A picture frame hung against a wall is suspended by two wires attached to its upper corners. If the two wires make the same angle with the vertical, what must this angle be if the tension in each wire is equal to 0.75 of the weight of the frame? (Ignore any friction between the wall and the picture frame.)

Video transcript

Hey, everyone in this problem, we have a crane that lifts the slab at a constant speed using two steel cables attached to its upper edges. The tension in each cable is equal to 0.8 of the slabs weight. If the two cables have the same angle relative to the vertical, we're asked to determine the value of the angle that makes the tension in the cable equal to 0.8 of the slabs width. OK? We're given four answer choices. Option A 38.7 option B 51.3 option C 57.9 and option D 32. And we're gonna start by drawing out this situation. So we have our slab and then the two steel cables attached to its upper edges and there at an angle. So we have the first one and we're gonna say that this has attention to what and we have another coming from the right end which we're gonna say has attention T two and these are at angles relative to the vertical. We're gonna call those angles, theta one and theta two respectively. All right. So we can break up these tensions. T one T two into the vertical and horizontal components. So we're just gonna write that out. So we have those if we need them. So we have T one Y T one X and then we have T two X and T two Y. All right. Now, we've written all of the information we have about tension in this diagram. Let's make sure we include all of the forces and the other force we have acting is the weight of our slab and that is gonna act downwards. OK? So that is our complete diagram, that is all of the forces acting on this slab. Now, I wanna think about what we're told. OK. Now we're told that the cables have the same angle relative to the verbal. What that means is that they don't want, it is equal to theta too. And we're just gonna call both of those theta we know that those angles are the same and we're told that the tension in each cable is equal to 0.8 of the slabs weight. Now each cable has the same magnitude either both the 0.8 of the slabs weight, then the 10 T one must be equal to the tension T two and that must be equal to something. And we're just gonna call that T OK. So since the magnitudes of the tension are the same and the angles are the same, then the components are going to be the same as well OK. So this tells us at T one X is equal to T two X and T one Y is equal to T two Y. And for the X component, when I say this, what I mean is the magnitude. So let's put magnitude symbols around that just to be very clear because those two tensions in the horizontal component do point in opposite directions. All right. So what we wanna find is the angle that makes this happen. OK. So let's think about Newton's second law that's gonna allow us to relate our tensions, our acceleration, our mass, all of those things and get closer to this answer. So Newton's second law tells us that the sum of the forces and we're gonna look in the Y direction since this crane is lifting this slab upwards, the sum of the forces because it's gonna be equal to the mass multiplied by the acceleration. Now, in this case, this is gonna be equal to zero. Why? Well, we're told that this is a constant speed. If this is moving at a constant speed, it means the acceleration is zero. And so the sum of the forces is going to be equal to zero. Now, what forces do we have acting well in the Y direction, we have the Y component of T one, the Y component of T two in the positive direction. And we're gonna take up to be positive and then we have the weight W acting downwards So we get T one, Y plus T two, Y minus W is equal to zero. Now, T one Y and T two Y are equal and we just said that they're equal. So we can write them just as Ty and then we can combine those two terms and write it as two ty minus W is equal to zero. OK. Now, we might be thinking at this point, we're looking for an angle, This equation does not have any angle theta in it. So what are we doing here? Well, what we're doing is we're gonna introduce the angle now because we're looking at the Y component of the tension because we're looking at the Y component, we need to figure that out in terms of the tension T one. OK? Or T two. They are the same now because the angle is relative to the vertical. When we want the Y component, we're gonna take cosine of the angle. And so Ty, it's just gonna be T cosine of theta. So we have two T cosine theta minus W is equal to zero. OK. That's great. We've introduced our angle theta and that is what we are solving. Now, when do we wanna find this angle for? Well, we wanna find the angle when the tension T is 0.8 of the slabs wait, right. So the tension T, 0.8 of the slabs weight. So we have T as 0.8 multiplied by W our equation becomes two multiplied by 0.8. W multiplied by cosine of theta minus W is equal to zero. We can simplify 1.6 W multiplied by cosine of theta is equal to W dividing both sides by 1.6. W we get cosine theta is W divided by 1.6 W. And you can see that that weight is going to divide it. OK. So it doesn't matter what the weight is, that's going to divide it. And this is going to work out the exact same way finally to isolate the, we're gonna take the inverse cosine, we get that theta is coast inverse of one divided by 1.6 which gives us an angle theta about 51.32 degrees. All right. Now, it's always important to remind ourselves of what that angle represents. So let's go back up, remember that we took the Y component of tension to be t multiplied by co state. OK. So when we did that, that means we're talking about an angle relative to the vertical. And that's exactly what the question was asking us to do. OK. So always just double check that the angle you found is the correct angle. OK? Whether it's relative to the horizontal or the vertical or whatever the question is asking. OK. So in this case, relative to the vertical and the correct answer is going to be option B 51.3 degrees Thanks everyone for watching. I hope this video helped you in the next one.