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Show that the system of hydrostatic forces acting on a submergedplane area $A$ can be reduced to a force $P$ at the centroid $C$ of the area and two couples. The force $P$ is perpendicular to the area and has a magnitude of $P=\gamma A \bar{y} \sin \theta,$ where $\gamma$ is the specific weight of the liquid. The couples are $M_{x^{\prime}}=\left(\gamma \bar{I}_{x}^{\prime} \sin \theta\right)$ i and $M_{y}=\left(\gamma \bar{I}_{x^{\prime} y} \sin \theta\right)$,where $\bar{I}_{x y^{\prime}}=\int x^{\prime} y^{\prime} d A$ (see Sec. 9.3). Note the couples are independent of the depth at which the area is submerged.

Physics 101 Mechanics

Chapter 9

Distributed Forces: Moments of Inertia

Section 2

Parallel-Axis Theorem and Composite Areas

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

Hope College

University of Winnipeg

McMaster University

Lectures

04:30

In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. In the case of a constant force, the resulting change in momentum is equal to the force itself, and the impulse is the change in momentum divided by the time during which the force acts. Impulse applied to an object produces an equivalent force to that of the object's mass multiplied by its velocity. In an inertial reference frame, an object that has no net force on it will continue at a constant velocity forever. In classical mechanics, the change in an object's motion, due to a force applied, is called its acceleration. The SI unit of measure for impulse is the newton second.

03:30

In physics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Given a force, F, applied for a time, t, the resulting change in momentum, p, is equal to the impulse, I. Impulse applied to a mass, m, is also equal to the change in the object's kinetic energy, T, as a result of the force acting on it.

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who asked to show that the system of hydrostatic forces acting on a submerged playing area A Can we do? Can we reduce the air Force at the central oId of the area and two couples? So the force P is particular to the area and has a magnitude of that's given there and the couples are given. And we wanna see how those equations were derived or came about. So we have our, um, water level here and then this is the Y axis that was given. They're playing on which the pressure acts. So I guess if this was a you know where the water was holding the water up, I have to find another axis here. Why prime? Just so that it's vertical. And then we have the centrally to some distance from here. Now it's two D So X in our cases, out of the is out of the page here and then. So here's our surface, just a cross section of it, and we can see that we have a linearly increasing pressure acting over that surface. So generally, why prime equals. Why sign of theater, where state is this angle that, um that the bottom makes with the water level. And then we know that the pressure is the, um what specific weight or the density times, gravity times. Um, why prime? And that we substitute in. And so we get that the pressure is the specific gravity times Why sign of data? So what we need to do now is, um, calculate the total force, so we need to integrate over the area. OK, so obviously this is to d. So So we integrate over the area of the pressure, okay, And that will give us the total of reaction result in force. And a lot of these things don't depend on the area. So the specific gravity is independent of various sign of theater isn't imprint of the very best. So we can pull those out and we get an inter grow of why over the area and that, by definition, is the central eight times the area. So we get the expression that, um is given in the book that the four total force is there a specific gravity times, um, the distance along this axis to the century times a time sign of data. And so obviously this, um, value here is the depth okay of the century. And now they're asking about the moments. So what we need to do to find the moments it is, Integrate the pressure times a given moment, arm over the area. So for the moment about the X axis, which is out of the board, we have the century here at the moment about the sense. Right. So we need the moment arms, Um, in the in this direction here. And so those are why minus y bar eso Wherever we are here, we're at a certain why, and we subtract white bar off of it. So we give specific gravity times, Um, the depth times a quietly why minus y bar and we integrate that over the total area, you can expand that out and we see that we get pull out this factor, which is a constant at that. This factor, which is a constant, doesn't depend on the area. And so we get an integral over the area. Ah, why squared and then minus Why bar, which is independent of the area, um, or where we are in the area and times integral of why over that area now we can recognize thes things. This is the area moment about the X axis. So the area moment about this point here, but this axis coming out of the board other, the paper and then minus um, this is a times why bar, So we get minus a wide bar squared, and that is the area moment about the central. It'll X axis. So again, we can use the parallel access their, um And this provides us with the area moment about the access out of the paper at the center right here. Now we have they ask us for the moment in the other direction about the why Prime access. Okay, so we have, um, moments coming about this access so kind of out of the page twisting, I guess if you wanted to do a double a roll thing, moments like that. Um, so those are those we need to figure out where the the X position is of the pressure. That's acting. Minus why bar? Hey, so we need we need, um to integrate this. The difference here. This is the moment arm that's generate creating, causing this moment. And then we didn't have to integrate that multiply that by the pressure and then integrate over the whole area so we can proceed like we did before. And we pull this stuff out, which is not dependent on the area. Why? Bar is also not dependent on well, at least where we are in the area. Obviously, it depends on what the area actually is, but it is a constant, um, and so we get a ninja grow over the area of X times y de a minus y bar times an integral over the area of ex D A. Now again, we can recognize some things here. This is the air is the cross of the area Moment. The cross area moment about the X Y axis. Okay, so and then this is minus. Why bar times in this by definition, this is a X bar. And so if we look ahead in the chapter, we can see the parallel actresses here. Um, extended to these cross moments is defined by this. And so this winds up being the area moment. I hear this runs at being the cross area moment about the ex prime. Why prime access or, um, basically, the axis. Well, this is still I shouldn't call it. Why prime? I suppose, because I've called. But anyway, this axis from the central and then this access out of the board amount of the pick paid for the X axis. So let's actually call this, um, you should actually call this. Why? Double price To make it distinguished from the axis that I defined vertically.

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