go back and remind ourselves what we're trying to do. We're trying to calculate the period T and that is equal to two pi multiplied by the square root of I divided by MGD. Now we have all the values we needed. We can go ahead and calculate that tension T we were looking for, I'm gonna do it. On the right hand side here we have the, the tension T is equal to two pi multiplied by the square root of I divided by MGD. OK. This is equal to two pi multiplied by the square root of 0.0066792 kilogram meters squared divided by the mass. And in this case, the mass of our pendulum is that total mass of the bar and the bob. So we have that total mass which is 0.27 kg plus 0.048 kg multiplied by the acceleration due to gravity 9.8 meters per second squared multiplied by that distance D we found of 0.1266 m. And I've written that down below because we ran out of room. But in the denominator, all three of those values are multiplied together. OK? And when we work this out, we get a period tea 0.81752 seconds. Yeah, we had kilogram meters squared divided by kilogram meters per second squared, multiplied by meters under the square root. So the units of kilogram and meters squared divide out were left with the unit of second squared inside of our square root. We take the square root and we end up with seconds and that is the period of that pendulum that we were looking for. If we compare this to our answer choices, we see that this corresponds with answer choice, D 0.82 seconds. Thanks everyone for watching. I hope this video helped see you in the next one.