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Problem 110 Hard Difficulty

A mountain climber, in the process of crossing between two cliffs by a rope, pauses to rest. She weighs $535 \mathrm{~N}$. As the drawing shows, she is closer to the left cliff than to the right cliff, with the result that the tensions in the left and right sides of the rope are not the same. Find the tensions in the rope to the left and to the right of the mountain climber.


Left: 919 $\mathrm{N}$ Right:845 $\mathrm{N}$


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Video Transcript

to solve this question, we have to Catholic the magnitudes off the tensions. Do you want as teach you? For that? We have to use Newton's second law on. I used the following reference frame a vertical axis pointing upwards on a horizontal axis, pointing to the right. Let me call this X and liver to relax is white. Then we have to apply Newton's second law to the person. By doing that, we get the following for the X axis In that force. Acting on the person is because of the mass off the person times, the acceleration off the person next direction. The person is resting there, so it's not moving and not going to move them. The acceleration is the question zero, so the net force next direction is equal to zero. But the Net Force index direction is composed by true forces. The Axe component off the tension number. Choo 32 Component X and the X component off potential. Number 12 U one x notice that t one X points to the right so it's positive while you'll teach you acts points to the left. So it's negative in my reference fring and these easier. Now we can find a relation between teacher on Teacher X and T one and t. One x for that I only have to do is use the triangles that the question is giving us well, Try and go is this one. So this is a 90 degree angle, and the other triangle is this one where these is also a 90 degree angle. With this triangles, we can calculate the missing angle, these one on these one. It's easy to see that the angles inside the triangle shoot some 180 degrees before 65 degrees. Plus these angle must be 90 degrees, and then you conclude that this angle is an angle off 25 degrees on outside. Doing an analogous calculation, you find that this angle is a 10 degrees angle. Now you can decompose the tensions two and one to discover what is their relation between their ex components on their magnitudes. We have to use order triangle for that we use the triangle formed by the force on their components. So 42 it's the following triangle like this where the high pot a news is the magnitude teacher. This side is teacher component acts decide is teacher component white and we know that the angle between teacher and Teacher X is 25 degrees. So these is 25 degrees. Then, by using the co sign off 25 we can calculate the following the girl sign off. 25 degrees is equals to they are just inside teacher acts divided by the high Ponta News t chew. Then the tube component acts is equals to teach you times the co sign off 25 degrees doing the same 41 We get another triangle which is these one? This is the high part Unused business t one Why component resisted one x component on Do we know that this angle is an angle off 10 degrees? No. Using the co sign off 10 degrees, we get the following information The curse sign is that just inside So t one x divided by the magnitude 31 then to you on component X is given by t one times the co sign off 10 degrees that we can use these results in this equation to get a relation between t one and T choose gets relation is the following to you on X equals to t two component X Then do you want times The coup sign off 10 degrees is equals to teach you times the coup sign off 25 The Reese. And now we have one equation, but we're left with two variables for the other equation. We have to apply Newton's second law on the vertical axis. For that, let me organize my board so I don't need this triangles again. So I keep them there and let me do this calculation here. So for the why access than that force he's given by the mass times the Y acceleration, which is how subzero Because the person is resting, then the net force in the Y direction is a question zero, but the net force the weather actions composed by three forces the weight force and the white components off the tensions. So we have the y component off potential number two on the Y component off facial number one. Then, as the tensions points to the positive direction we have teacher, why was t one? Why minus the weight being equals 20 then teacher, why plus t one? Why is equals to the weight force then for the white components, we can do one analogous calculation in the triangles. But using the signs in the ministry triangles, for instance, in this triangle, the sign off 25 degrees he's given by the opposite side. So you teacher, why divided by the hard part in Usti? True then teacher, why is he close to teach you times the sign off? 25 degrees By the same reason we have the falling relation here t one Why is it Costa t one times this sign off 10 degrees Then we conclude that teach you times the sign off 25 degrees plus t one times the signs off 10 degrees is equal to the weight force. Now we have two equations and two variables. Let us solve them. So let me organize the board before proceeding. Okay? Using the equation on the left we know that to you one is equals to teacher times They call sign off 25 degrees divided by the cool sign off 10 degrees, then using the other equation We have t true times This sign off 25 degrees plus the truth times they co signed off 25 degrees divided by the co sign off 10 Degrees these eesti one. What I know from the other equation that he one is going by these relations. And then we have to move to apply to you one by the sign off 10 degrees in these results in the weight force, we can factor t True, they only got to be true times This sign off 25 degrees plus through sign off 25 released times Notice that we have signed Divided by Curve Sign here. So this is a tangent Times the tangent off 10 degrees is the cost of the weight force. Then teacher is equals to the weight force, which is 535 new times, divided by the sine off 25 degrees, plus the curse Sign off 25 degrees times the tangent off them degrees. And these results in the magnitude off approximately 919 Newtons is the tension number two. Then we can go back to this equation to calculate attention. Number one attention number one is then given by 919 times the cool sign off 25 degrees divided by the coup Sign off, then the grease. These results in attention off approximately 845 mutants on these is the answer to the question.