A person's center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the woman's feet to her center of mass?

Question

A person's center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the woman's feet to her center of mass?

Accepted Answer

Hey, everyone in this problem, we have a group of engineers doing this simple experiment to find the center of mass of a person. So they set up a 3 m long 6 kg wooden plank horizontally. And on the right end, we have a scale. And on the left end, we have a support for this wooden plank. We have our 55 kg participant laying down with their feet above the support. At the left end, the engineers read the scale and observe that it measures a mass of 22 kg. And we're asked to determine the distance from the person's feet to their center of mass. A, we have four answer choices all in meters. Option A 4.92 option B 3. option C 1.04 and option D 2. we have this problem and we can think about this problem as OK. We have this plank and we know it's an equilibrium, OK? It's not moving, it's positioned so that it's stationary. And so we have the sum of torques equals to zero. And again, this is an equilibrium. So we know that the sum of the torques is equal to zero. All right. Well, is that gonna help us find what we want? We want to find the distance from the participants feet to their center mass and we know that the torque and we can write the torque in an equation that's gonna bring that distance in. And we're gonna do that in a few steps. So let's go ahead work with our sum of torques. Ok? And we're gonna consider the left end as our pivot. OK. So that when we're looking at the distances in our torque, it's gonna be that same distance that we're interested in from the participants fee. All right. So what forces do we have acting in this case? Well, on the left end, we're gonna have some normal force. We're just gonna call it N one acting upwards from our support. On the right end, we're gonna have the same thing, another normal force. We're gonna call it N two acting upwards from the scale. OK. Pushing on that plank. We have the weight of the person acting downwards from their center of mass and we have the weight of the wooden plank also acting downwards from its center. OK. So we're calling that WP and WW for the person and the wood point respectively. All right. So we have these four horses, which of them are gonna call us torque. Well, the first horse and one we can ignore that. OK? It's acting right at the pivot. So that is not going to produce a torque. So we have the other three to worry about. Now, the weight of the person, the weight of the plank, they're acting downwards. And so we can imagine that those forces are gonna cause a clockwise rotation. So that's going to be a negative torque. OK. So we have negative the torque from the weight of the person minus the torque from the weight of the wood plank. And then our normal force and two at the right hand end that's pointing upwards, that's gonna cause a counterclockwise rotation, that's a positive torque. And so we have plus the torque due to the force N two. And all of this is gonna be equal to zero again because we're an equilibrium. So now we're at the point where we're gonna introduce that distance and we've written our torque equation. Now let's recall what the torque is actually equal to and the torque is going to be equal to the force multiplied by the distance and multiplied by sign of the angle between the four vector and that position vector. All right. So writing this up, we're gonna have negative WP and that's the force that the weight multiplied by RP, the distance from the left end to the center of mass of the person multiplied by sine of beta pete the angle between those two vectors. OK. Then we have minus WW RW sign of beta W OK. So the exact same as the first term. We're just using subscript W for the wooden plank. And then we're adding, and then this last term we're gonna have N two in that normal force multiplied by R two, multiplied by sine of the two. And this is all equal to zero. Now, what we're looking for here, OK. We're actually looking for RP. That's exactly what we want. OK. RP is gonna measure the distance from our pivot axis of rotation to the center of mass of the person. What pivots at the left end where their feet are. And so that's exactly what the question was asking the distance from their feet to the center of mass. And so RP is what we're trying to copy. Now, we can calculate the weight WP, we can calculate the weight WW, we can calculate this normal force and two because we're told the mass that that scale is reading, what about these data bytes? OK. Well, it turns out the P beta W and theta two are all equal, theta P equals theta W equals theta two. And they're gonna be equal to 90 degrees. Looking at our diagram, we can see that if we look at the position vector from our pivot to any of those forces that's gonna be horizontal and the forces are acting vertically straight up or down. So they are acting perpendicularly, that angle is gonna be 90 degrees. And if the angle is 90 degrees recall that sign of 90 degrees is just equal to one. And so each of these sign terms is gonna go to one and we're left with just that force and the distance. All right. So let's fill in some more information here. Now, the weight recall that the weight can be calculated as mass multiplied by the acceleration due to gravity. And let me just write that on the side, the weight W is equal to the mass multiplied by the acceleration due to gravity G. So for each of our weight terms, and we have negative WP that's gonna be the mass of the person 55 kg multiplied by the acceleration due to gravity 9.8 m per second squared multiplied by this distance. RP that we're looking for. Then we do the same for the wooden plank. We have minus 6 kg multiplied by 9.8 m per second squared multiplied by 1.5 m. And why? 1.5 m? Well, we're gonna assume that this plank is nice in uniform. And so the distance from the pivot to the center of mass is gonna be exactly half of that length of the plank. The length of the plank is 3 m. And so the distance we're interested in is 1.5 m. Right now. For our last term, we have this normal force and two. OK. And that normal force is actually going to be front scale, pushing upwards on our plank. OK. We know that this force because we're in equilibrium is going to be equal in magnitude opposite in direction to the force that's being pushed down on the scale by the plank. Now that force is the weight. OK? We know that weight, we're told the mass so we can calculate it. OK. So same thing for this term, the only difference is that it's acting in the opposite direction as the other ones. And so we have the mass that is measured on the scale kg multiplied by the acceleration due to gravity 9.8 m per second squared multiplied by the distance. And because this is at the far right end of our plank, it's that entire length of a plank about 3 m and all of this is equal to zero. All right. So now what can we do? Well, we have this 9.8 m per second in every single one of our terms, we have just zero on the right hand side. So we can divide the entire equation by 9.8 m per second squared. And that term is going to divide out all the way through. We're gonna keep our constant terms on one side, we're gonna move our term with RP to the other side. So we can try to isolate for RP. And what we're gonna have is 55 kg multiplied by RP is going to be equal to 66 kg meters minus 9 kg meters. All right, last step, we wanna divide by 55 kg. So we have 66 kg meters minus 9 kg meters. That's gonna give us 57 kilogram meters all divided by 55 kg. Ok. The unit of kilogram is going to divide it. We're gonna be left with the unit of meters, which is exactly what we want when we're talking about this distance and we get that the distance is about 1.036 repeated meters. And again, this is the distance from the pivot to the center of mass of the person. But when we look at our diagram, we've picked the pivot to be at the person's feet. And so it's the exact same as the distance from their feet to their center of mass that we were asked to find. If we round this to two decimal places, we can see that the correct answer is gonna be option C 1.04 m. Thanks everyone for watching. I hope this video helped see you in the next one.

Verified Solution

Key Concepts

Center of Mass

Torque

Equilibrium