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Numerade Educator

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Problem 41 Medium Difficulty

To immobilize a fractured femur (the thigh bone), doctors often utilize the Russell traction system illustrated in Figure $6-29 .$ Notice that one force is applied directly to the knee, $\overrightarrow{\mathbf{F}}_{1},$ while two other forces, $\overrightarrow{\mathrm{F}}_{2}$ and $\overline{\mathbf{F}}_{3},$ are applied to the foot. The latter two forces combine to give a force $\overrightarrow{\mathrm{F}}_{2}+\overrightarrow{\mathrm{F}}_{3}$ that is transmitted through the lower leg to the knee. The result is that the knee experiences the total force $\overrightarrow{\mathrm{F}}_{\text {total }}=\overrightarrow{\mathrm{F}}_{1}+\overrightarrow{\mathrm{F}}_{2}+\overrightarrow{\mathrm{F}}_{3}$. The goal of this traction system is to have $\overrightarrow{\mathbf{F}}_{\text {total }}$ directly in line with the fractured femur, at an angle of $20.0^{\circ}$ above the horizontal. Find (a) the angle $\theta$ required to produce this alignment of $\overrightarrow{\mathbf{F}}_{\text {total }}$ and (b) the magnitude of the force, $\overrightarrow{\mathrm{F}}_{\text {total }}$ that is applied to the femur in this case. (Assume the pulleys are ideal.)

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Video Transcript

problem. We have given diagram like this. So for this problem, I suppose this line this is supposed positive X axis. And this is positive y axis. So we can say there are two component of frost fn, fn X will be equal to First of all, we will try to find the magnitude of F one F 23. Okay, we know that everyone f two and F three r are the forces in the same rope so we can say F three is connected here. Suppose the attention is still so we know that this F three should be equal to 4.25 and 29.80 That means it is equal to 41.65 Newton or we know that after it should be equal to F two and F one also So we can write F one magnitude and F two and F three should be equal to 41.65 Right, So now we will right F one x everyone X will be equal to 47.65 because theater minus 20 right? We can read this now the direction of these forces towards negative X axis in this direction. So we we write minus and F one white will be equal to 41.65 Signed heater minus 20. Yeah, like this after will be equal to minus 41.65 because it is already in the 20 degrees above the horizontal on this plane. So this is minus 41 went 65 F two y will be equal to zero and F three x should be equal to this is F T. This is equal to minus 41.65 because of this angle is 20 degrees, 30 40 degree cost 40 and f three. Why should be equal to 41 minus 41 65 signed 40 Because it is in the downward direction in this direction three by. So these are the component of F one, F two and F three. So now we will equate. You will quit the vertical component because f. N y in that positive y axis in this direction and f three y is in this direction. So these two should be equal. So we will right 41.65 Sign off. Tita minus 20. Should be equal to 41.65 Sign off 40 degree. So from here, we can see if he does sort of equal to 60 degree. No, this is the value of pita. So now we will right F net will be equal to Afghan X plus f two X plus f three x. I kept less f one way f two y plus f three way Jacob So f one x should be called minus 41.65 and this will be equal to cost 14 minus 41.65 Cost 40 and F two x is minus 41.65 Mhm and F three X is minus 41.65 cars 40. Mhm. Mhm. Okay, Graham, the icon direction now Vanuatu is 41.65 signs 40. Yeah, after five years 035 years minus 41.65 Signed 40 Mhm. Okay, so this is equal to zero. The objective direction. If we saw this, this is equal to minus 41.65 Yeah, two costs 40 last one. So if we saw this, the net force net force is equal to minus 106 Newton. The negative X direction. If we talk about the magnitude, this is equal to 106 Newton.