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Problem 68 Hard Difficulty

mmh A kite is hovering over the ground at the end of a straight $43-\mathrm{m}$ line. The tension in the has a magnitude of 16 $\mathrm{N}$ . Wind blowing on the kite exerts a force of $19 \mathrm{N},$ directed $56^{\circ}$ above the horizontal. Note that the line attached to the kite is not oriented at an angle of $56^{\circ}$ above the horizontal. Find the height of the kite, relative to the person holding the line.

Answer

32 $\mathrm{m}$

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Video Transcript

to solve this question, we have to discover what is the angle, Tita. In order to do that, we can extend this horizontal line here to the other side. So that's now we can decompose the tension force in two components went or his own toe component and one small vertical component. We can do the same with the wing force. It has one vertical component and one horizontal component. Let me call this tension forced important. Why the ancient force component X weaned force component. Why and wind force component X Note that this angle here, in between the tension force component X and the original tension force, is equal to Tito because we have to poorer lines that has one other line going through them. The reform. This angle Tita is equals to this angle. Teeter by symmetry, so we can now exploit the symmetry to discover what is the value off Peter. For that, we have to apply Newton's second law. We have two directions. The y direction which the vertical direction on dax direction, which is the result direction. Note that on the Y direction we have the weight force, but we don't know what is the mass off the kite surfing. There isn't sufficient information to that. Remind the weight. So in order to go around this, we will use the horizontal axis because on the horizontal axis there are only two forces acting the wind force Invitation Force, then Newton's second law. Thousands. The following on the X direction. The Net force Is it close to the mass off the kite times its acceleration in the X direction, which is close to zero because the kite is standing still. So the Net forest component X is equal to zero, but it's composed by two forces. We have the wind force component X, which is pointing to the positive extraction minus the tension force component acts which is pointing to the negative X direction. Then the wind force component X is the close to detention forest component X and now new to the following. We can use triangles to better mind. What is the relation between the acts component and the magnitude of the force for doing force? We have this triangle here is the wind force. Here we have the white component and nearly half decks component and 56 degrees. Then note that the axe component is the address and side to the 56 degrees angles. Therefore, half w component X is it goes to after you times the co sign off 56 degrees because it's the address in sight. So weaned Force Times Co. Sign off 56 degrees is he goes to the sanction for its component X For attention force. We have another triangle which is thes one. Here we have the tension force attention forced Component X and attention for its component. Why and here is data note that the tension force component acts is also address ST to dangle. Tita therefore contention for us component X. Is it close to the full tension force? Times the co sign off Rita because it's address into the enter. So we have attention. Forced times co sign off Dita No, we can calculate What is the co sign off? Data on these is given by the wind force 19 times the co sign off 56 degrees divided by the tension force, which is 16. And then we can calculate what it's teeter Tita is he goes to the inverse co sign off all these and then this is approximately 48 degrees. So now we can calculate what is the height off that kite? Let me organize. Deport to do that. Clean the board. Now let's do it. So the height off the kite is these Orange line here. Note that we have erected will try and go here, and the height off the kite will be the opposite angle to teeter, which new? That is approximately 48 degrees. Dan. Using this information, let's call the height age. We have the following this high off Tita, which is 48 degrees. Is it close to age, which is the opposite side, divided by 43 meters, Then the height is the cost of 43. Find the sign off 48 degrees. These give the height off approximately 32 meters.