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As part $a$ of the drawing shows, two blocks are connected by a rope that passes over a set of pulleys. One block has a weight of 412 $\mathrm{N}$ , and the other has a weight of 908 $\mathrm{N}$ . The rope and the pulleys are mass-less and there is no friction. (a) What is the acceleration of the lighter block? (b) Suppose that the heavier block is removed, and a downward force of 908 $\mathrm{N}$ is provided by someone pulling on the rope, as part $b$ of the drawing shows. Find the acceleration of the remaining block. (c) Explain why the answers in (a) and (b) are different.

a) 3.68 $\mathrm{m} / \mathrm{s}^{2}$ b) 11.8 $\mathrm{m} / \mathrm{s}^{2}$ c) There is less inertia on the second setup, therefore, the system accelerates more under the influence of a similar force.

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Lesanewerk Y.

May 14, 2021

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in the first item. We have to calculate the acceleration off the lighter block in this first situation. So for that we have to use Newton's second law. And for that I will use a weird reference frame, which is these one? These is my access. Okay, then, using this crazy reference frame, lettuce sold this question. So let me say that these lighter block is block number one and they have your blocks Block number two applying you turn second allowed to block number one. We get that the net force is equals to m one times eight. But the net force acting on block number one is composed by two forces. The tension pointing to the positive direction on the weight, pointing to the negative direction show you the tension. Minus 412 is M one times eight. Now we repeat for block number two for block number two. We have that the net force is equals to its mass times Its acceleration notice that both blocks have the same acceleration because they move together. Then the net force acting on the second blocks composed by true forces, the station force and the weight force. But Now the tension force points to the negative direction while they wait for us points departs direction. So 908 minus t is equals True m true times eight. And then we have these two equations with two unknowns which are detention as the acceleration. We can calculate the mass s using the weight force that was given on the acceleration of gravity. So it's not a big deal. Then we have to discover what is the acceleration for that Notice that if we add these equations and these equation, the tensions will cancel. Therefore, let's do it So adding equations number one and chew results in the following equation T minus 412 plus 908 minus T is equals to m one times a plus m two times a. Then, as I said, there is a cancellation of the tensions and we got 908 minus 412 is equals to I am one plus M Chu times eight. Now notice that the weight force is the mass times acceleration of gravity Sue the mess is the weight force divided by the acceleration of gravity. And then we can write it as 908 minus 412 being equals true Wait number one plus wait Number two and the whole thing is divided by G and multiply it by a then Deasy's 908 minus 412 equals to 908 plus 412 divided by G. Multiply it by a there. We send these terms to the other side to get G times 908 minus 412 divided by 908 plus 412 is equals to a remember that G is approximately 9.8 meters per second squared, then a is equals to 9.8 times 908 minus 412 divided by 908 plus 412. And these results in an acceleration off approximately three 0.68 meters per second squared. So this is the acceleration off the system. In the first situation on the answer for the first item for the second item, let me organize my board. In the second situation, someone's pulling down to hurt with force off 908 new terms. And this produces a reaction force, which is attention force there for in the second situation, that tension force is 908 Newtons. Now, knowing this, we can calculate what is the acceleration off the lighter block in the second situation? For that, we have to use Newton's second law again. I'm choosing now. These reference frame show that the Net force acting on that block he's given by its mass times its acceleration. But the net force is given by the tension minus 412. Certain 908 minus 412 is equals to the mask off this lighter block times its acceleration. So remember again that you can write the mess as the weight forward, divided by the acceleration of gravity. Show you 908 minus 412 is the weight force off the lighter block divided by G. Times eight. And now we can solve this equation for a so eight is equals to G. Times 908 minus 412 divided by the weight off the lighter block, which is 412 g we know that is approximately 9.8. So we have 9.8 times 908 minus 412 divided by 412. This results in an acceleration of approximately 11 0.8 meters. 1st 2nd squared. So this is the answer for the second item. So there's a difference between the acceleration. There is a huge difference, actually. Why is this the case? This is the answer for the last item. So the result is that before we had true masses so we had more inertia on the world system. The second situation, we have only one mass. So we have less inertia on the second system. This is why the second system accelerates much more under the effect off a similar force.

Brazilian Center for Research in Physics