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Problem 59 Hard Difficulty

A 1.40-kg bottle of vintage wine is lying horizontally in a rack, as shown in the drawing. The two surfaces on which the bottle rests are $90.0^{\circ}$ apart, and the right surface makes an angle of $45.0^{\circ}$ with respect to the horizontal. Each surface exerts a force on the bottle that is perpendicular to the surface. Both forces have the same magnitude $F .$ Find the value of $F$ .


9.70 $\mathrm{N}$


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Video Transcript

in this question. We have this view that is presented by the book off the situation. But I think that this question is much easier when you look at it from this point of view from the front, our point of view because like this, it is clear where the forces are pointing and what do we need to do? Sir, we have to forces which are contact forces that are counterbalancing the weight force. So the weight for most points in the vertical direction So what we have to do in order to complete this forces? It's what they compose them and calculate their vertical component. And why does it is the case Because of the following? This is the horizontal component off this force half on the left. So let's call this F one x and this is their horizontal component off the order force which has the same magnitude the same Unho with the plane, F X number two note that f has the same magnitude off f. Therefore, the horizontal component off this force F has a six magnitude as these a result, a component off this force f. These implied that these are counter balancing one another. And so the exactly council ought the reform to counter balance the weight force. The only component off the force after please a rule is the vertical component. So these these f why and this is F why truth are not giving them names like F Y 1.5 way, too. Because they are completely equivalent. They are exactly the same. They have the same direction and the same magnitude. Then the next step is applying Newton's second law off on the vertical axis, which I will call bite. So Newton's second law tells us that the net force on the Y direction is equal to the mass off the boto times that celebration on the Y direction. But the bottle is a standing still. Therefore, acceleration is it was 20 Then the net force in the Y direction is He goes to zero. But the net force on the Y direction is composed by three forces F white F y again on W. Then we have F white plus half white plus not plus minus because W is pointing to the negative direction off the way Access so miners w easy close to zero, then two times f Why is equals to W? And finally f Why is equals to double your divided by True, then all we have to do to complete this white component off the force half Easter completes the weight force and the weight forces given by the mass off the bottle times the acceleration of gravity divided by true. And these is 1.4 times. Remember that G is approximately 9.8 meters per second squared. So we have 9.8 year divided by true, which is a close to 6.86 new terms. So what, Harry calculated we calculated the Y component off the force f But there is a relation between the white component off the force F and the force F on the relation can be seen by using a rectangle triangle. So we have the following these these directing the triangle. The high party news is equivalence. The magnitude after force f this side is it will be the y component on this side will be the X component for this case acts component one. Then this is a right angle and these are no. Here is an angle off 45 degrees, so these other angle is also 45 degrees. So we know everything about this rectangle triangle. Then how can we relate? F Why we've f by using the co sign or this sign off 45. So true is one. Let me choose that one, then the sign. The co sign off that angle is equals to They are just inside f y divided by the magnitude. Or did I punt the news? F Then if y is equal to half times Liko sign for 45 degrees, then the force F is equals two f y divided by the co sign off 45 degrees by using the value for off F Y we had just completed to get that F is equals to 6.86 divided by the co sign off 45. And these results in approximately nine 0.7 neutrons. These is the final answer off this problem magnitude. Off each of these forces F is 9.7 new tubes

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