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The masses of blocks $A, B, C,$ and $D$ are $9 \mathrm{kg}, 9 \mathrm{kg}, 6 \mathrm{kg},$ and $7 \mathrm{kg},$ respectively. Knowing that a downward force of magnitude $120 \mathrm{N}$ is applied to block $D$, determine $(a)$ the acceleration of each block, $(b)$ the tension in cord $A B C$. Neglect the weights of the pulleys and the effect of friction.

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13:23

Khoobchandra Agrawal

Physics 101 Mechanics

Chapter 12

Kinetics of Particles: Newton’s Second Law

Newton's Laws of Motion

Cornell University

University of Michigan - Ann Arbor

Hope College

University of Sheffield

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

09:37

Isaac Newton (4 January 1643 – 31 March 1727) was an English mathematician, physicist, astronomer, theologian, and author (described in his own day as a "natural philosopher") who is widely recognised as one of the most influential scientists of all time and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy"), first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and he shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

07:08

Block $A$ has a mass of $1…

03:36

The two blocks shown are o…

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03:33

05:00

04:45

Assume the three blocks po…

05:35

here we have this little contraption of three police and three hanging masses or blocks. And so we have 12 cables. So this cable comes over these poisoned patches to the center of this Pulliam. And then there's another cable room over that point. So we need to do a little bit of kina Matics analysis here. But that's not too difficult. Um, we know that. Let's see here, we're gonna call it positive down. And so I have, you know, the forces acting on all of our blocks and I'm here. So we'll go through that. So let's say, we know they're told that this block weighs 10 kg in. These each weigh five. We know that this block here drops uh three m. I think I should need that should be plus. Um, yeah, let's see here. Um yeah, that should be a plus because we have because I said positive was down. I think let me look up the problem. I'm sure that it said it was you couldn't raise up. Mm. Yeah it could rise up. But let's take a look and see what the problem actually says. I can find the problem this year. Number 28. Let's see here. Uhh uh 28 moves through meter. Okay. It just says it it moves moves through 63 m in two seconds. So it doesn't say um well we got a low, I forgot about this extra load, this extra load acting on it. So we're assuming that it's probably going to move down. So I'm gonna say it's plus three years. Um so let's see here, we can write Newton's laws for each of these guys. And so for this guy we just have the weight minus the T. One equals the mass times its acceleration. And we know since this cable is doesn't change the length, we know that A. D. Is minus A. So um that's the acceleration at this point. And so then we also know that A. B. With respect to T. And A C. With respect to the those are absence of each other because of this case. Right? But that's just respect to this point here. So that doesn't give us the absolute accelerations of these because this is moving in, this cable is shifting. So when we look at this one here we get W. B. Uh looks like I'm just I didn't write a minus sign in there um W B minus two to plus P equals M. V. Transit. This is the absolute acceleration of feed. And let's see here this doing a analysis at this where this cable, this polio seemed to be massless just gives us this which just tells us that T one is twice too. And then this guy here we get minus T two plus the weight of this block equals the mass times the absolute acceleration of this block. So then we have, let's see here we look at cinematics um we know this distance here basically this year and this year. Um which is, are these things right? That has to be a constant because this cable has it constantly. And so that tells us that a B plus a c minus two, A. D. Is zero. Now, let's see here and then we can take that this and plug that into there. And we get this expression for our cinematics relating the acceleration of the absolute acceleration of each block. Now we know um why, you know from kinda maddox that change in displacement of this guy? Because it starts from rest is one half the accelerates acceleration times T squared. So A B why did uh yeah, so a B winds up being three halves meters per second squared. Okay, because this was three and we have it too and then we square this time, so that's what this winds up being now, we can say A. S. S. A. We can solve this one for a some a this equation and then we can also, let's see here, what did I, where did I get this one to T. Two? Um Oh, oh yeah, I just, I just substituted in um in the Tijuana because to T. Two, right, that's all I did and then distributed the minus side through there and then we know a C. S. F. C. We can get that from from here. Yeah. And then we can basically, so now we can plug, listen to hear this into here. We know a B already. And so now in this equation here, all we don't know is T. Two. So you can solve for T. Two and plug in values and it comes out to be 51 6 news now we can figure out what P. Is, which was another unknown in our in the problem. And we can get that by just using this equation here and now we know all of these values. And that turns out to be just 10 newtons. So putting a 10 newton load for load on that, this could pull it down and then we can figure out what T. One is and that's simply twice T. Two. So that's 103 newtons. So that's the tension in this cable. And I think that's all they asked for. Yeah. And so again just, you know, simple newton's laws and then some kinda Matics and then just some algebra basically we'll get the answer for that one.

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