Learn about a more complicated friction/inclined plane problem Video

By: Guest More than a year ago

hi there, i found this video very helpful thanks alot. but i was wondering if u just ignored the friction between the rope and the pulley or whether u were asked to ignore it in the question. thank you

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Learn about a more complicated friction/inclined plane problem

I've been doing a lot of problems where I kind of made up the numbers so everything worked up nice and neat and clean. But now I got this problem off the internet. I think this will be instructed because the numbers aren't so easy and it’s not like sine and cosine of 30 degrees. We’ll going to have to deal with 35 degrees. And this is so that I can make sure that we got the same answer that they got, I'm going to use 9.8 meters per second squared as acceleration of gravity instead of 10. So let's do this problem. So let's see what's going on here. We have this inclined plane or this wedge or this ramp. Whatever you want to call it. And you have a 10 kilogram mass sitting on the ramp. Then there's this mass less rope or wire that goes over this pulley and then that same wire is attached to a 20 kilogram mass that’s just hanging there. So what's going to happen just very intuitively? Well here you have the whole force of gravity pulling down on this 20 kilogram mass. And then here pulling back on this, you’re going to have the mass of this. Let’s just assume, you know, just to get the intuition intentional friction just to figure out which direction things moving. You're going to have the parallel component, the component of this object’s weight that’s parallel to the ramp pulling back. So that’s going to be something less than this object’s weight. This object’s weight is roughly 98 Newtons. So what's pulling back while it’s going to be something less, I don’t know, 50 Newtons or 60 Newtons. We’ll figure that out. But clearly, the weight pulling straight down on this object is greater than the force pulling back parallel to the ramp on this object. So the whole system will move to the right. So the whole system will move to the right, we know that the force of friction is going to act backwards. Because it always acts against you. So let's do the problem, let's do the math. Now that we have the intuition of what's going to happen. And of course this object will fall or it will accelerate downwards slower than it would have if it wasn’t attached to this whole mechanism. Right. This will be pulling back on a little bit. So the problem tells us that the coefficient of friction between this mass and the ramp is .2. So we can just go into our mode, because this is a hard one just to figure out what happens with this object because it has friction, it’s on an inclined. So let's see what happens. So first we know that there's the force of gravity pulling straight down on it. Right, that’s the force of gravity. And then we’ll have, I know I always change words because I actually don’t know the correct. Well, it seems like everyone uses different convention for this, but you have the perpendicular component of gravity, which is that force. And then you have the parallel component of gravity. Let me switch colors. The parallel component of gravity, which is that force. And of course we have to figure out the perpendicular component, which is same in magnitude to the normal force. And that helps us figure out what the force of friction is. And then we use the parallel force, so we just figure out the forces, how much of gravity is pulling parallel to the ramp backwards. So let's do that. So first of all, what's the weight of the object? Well, its 10 kilograms and the acceleration of gravity are 9.8 meters per second squared. So its 9.8 times 10 is 98 Newtons. And we've learned from previous videos that if this angle is 35 degrees, this angle is 35 degrees as well. So I will reprove it here. So if this is 35 degrees and this is 98, what is this side equal to? This side that is perpendicular to the surface of the ramp. Well, adjacent over hypogenous is equal to cosine. So this side is equal to 98 cosine of 35 degrees. And I do not know what the cosine of 35 degrees is, so I will use my calculator. Cosine of 35 degrees is roughly .82 and then I multiply that to 98. That equals roughly 80. So this equals 80 Newtons. This vector right here. The part of gravity, the component of gravity, that is pulling perpendicularly down to the surface