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Problem 64 Hard Difficulty

Integrated Concepts
(a) What force must be supplied by an elevator cable to produce an acceleration of 0.800 $\mathrm{m} / \mathrm{s}^{2}$ against a $200-\mathrm{N}$ frictional force, if the mass of the loaded elevator is 1500 $\mathrm{kg}$ ? (b) How much work is done by the cable in lifting the elevator 20.0 $\mathrm{m} ?(\mathrm{c})$ What is the final speed of the elevator if it starts from rest? (d) How much work went into thermal energy?

Answer

(a) $F_T=1.61 \times 10^{4} \mathrm{N}$
(b) $W_T=3.22 \times 10^{5} \mathrm{J}$
(c) $v_{\mathrm{f}}=5.66 \mathrm{m/s}$
(d) $W_{\mathrm{th}}=4.0 \times 10^{3} \mathrm{J}$

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Physics 101 Mechanics

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Chapter 7

Work, Energy, and Energy Resources

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Video Transcript

Yeah, In this problem, we have an elevator set up. So the elevator has a massive 1500 kilograms that has an acceleration of 15000.8 meters per second squared. It traces a frictional force of 200 Newtons. And you see the other two forces that I had you on here is the gravitational force and the force from this cable. So the first part is to find the force applied by the elevator cable. And since we're given what the acceleration is, this is really just the application of you in a second mom to this toast us that our F net is equal to our mass times acceleration. You see here that that's ah tension in the cable minus our frictional minus our gravitational. So if we saw for attention, we have that for supplied by the elevator cable bang, the mass times acceleration. Thus, our fictional plus our gravitational or in a little bit nicer form. So we have our fictional being 200. The mass is 1500 gun acceleration of 0.8 and this gives a 16,100 news. The second part asks us to find the work done by the cable to lift the elevator too. 20 meters. So with this acceleration being constant, that that force applied by the elevator cable that we just found remained constant over this whole distance. So we know that the work done by the cable is equal to the force of the cable mold blood by the distance of her Richard acts, which is denoted h here and then co sign of the angle between those two here, our attention is pointing up, and the distance that is acting is also in that same direction. So the angle between 20 on that coast on this one to the work done is just equal to the force multiplied by the distance got 16,100 multiplied by 20. So this gives us 322,000 jewels of work. The next part. We want to find the final speed if our initial speed is equal to zero and this we can just do with cinematics. So we know that our final speed is gonna be equal to our acceleration times time. Well, we don't know is the time, But we also know that with the constant acceleration in a zero initial velocity that that distance that vertical distance Our height is equal to 1/2. What's blood by acceleration times of time squared. Anna, your time is equal to the square root of two age over our acceleration. So we can plug that into here for our final velocity. We have our acceleration. Want to buy with the square root of two age over a. We've got the square root of two h A. Our age is 20 meters are acceleration. Is this 0.8 meters per second squared? So this gets us 5.66 meters per second. The last part is asking us the amount of work that went into the thermal energy, this energy that's lost and that is going to be given by the work done by this fractional poor. So that's gonna be or until the F, which is our frictional, that same distance over which is acting, which is the height co sign of the angle between the two. But here we see that that angle between our direction of displacement in the direction of our frictional forces. 180 says it's gonna be negative. Which makes sense. Recall that a negative on our system means energy is leaving the system. And here energy is leaving this system in the form of heating up these walls in, uh, elevator shaft. So here we have frictional forces. 200 over 20 meters. So you got minus or 1000 jewels. That is the work that went into the thermal energy.

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