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Problem 51 Hard Difficulty

While moving in, a new homeowner is pushing a box across the floor at a constant velocity. The coefficient of kinetic friction between the box and the floor is 0.41. The pushing force is directed downward at an angle $\theta$ below the horizontal. When $\theta$ is greater than a certain value, it is not possible to move the box, no matter how large the pushing force is. Find that value of $\theta .$

Answer

$68^{\circ}$

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Video Transcript

in this question. There are 1234 forces acting on the box. The weight forced the normal force, the force that is, apply it to push the box and the frictional forces. Then, to begin serving this question, we have to use Newton's second law and we have to use that law both in the vertical direction, which I'm calling. Why on on the horizontal direction which I'm calling X then that is applying Newton's second law on the vertical direction. So that direction, what do we have the net for force is equals to the mass times acceleration on the Y direction. Note that the box is standing still in that direction before the acceleration is equals to zero. So the net force on the Y direction is the of course to zero. Then what are the components off these net force? Well, we have the normal force. We have the weight force and apart off these Apply it force. So we have the normal force that is pointing upwards miners the weight force which is pushing downwards minus something that is also pointing downwards. But what is that? So note the following here these force F can be interpreted as two forces one force pointing downward and the mother force pointing to the right. This would be the Y component off the force f and these would be the axe component off the force s. Then we see that what is pointing in the vertical direction is the Y component off that force f. So here we have f component white and these easy question zero. Then we can relate that. Why component off the force f to the full force f and how can we do that? Note that these rectangle here that arises by the composing the force f is a rectangle triangle. Then these means that this angle here, let me call it out for is such that when we some offer plus teeter, these is equals to 90 degrees. Then as the angles off a triangle must add to 180 we have the following conclusion these easy close to 90. So this angle from a plus this angle must be close to 92. Therefore, the only possibility is that these angle is also equal. To think that then we can write that directing with triangle as follows so let me draw it here. And the bigger drawing. You have the full force, the Y component and the X component right under here. And Tita is right here. Then how can we calculated the Y component? We can use the sign off, Peter. The sign off Peter is equals to the opposite side off the triangle, So f why Divided by the doctor News F the reform f white using close to half finds this sign off, Tita. Then we have the following here. Normal force minors The weight force minus f times this. Sign off, Peter, using close to zero. Okay, Now letters applying Newton's second law horizontal axis and see, What can we do show before the horizontal axis? We have the following the net force on that direction. Is it close to the mass? Off the box times acceleration off the box after box is also not moving by a section on the X direction is equal to zero. So the net force in next direction is equal to zero. But the net force in the X direction is composed by two forces. The future all force and these components off the plate force. So the frictional force is pointing to the left so it carries a minus sign and they apply it. Forced component acts is waiting for the right. So it has a plus sign. And this is because zero. Then you get the following conclusion. They play it forced in the X direction most of equal to the frictional force. But then, how can we calculate the fictional force where remember that the frictional force is equal to the connected traditional coefficients times the normal force, Then half component acts must be equal to the kinetic frictional coefficient times the normal force. But then, what is the normal force? We can use these other expression to calculate it. The normal force is equal to the weight force, plus f time sign off detail. Then they apply it. Forest component acts must be equal to the kinetic friction coefficient. Times the wait for us. Plus have times the sign off, Peter. But then what is the X component off the player force? We can use a similar idea to that one. The co sign off Tita is equal to the address inside off the triangle. So half X divided by the hyper two news half. Therefore, half component X is equal to the hypotenuse times nickel sign off detail. Then we have the same The following conclusion f times the co signed off Peter is equal to the kinetic friction aquisitions times the weight force which is the mask off the box times the acceleration of gravity blows f times the sign off Ito And then we want to discover what is the neccessary force f for the situation toe hold. Then we must solve this equation for F. And to do that, I have to organize the board. Okay, now we can do as follows f go sign off. Tita is equal to the kinetic frictional coefficient times the mass times the gravity plus the kinetic which no PlayStation times f times this sign off detail Then we send this term to the other side to get f times the co sign off Peter Miners Mu K times, half time. The sign off Peter is he quote your UK times the mass times gravity acceleration. Then we can factor out f so f times the co sign off detail minus relocate Hangs This sign off. Pita is he course from UK times the mass off the box times the acceleration of gravity, Then the neccessary force to push that box f is equal to communicate times the must off the box trying to the acceleration of gravity divided by the co sign off detail miners Mu Kate. Don't the sign off, Tita. Now note the following. If the denominator off this fraction is equal to zero, then that fraction will become infinity. Therefore, if these easy cost zero the neccessary force to push the box would be close to infinity and in the situation, it's impossible to push the box. Then then we most of the following equation the go sign off the tip Miners were Kate Times the sign off Tito must be equal to zero so that the necessary force to push the box will be equal to infinity. Then we solved this new equation for Tito. We send this term to the other side to get the curse Sign off Tita Z Culture muche eight times this sign Dita. Then we sent this term to the other side of these terms to the other side. To get one divided by Mu Kate is equal to the sign off. Tita divided by the co sign off Peter signed. Divided by the coastline is the tangent off Dita before the tangent off Dita is equal to one divided by mu Kate. So Tita is equal to the inverse tangent off One Divided by you, Kate. And these is the inverse tangent off one divided by 0.41. And this gives us an angle. Teeter off approximately 68. The grease. So when Tita approximately 60 degrees, it doesn't matter how hard you put the box, it will not move.