In a certain library the first shelf is 15.0 cm off the ground...

Question

In a certain library the first shelf is 15.0 cm off the ground, and the remaining four shelves are each spaced 38.0 cm above the previous one. If the average book has a mass of 1.25 kg with a height of 22.0 cm, and an average shelf holds 28 books (standing vertically), how much work is required to fill all the shelves, assuming the books are all laying flat on the floor to start?

Accepted Answer

Welcome back everyone in this problem in a construction project, the first layer of bricks is to be placed 30 centimeters above the ground and each of the remaining three layers is sacked 25 centimeters above the previous one. If the average brick has a mass of 2.5 kg with a height of 7.5 centimeters and each layer requires 100 bricks to complete how much work is required to stack all the bricks. Assuming the bricks are initially all underground. For our answer choices. A says it's 6.6 multiplied by 10 cubed jewels. B 2.9 multiplied by 10 cubed jewels, C 2.6 multiplied by 10 cubed jewels and D 7.4 multiplied by 10 squared jewels. Now, if we're going to figure out the work needed to stock all the bricks, OK. That is the total work. Let's first draw a diagram to visualize what's really going on here. OK. So let's say that this is the ground, right? And our first layer of bricks is 30 centimeters above ground. OK. So I'm gonna draw that layer here is our bricks and let's say that it's 30 centimeters above ground. So that's our first height. H one equals 30 centimeters which in si units, let's write that as 0.3 m. Next, we have another layer that it says is stacked 25 centimeters above the previous one. OK. So we have our other layer here where if it's 25 centimeters above in the previous one, that's going to be 30 plus 25. OK. Which is 55 centimeters. or we can write that as a 0.55 m. So H two equals 0.55 m. OK. Next, we have another layer because notice we have four layers. So we have another layer that is 25 centimeters above again. So if we come out here, that means the height of that layer off the ground is gonna be 0.55 m plus 25 centimeters, which is gonna be 0.8 m. OK? So we can say H three equals 0.8 m. That's the height of the ground. I should probably fix my ground here. It doesn't look the heights. Yeah, let's say that this is the ground. And then finally, we have our first layer of bricks, OK? And for our fourth layer of bricks, it's also 25 centimeters above our last height of 0.8 m. So that means our fourth layer, the height of our fourth layer four is going to be equal to 105 centimeters or 1.05 m. So, and, and let's say that we have some pillars here that uh are holding up our bricks. OK? Because we're just interested in the work that's done to lift up these bricks. Now, let's first ask ourselves, what do we know about work? What do we know about work over a vertical distance? Well, recall, OK, that the work done over vertical distance W equals the mass of the object multiplied by the acceleration due to gravity multiplied by the height that we are lifting our object through. No, we are assuming that the bricks are lifted at a constant velocity. Meaning the kinetic energy change from the ground to each layer is zero is zero. So that's why we calculate the work done. We only calculate the work done against gravity. We are also assuming that the effect of air resistance on the bricks as they are lifted is negligible. So the work done to lift one brick would be equal to the its mass multiplied by gravity multiplied by height. Now remember we're talking about 100 big bricks here for each layer. So at 100 bricks, OK, the work done for a layer is going to be equal to 100 multiplied by MGH. Now notice that each layer has a different height. So if we can find the work done at each layer, then we should be able to find the total work done. Now notice here we have four layers. So the work at layer one is W one, then we have W-2 W three and then W-4 at layer four. So what this tells us then is that our total work said W total is going to be equal to the work done in layers one or the sum of the work done in layers one through four. OK. Now what do we know? Well, the work done in layer one is mg multiplied by H one work done in layer two mg multiplied by oops it should be 100. Sorry because the work in the layer is 100 MGH my apologies. So the work done in layer one is 100 multiplied by MGH one in layer 2, 100 multiplied by MGH two layer three, 100 multiplied by MGH three because we have the same mass of the brick acceleration due to gravity is constant. And the work done in layer 4, 100 multiplied by MGH four. Notice that all of these terms have 100 mg in common. So if we factor out 100 mg, then the total we work done is 100 mg multiplied by H one plus H two plus H three plus H four, we know all these values. So we can substitute to figure out what the total work is. So by substituting that's going to be 100 multiplied by the mass fric 2.5 kg multiplied by the acceleration due to gravity we take to be 9.8 m per second squared multiplied by the sum of H 10.3 m plus H 20.55 m plus H 30.8 m plus H 41.05 m. Now, we can put this expression directly into our calculator. OK. And when we do that, eh we should find that our total work ends up being 6621.75 joules. If we write this in units of 10 cubed, that means it's going to be approximately equal to 6.6 multiplied by 10 cubed jewels that the hotel work done the stock, all the bricks is 6.6 multiplied by 10 cubed jewels. If we look back at our answer choices, A is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Work in Physics

Gravitational Potential Energy

Kinematics and Height Calculation

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