A chandelier is supported from the ceiling by 2 chains. Both chains make a 30° angle with the vertical. The tension in each chain is the same because of symmetry:T 1 =T 2 =50 N. What is the mass of the chandelier?

Alright guys. Welcome back. Let's check this one out. So we got the chandelier that's supported from the ceiling by 2 chains. So I've got the ceiling like this. I've got these 2 chains that are gonna be connected like this diagonally. Now I'm drawing them diagonal because I know the angles with respect to the vertical are gonna be 30 degrees for each one of them. Right? So they're symmetrical. So both these angles here are gonna be 30 degrees. Notice how I'm drawing it with respect to the vertical. What we wanna do is we wanna find out the mass of the chandelier here, and we're only told that there is, that these tensions in these cables are gonna be symmetrical. So let's go ahead and figure this out. We we gotta draw the free body diagram. So I'm gonna go ahead and do that. So the free body diagram still look like this. We've got the weight force that acts straight down. That's our mg. And then we basically have any applied forces or tensions. We don't have any applied forces, but we're gonna have 2 tensions. So I'm gonna call this 1, t one, and I'm actually gonna call this 1 t 2. We know that they're both gonna be 50 newtons. We're told that the magnitudes of each one of these things is gonna be 50 newtons. Right? And, because this thing is suspended in the air, right, there's gonna be no normal forces or frictions or anything like that. So really this is our free body diagram. Alright? So we wanna figure out the mass of the chandelier. So how do we go about doing that? Well the next thing we would do is we would decompose any 2 dimensional forces. We've got these 2, tensions that point diagonally like this and they're also symmetrical so it's gonna be very easy to decompose them. So basically we've got this t one x and then we've got this t 1y. And then this one, we've got t2x. And then here we've got t2y. Alright. So we've got these 2 dimensional forces here. We can go and decompose them but first we actually wanna draw write out the F equals m a So that's the second step. Right? Oh, sorry. The 3rd step. We're gonna write f equals m a. We're trying to figure out the mass of the chandelier. So what happens is in the x and y axis, we write our f equals m a. So this is a sum of all forces equals mass times acceleration. Sum of all forces in the y axis equals mass times acceleration. What happens in this problem here is if you take a look, the chandelier is supported by the ceiling by these two chains. It's supported and it's basically held in place. So what that means is this is actually a 2 dimensional equilibrium problem. So basically what happens is these 2, cables or chains are suspending the chandelier so that it doesn't move in the x and y axis. What that means here is that both of my x and y axis, my sum of forces are basically gonna cancel out. So what happens is we expanded all of our forces in the x axis, we're gonna pick a direction of positive. I'm gonna choose the up and to the right direction to be positive. So when we expand out the sum of our forces, what we're gonna get is t one x minus t one, Oh, sorry. Minus t two x is gonna equal 0. But this equation doesn't really give you the mass. And if I wanna figure out the mass, what I'm gonna have to do is I'm gonna have to look at the y axis instead. Because what happens is in the y axis, I've got my t one y plus t two y minus mg equal to 0. So here's where I have my unknown variable. So I'm gonna work with the y axis. Right? Now what actually happens is because of symmetry, both of these components are actually going to be the the same. Right? Both of these tensions here are 50 newtons. So what happens is if I have a tension that's equal to 50 newtons, then I can figure out t y by just doing 50 times the sine of the angle. Alright? So I've got this angle here that's 30 degrees. So what happens is if I were to complete this triangle, remember, this angle is with the vertical, and I don't want that. I actually want the angle that's relative to the horizontal. So what I can do here is I can make a little triangle, and if this is 90 degrees and this is 30, then that means this has to be 60. So I'm actually not gonna use 30 degrees and I am gonna use 60 because that's the one that's with respect to the horizontal. So when I use my sine I'm gonna actually have to use 60 degrees and if you go ahead and work this out, you're gonna get 43.3. Now what happens is because of these two tensions, they're actually the same. Right? So you can actually just say this is 2 t y because these t ys are gonna be the same, magnitude and direction. So I've got 2 t y minus mg equals 0. So this 2 t y equals mg. And so therefore my mass is just gonna be 2 times 43.3 divided by 9.8. And I'm just gonna get 8.8 kilograms. And that's the mass of the chandelier here. Alright? So let me know if you guys have any questions and that's it for this one.

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A chandelier is supported from the ceiling by 2 chains. Both chains make a 30° angle with the vertical. The tension in each chain is the same because of symmetry:T1=T2=50 N. What is the mass of the chandelier?

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A chandelier is supported from the ceiling by 2 chains. Both chains make a 30° angle with the vertical. The tension in each chain is the same because of symmetry:T₁=T₂=50 N. What is the mass of the chandelier?