of this ramp is 80 Newtons. And of course the normal force, that’s the force that the weight is kind of pushing into the ramp, and the normal force is just the equal and opposite force that the ramp is pushing into the weight. That keeps the weight from falling into the ramp. So that’s also going to be 80 Newtons. So the normal force is also 80 Newtons. And the only reason why I point that out is just because I wanted to start referring to the normal force because I didn’t have a variable here. But anyway. So you could say that the force of friction now, so the force of friction. We already determined is going backwards. Because we determined that the whole system will move to the right. We just figure that out just by looking at the relative magnitude of the masses. So the force of friction is equal to the coefficient of friction, .2, you could either say times the normal force or times the force of gravity going perpendicularly down in to the surface. But either way, that’s going to be 80 Newtons. .2 times 80, that is equal to 16 Newtons. Excellent. So we figure out that the force of friction pulling back on this object is 16 Newtons. And what's the component of gravity pulling back on this object or this vector going parallel to the surface? Well, we've done this a lot. Now I will call this force parallel, just because I need something. The force parallel, this is the opposite side. This is the hypogenous. Opposite over hypogenous is equal to sine of 35 degrees. So this is equal to, we've done this a lot, I won't go into detail from that. 98 sine of 35 degrees. Go back to the calculator. Because I have not memorize sine of 35 degrees. 35, sine, is about .57. Times 98 is equal to 56. So the force parallel is equal to 56 Newtons. So there are two forces pulling back on this object. One is 56 Newtons, and that’s just the component of gravity parallel to the surface. And then you have the force of friction pulling back. And of course the force of friction is always acting against you. So the total forces pulling back on this object, if you add these two, 56 plus 16 is 72 Newtons. So there's 72 Newtons of force essentially trying to pull this whole system back. Right. You can almost view it, there's 72 Newtons of force pulling up on this object to. Right. That’s what's pulling this whole system back. And now what's pulling this whole system to the right? Well it’s the weight of this object. And what is the weight of this object? Well the weight of that object is 20 times 9.8 meters per second squared. So the downward force, the force of gravity, so that’s this force. Let me do it in different color because it’s getting messy. That downward force is 20 times 9.8, that’s 196 Newtons. So you have 196 Newtons pulling this way, that’s also pulling up here. So the net force of this system, think about it this way. You have 196 Newtons that want to go roughly in that direction. Then you have 72 Newtons that wants to roughly in that direction. I know it maybe makes it a little bit confusing with this pulley and all that. But hopefully that makes an intuition of there's two direction it can go to. It can go into the up and down direction or the up and to the left direction. So what is the net of this two? Well, what's 196 minus 72? So the force net trying to go in that direction is 196 minus the force of friction minus this parallel force which is 72. The combined force of the friction and the parallel force is 72, so its 196 minus 72 is 124 Newtons going in this general direction. 124 Newtons. And so what is the acceleration of the system? Well to know the acceleration of the system, we have to know the mass of the system. And what is the mass of the entire system? Is it just the 20 or is it just the 10? No, it’s both. Because the entire system is moving, the entire thing. This entire contraption, this whole mass and the wire and the other mass, it’s all being accelerated. So the total mass is 30 kilograms. 10 plus 20. So you have the force, 124 is equal to the mass, 30 times acceleration. So acceleration is equal to 124 over 30 and that is 4.13 meters per second squared. So this whole system will accelerate up into the right. And then it will accelerate downward at 4.13 meters per second squared. Which also makes an intuitive sense, because essentially this mass has slowed down this objects descent. Right. If this was completely unattached to this wire, it would have fall in 9.8 meters per second squared. I’ll see you in the next video.

