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Video transcript

let's now tackle Part C so they tell us block three of mass M sub three so that's right over here is added to the system is shown below there is no friction between block three and the table all right indicate whether the magnitude of the acceleration of block two is now larger smaller or the same as in the original two block system explain how you arrived at your answer so let's just think about the intuition here if you think about the net forces on the system itself they're the same as we had before now they're the internal forces are going to be different you actually we're going to have two different tensions now now that you have two different strings but the net force is the ones that are causing this thing to accelerate to in in the upward direction on the left hand side to the right on the top and then downwards on the right hand side it's still the the difference in the weight between the two blocks but now that difference in those in between the weights of the two blocks is moving more mass and we know that force is equal to mass times acceleration or acceleration is equal to force divided by mass and our force our net force is being is the differential between the weights or the difference between the weights of the block but now we're going to be moving more aggregate mass this is going to be M 1 plus M 2 plus M 3 and so you're going to have a smaller acceleration and so what you could write is is acceleration acceleration smaller because same difference difference in weights in weights between M 1 and M 2 is now accelerating more mass accelerating more mass and that's the intuitive explanation for it and if you wanted to dig a little bit deeper you could actually set up Freebody diagrams for all of these blocks over here and you would come to that same conclusion so let's just do that just just to feel good about ourselves so what are on mass 1 what are going to be the forces well you're going to have the force of gravity which is m1g then you're going to have the upward tension pulling upwards it's going to be larger than the force of gravity with in a different color so you're going to have whoops let me do it alright so you're going to have this tension let's call that t1 you're not going to have two different tensions here because you have two different strings now the tension there is t1 the tension over here is also going to be t1 so let me do the same magnitude t1 now since block two is a larger weight than block one because it has a larger mass we know that the whole system is going to accelerate is going to accelerate in on the right-hand side is going to accelerate down on the left-hand side it's going to accelerate up and on top is going to accelerate to the right and so at the top is accelerating to the right then the tension in this second string is going to be larger than the tension in the first string so we did that in another color so I'm having trouble drawing straight lines all right so that we could call t2 and if that is t2 then the tension through so then this is going to be t2 as well because the tension through the the magnitude of the tension through the entire string is going to be the same and then finally we have the weight of the block we have the weight of block 2 there's going to be larger than this tension so that is m2 G now I've just drawn all the forces that are relevant to the mag to the acceleration if I wanted to make a complete I guess you could say Freebody diagram where I'm focusing on M 1 M 3 and M 2 there are some more forces acting on M 3 M 3 in the vertical direction you have its weight which you could call m3g but it's not accelerating downwards because the table is exerting force it sit on it on an upwards exerting and upwards force on it so of the same magnitude offsetting its weight so that's if you wanted to do a more complete free body diagram for it but we care we care about the things that are moving in the direction of the acceleration depending on where we are on the table and so we can just use Newton's second law like we've used before saying the net force is in a given direction are equal to the mass times the magnitude of the acceleration in that given direction so the magnitude of that force is equal to mass times the magnitude of the acceleration and so we can do that first with block one so with block one actually let me just do this with specific so block one I'll do it with this orange color so block one what's the net forces well it is t1 minus m1g that's going to be equal to mass times acceleration so it's going to be m1 times the acceleration now what about block three well block three we're accelerating to the right we're going to have t2 we did it in a different color block three we are going to have T two minus T one minus T one is equal to M is equal to M three and the magnitude of the acceleration is going to be the same here we're accelerating to the right here accelerating up here we're slowing down but the magnitudes are going to be the same there all I can I can denote them with this lowercase a and then finally we could think about block three we could say that the net force is well that's m2 G minus t2 that's going against m2 G is equal to M two times its acceleration and now if we want to solve for acceleration this will be quite convenient we can just add up all of the left-hand sides to get a new left-hand side and add up all the right-hand sides to get a new right-hand side we can do that algebraically because they're all this is equal to that that is equal to that that is equal to that so if you add up these and then you add up those well then the the sums are going to be equal to each other and so what are you going to get so if you add up all of this this t1 is going to cancel out with subtracting the t1 this 2 2 is going to cancel out with the subtracting the t2 and you're just going to be left with an m2 G m2 G minus m1 G minus m1g m2g minus m1 G is equal to and just four well let me just write it out is equal to m1 a plus m3 a plus m2 a but we could of course factor the a out and so let me just write this as that's equal to a times m1 plus m2 plus m3 and then we could divide both sides by m1 plus m2 plus m3 so let's just do that so M 1 plus M 2 plus M 3 M 1 plus M 2 plus M 3 these cancel out and so this is your the magnitude of your acceleration and notice you have the same difference in weights that's providing the net force on the system but is now accelerating more mass so you can even say hey look I have more mass here so a more mass to accelerate more mass more mass to accelerate while I have the same net force acting on the system we're not talk about the internal forces those all canceled out when I when I when I added these equations and so if you're if you're taking the same net force and you're dividing it by more mass you're going to have a smaller smaller acceleration hopefully that all made sense to you
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