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Problem 37 Hard Difficulty

A block-and-tackle pulley hoist is suspended in a warehouse by ropes of lengths $2 \mathrm{~m}$ and $3 \mathrm{~m}$. The hoist weighs $350 \mathrm{~N}$. The ropes, fastened at different heights, make angles of $50^{\circ}$ and $38^{\circ}$ with the horizontal. Find the tension in each rope and the magnitude of each tension.

Answer

$$\begin{aligned}
&\text {} \approx-177.39 \mathrm{i}+211.41 \mathrm{j}, \approx 177.39 \mathrm{i}+138.59 \mathrm{j}\\
&=275.97 \mathrm{N}, \approx 225.11 \mathrm{N}
\end{aligned}$$

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Video Transcript

for this problem. We have the following diagram Where and there are two roads of land two m and three m. And the ropes are fascinated. Different and different hates making an angle of 5th 50° and 30° as shown in this Figure. So to solve the problem, we need to uh we need to find we are going to call this tension one and this tension to and we need to find the values of tension one and tension too. So because we need to find attentions, it is very straightforward that we need to draw our and our diagrams of forces and see the forces that are acting in the ads. And in the white company then apply Newton's second law. We can obtain the values of the forces that we are interesting in. So um and here we are going to apply the the diagram to the whole list which is this. So we can see that are and this object they drawing our access. We call these ads. We call this wine. And the forces that are being applied in these particular objects are the weight of the hoist that we call. Um The weight. We also have detention too. And what we call the tension one we kills. We can see that from this diagram that this angle in here corresponds to this angle in here. So this Fungal gives 50° and this uncle is 30°. So knowing all of this, we can apply Newton's second law to obtain the suma forces in both components the white company and the X company. So for the why company we have some of the forces in the white company we have that this is equal to detention too attention and one first detention one and the company of the white part is given by design of 15 decreased. And for the attention to we also have this is positive because we are working and we said that this in here in the Upper one is the positive part of the wide companies. So T two will be again signed off in this case purity degrees. And the weight the weight is just laying in the negative part of the Y axis. It's so it's minus Um the weight and this needs to be equal to zero because this is the the toys is addressed it is not moving so it has 00 acceleration. And for the ex parte we have the companies of the tension in the expert which are in here we can see that T. one is indeed in them in the negative direction of ads. Meanwhile T two is in the positive reaction of ed's. So we will have T two positive courtesy of 30 degrees minus T one. The scene of 50 degrees and this is also equal to zero because the object is not moving in the X direction neater. Um from this is can be easily obtained attention to is equal to Attention one cosine of 15 Over Casino of 30. So knowing this that we are going to co want and calling this question too, we can substitute this value in here in the second one so that we obtain the value for for for one of the tensions. And then we can substitute that body of the tension in the other equation. And to obtain the other tension T. One or T. Two in this case. Since we solve 42 we are going to substitute T two in the second equation. So um uh plug in wine and to to we obtain the following um Taiwan just live it like that. The one sign of 15° loss the two that we obtain that this is equal to This expression. She's 51 cosine of 15 Over casino of 30 and minus the way it equals to zero. Um in here we can fact arise, we can factor out T one Soldier, we obtain Taiwan Time sign of 18°. Last A sign of 50° over the sign of 30°. This we can pass the weight to the right hand side of the equation. So we obtain that. This is should we go to the wake. So Detention one is the weight that we are given in this problem that is 350 newtons. And over the sign of 15° laws. The older term. Um we know that the weight As I said, the weight of the more the highest is equal to 350 Newtons. So we're putting in here and using our calculator, we can find a value. Remember to use your calculator in Greece. So I obtain a body of 200 and 32 .19 items .19 items for tension T. One. Now we can simply use equation one to find the value for the attention to. So that's what we are going to do, attention to is equal to tension one. Because scene uh If the degrees over casino of 30° the institutional by the value that we just have found is to 132.1 Newtons putting this in the calculator, we obtained a value for T. Two oh Mhm. One 172 .27 New Gyms. And this are the values for the tensions in the ropes.