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I've been doing a lot of problems where I kind of made up the numbers so everything worked up nice and neat and clean. But now I got this problem off the internet. I think this will be instructed because the numbers aren't so easy and it’s not like sine and cosine of 30 degrees. We’ll going to have to deal with 35 degrees. And this is so that I can make sure that we got the same answer that they got, I'm going to use 9.8 meters per second squared as acceleration of gravity instead of 10.
So let's do this problem. So let's see what's going on here. We have this inclined plane or this wedge or this ramp. Whatever you want to call it. And you have a 10 kilogram mass sitting on the ramp. Then there's this mass less rope or wire that goes over this pulley and then that same wire is attached to a 20 kilogram mass that’s just hanging there.
So what's going to happen just very intuitively? Well here you have the whole force of gravity pulling down on this 20 kilogram mass. And then here pulling back on this, you’re going to have the mass of this. Let’s just assume, you know, just to get the intuition intentional friction just to figure out which direction things moving. You're going to have the parallel component, the component of this object’s weight that’s parallel to the ramp pulling back. So that’s going to be something less than this object’s weight. This object’s weight is roughly 98 Newtons. So what's pulling back while it’s going to be something less, I don’t know, 50 Newtons or 60 Newtons. We’ll figure that out. But clearly, the weight pulling straight down on this object is greater than the force pulling back parallel to the ramp on this object. So the whole system will move to the right. So the whole system will move to the right, we know that the force of friction is going to act backwards. Because it always acts against you.
So let's do the problem, let's do the math. Now that we have the intuition of what's going to happen. And of course this object will fall or it will accelerate downwards slower than it would have if it wasn’t attached to this whole mechanism. Right. This will be pulling back on a little bit.
So the problem tells us that the coefficient of friction between this mass and the ramp is .2. So we can just go into our mode, because this is a hard one just to figure out what happens with this object because it has friction, it’s on an inclined. So let's see what happens.
So first we know that there's the force of gravity pulling straight down on it. Right, that’s the force of gravity. And then we’ll have, I know I always change words because I actually don’t know the correct. Well, it seems like everyone uses different convention for this, but you have the perpendicular component of gravity, which is that force. And then you have the parallel component of gravity. Let me switch colors. The parallel component of gravity, which is that force. And of course we have to figure out the perpendicular component, which is same in magnitude to the normal force. And that helps us figure out what the force of friction is. And then we use the parallel force, so we just figure out the forces, how much of gravity is pulling parallel to the ramp backwards. So let's do that.
So first of all, what's the weight of the object? Well, its 10 kilograms and the acceleration of gravity are 9.8 meters per second squared. So its 9.8 times 10 is 98 Newtons. And we've learned from previous videos that if this angle is 35 degrees, this angle is 35 degrees as well. So I will reprove it here. So if this is 35 degrees and this is 98, what is this side equal to? This side that is perpendicular to the surface of the ramp. Well, adjacent over hypogenous is equal to cosine. So this side is equal to 98 cosine of 35 degrees. And I do not know what the cosine of 35 degrees is, so I will use my calculator. Cosine of 35 degrees is roughly .82 and then I multiply that to 98. That equals roughly 80. So this equals 80 Newtons. This vector right here.
The part of gravity, the component of gravity, that is pulling perpendicularly down to the surface of this ramp is 80 Newtons. And of course the normal force, that’s the force that the weight is kind of pushing into the ramp, and the normal force is just the equal and opposite force that the ramp is pushing into the weight. That keeps the weight from falling into the ramp. So that’s also going to be 80 Newtons. So the normal force is also 80 Newtons. And the only reason why I point that out is just because I wanted to start referring to the normal force because I didn’t have a variable here.
But anyway. So you could say that the force of friction now, so the force of friction. We already determined is going backwards. Because we determined that the whole system will move to the right. We just figure that out just by looking at the relative magnitude of the masses. So the force of friction is equal to the coefficient of friction, .2, you could either say times the normal force or times the force of gravity going perpendicularly down in to the surface. But either way, that’s going to be 80 Newtons. .2 times 80, that is equal to 16 Newtons. Excellent.
So we figure out that the force of friction pulling back on this object is 16 Newtons. And what's the component of gravity pulling back on this object or this vector going parallel to the surface? Well, we've done this a lot. Now I will call this force parallel, just because I need something. The force parallel, this is the opposite side. This is the hypogenous. Opposite over hypogenous is equal to sine of 35 degrees. So this is equal to, we've done this a lot, I won't go into detail from that. 98 sine of 35 degrees. Go back to the calculator. Because I have not memorize sine of 35 degrees. 35, sine, is about .57. Times 98 is equal to 56. So the force parallel is equal to 56 Newtons.
So there are two forces pulling back on this object. One is 56 Newtons, and that’s just the component of gravity parallel to the surface. And then you have the force of friction pulling back. And of course the force of friction is always acting against you. So the total forces pulling back on this object, if you add these two, 56 plus 16 is 72 Newtons.
So there's 72 Newtons of force essentially trying to pull this whole system back. Right. You can almost view it, there's 72 Newtons of force pulling up on this object to. Right. That’s what's pulling this whole system back. And now what's pulling this whole system to the right? Well it’s the weight of this object. And what is the weight of this object? Well the weight of that object is 20 times 9.8 meters per second squared. So the downward force, the force of gravity, so that’s this force. Let me do it in different color because it’s getting messy. That downward force is 20 times 9.8, that’s 196 Newtons. So you have 196 Newtons pulling this way, that’s also pulling up here. So the net force of this system, think about it this way. You have 196 Newtons that want to go roughly in that direction. Then you have 72 Newtons that wants to roughly in that direction. I know it maybe makes it a little bit confusing with this pulley and all that. But hopefully that makes an intuition of there's two direction it can go to. It can go into the up and down direction or the up and to the left direction. So what is the net of this two? Well, what's 196 minus 72? So the force net trying to go in that direction is 196 minus the force of friction minus this parallel force which is 72. The combined force of the friction and the parallel force is 72, so its 196 minus 72 is 124 Newtons going in this general direction. 124 Newtons.
And so what is the acceleration of the system? Well to know the acceleration of the system, we have to know the mass of the system. And what is the mass of the entire system? Is it just the 20 or is it just the 10? No, it’s both. Because the entire system is moving, the entire thing. This entire contraption, this whole mass and the wire and the other mass, it’s all being accelerated. So the total mass is 30 kilograms. 10 plus 20.
So you have the force, 124 is equal to the mass, 30 times acceleration. So acceleration is equal to 124 over 30 and that is 4.13 meters per second squared.
So this whole system will accelerate up into the right. And then it will accelerate downward at 4.13 meters per second squared. Which also makes an intuitive sense, because essentially this mass has slowed down this objects descent. Right. If this was completely unattached to this wire, it would have fall in 9.8 meters per second squared.
I’ll see you in the next video.

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