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A block is pressed against a vertical wall by a force $\overrightarrow{\mathbf{P}},$ as the drawing shows. This force can either push the block upward at a constant velocity or allow it to slide downward at a constant velocity. The magnitude of the force is different in the two cases, while the directional angle $\theta$ is the same. Kinetic friction exists between the block and the wall, and the coefficient

of kinetic friction is 0.250 . The weight of the block is $39.0 \mathrm{N},$ and the directional angle for the force $\overrightarrow{\mathbf{P}}$ is $\theta=30.0^{\circ} .$ Determine the magnitude of $\overrightarrow{\mathbf{P}}$ when the block slides $(\mathbf{a})$ up the wall and (b) down the wall.

52.6 $\mathrm{N}$

39.4 $\mathrm{N}$

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in the first situation where the box is lighting up the wall. We have the fiddling forces acting. We have the weight force. We have a frictional force that points no works because the block is moving upwards. And then we have this press and force that is pressing the block against the wall and because the bloc's being press it against the wall, the wall response by pressing the box against the pressing force and then it results in a normal force that act on the block. Then in the situation, in order to discover what is the value off the Force P, we have to use Newton's second law. So we will have to use Newton's second law on both directions. The vertical direction, which I'll call white on the horizontal direction. We shall call X then on the vertical direction. Newton's second law tells us the following the net force on the vertical direction is it goes to the mass off the block times its acceleration in the vertical direction, which is the cost of zero. Because the blocks is not moving with a variety velocity, it's moving with a constant velocity and because the velocity is not changing. Acceleration is close to zero. Then the net force in the wider actually is. It goes to zero. But the net force in the wider action is composed by several forces. It is composed by the weight force, the frictional force and one component off. This pressing for speed on the components that composes the net force in the Y direction is these one is the vertical component B. Why and then there is the order component which are called B X. So in the vertical direction, we have plus the white because it points to the positive direction minus the weight force minus the frictional force is equal to zero. Then B Y is equal to the weight force, plus the frictional force. Then the weight force is the close to 39 new times, and the frictional force is given by Remember the equation the genetic frictional coefficient, times the normal force, the kinetic frictional coefficient in these questions, Is it close to 0.25? And then the only thing we have to discover is what is the value off the normal force? For that? We're playing Newton's second law on the horizontal axis By doing that, we get the following The net force on that axis is the cost to the mass times acceleration off the block in that access, which is also equal to zero because the block isn't moving away from the vault or past the wall, then the net force next direction is it close to zero, and the forces that composed the net force Next interaction are the following minus the normal, which points to the negative direction off the X axis, plus the X component off the press and force which points the positive direction off the X axis and the business goes to zero. Then we have that X is equals to the normal force. Therefore, the normal force is it goes to the axe component off this P force, then going back to the equation. For P white, we have two following a few white is equals to 39 plus 0.25 times be X. But what is the value off PX And what is the value off the wife for that? We can use a little bit off geometry in this triangle right here. So in this triangle disease, a right angle and also Tita is egos to 30 degrees, as given by the problem, then is 30 degrees. Note that this angle is also a right angle reform. These angles should be caused to six degrees and these order 30 degrees. Then how can we calculate P X and P y knowing that the hypothesis is it was to be This is not that difficult. Just use the sign on the coastline. So let me use the sign off 30 degrees. So the sign off 30 degrees is equals to the opposite side off the triangle. In this case, P X, divided by the headquarter news then p X is equals to be times this sign off 30 degrees. Now, using the co sign off 30 degrees, you get the following the coastline is it goes to the I just inside off the triangle, so b y divided by the Ponting who speak then you Why his ego stripy times the co sign off. 30 The Reese Now using these equations that we had just arrived in our equation for P, we get the following times the co sign off 30 Zico's to 39 plus 0.25 time speak times their sign off 30. Then we can send these terms to the other side to get the following be times the cool sign off 30 minus 0.2 point 25 times speed times this sign off. 30 Easy of course to 39. Then factoring out beat, we have that be time The co sign off 30 degrees minus 0.25 times the sign off 30 degrees Easy goes to 39 Finally be. Is it close to 39? Divided by the co sign off 30 degrees minus 0.25 times this sign authority degrees and these givers magnitude off approximately 52 0.6 neutrons. These is the answer for the first item off our question. Let me put this answer. Here is a question 52 0.6 neutrons. Then, to solve the second item, I have to clear the board. But keep in mind that the second item is very similar to the first item. So the only thing that changes is the direction off the frictional force. So let me clear the board now and Soviet in the second item we have the same forces acting, but now the box is going down the wall so the frictional force is going upwards. Then, in order to discover they've AII off be. We have to a plane. You don't second law on the same referential we have used before. Then on the Y direction we have the following the net force in the Y direction is it goes to the mass off the box times acceleration off the box in the Y direction which is equal to zero because the box is moving with a constant velocity. Then in that force on the Y, direction is close to zero. But the net force on the Y direction is going to lose it by several forces. Again, we have the components off the pressing force on the only components that is pointing in the vertical that actually is P y. And therefore the net force in the Y direction is given by P. White plus the frictional force minus the weight force. And this is a question zero. Then B Y is equal to the weight forced my news, no frictional force. But remember that the frictional force easy goes to the kinetic frictional coefficient times the normal force then the white is equals to 39 which is the wait for us minus zero points 25 which is the kinetic friction coefficient times the normal force. Then we have two company, the normal force, And how can we do that? We can do that by using Newton's second law on the X direction in the direction you don't. Second Law tells us that the Net force is, of course, to the manse off the box times acceleration off the box in the direction these acceleration is again equals to zero. Because the box is not moving past the wall are moving away from the wall then and that force next direction musicals to zero. No notes that the net force in the X direction is composed by two forces. We have the normal force pointing to the negative direction, and we have PX pointing to the positive direction. So PX mind was the normal forced Is it close to zero, then the normal force Easy close to P X. Now let us use the result in the equation. We had four p wife, then the white is It goes to 39 minus 0.25 times p x They remember that B y Z goes to be times they go. Sign off 30 degrees and PX is a goes to B times This signed off 30 degrees. Then, by using these two equations, we can write the following p y, which is times the co sign off. 30 degrees is Equals to 39 minus zero points 25 times speak times this sign off 30 degrees, which is BX. Then we can send this term to the other side. So be times the co sign off 30 plus 0.25 times speed. Thanks. This sign off 30 is it goes to 39. Then, factoring out, we guessed that Pete times the co sign off 30 degrees plus 0.25 times the sign off 30 degrees is equals to 39. Finally, be easy cause too 39 divided by the co sign off 30 degrees plus 0.25 times this sign off 30 degrees. And these results in oppressing force before magnitude of approximately 39 0.4 Beautiful. These is the answer to the second item

Brazilian Center for Research in Physics