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Determine the moment of inertia for the shaded area about the $x$ axis.

$I_{x}=0.267 \mathrm{m}^{4}$

Physics 101 Mechanics

Chapter 10

Moments of Inertia

Motion Along a Straight Line

Motion in 2d or 3d

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Cornell University

University of Michigan - Ann Arbor

Hope College

Lectures

04:34

In physics, kinematics is the description of the motion of objects and systems in the frame of reference defined by the observer. An observer has to be specified, otherwise the term is meaningless.

04:12

In physics, potential energy is the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule (J). One joule can be defined as the work required to produce one newton of force, or one newton times one metre. Potential energy is the energy of an object. It is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force which works against the force field of the potential. The potential energy of an object is the energy it possesses due to its position relative to other objects. It is said to be stored in the field. For example, a book lying on a table has a large amount of potential energy (it is said to be at a high potential energy) relative to the ground, which has a much lower potential energy. The book will gain potential energy if it is lifted off the table and held above the ground. The same book has less potential energy when on the ground than it did while on the table. If the book is dropped from a height, it gains kinetic energy, but loses a larger amount of potential energy, as it is now at a lower potential energy than before it was dropped.

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in this question, you have to Dr Mind What is the moment off inertia off this shaded area when it's rotating about the X access. So this area, we rotate about these access. Okay, keep that in mind. Now, in order to event with the moment of finger share, you have to use this equation, and the equation tells us that the moment of inertia is a cost to the integral, off y squared elements off area. So I only have to do in this kind of question. Is that her mind? What is the expression for the element off area? To do that? It's nice to begin by drawing what will be an element off area in the situation. And there are two possibilities. We can draw a horizontal element off area, which will be something like that, or we could draw a vertical element off area, which will be something like that. So, in order to the site, if you are using horizontal element off area or a vertical element off area, you have to take a look here at the expression for the moment of inertia and this expression the variable is why so you would like to have something that has a d y in it. On what? Have a d y in it. This version off the element off area are these version. Well, it happens to be this version, because in these version, you have d white and then you have the X coordinate. So the expression for the element off area is the following d A. Is equals two x d white. And that's it. So let me raise the order element off area, which will be something like why the X and we are not interested in that situation and continue serving the equation from here. Okay, It will be very nice to draw that element off area in the figure so that you can have a better visualization of the situation so that control it. So an element off area will be something like this. Okay. And these will be read, so let's be painted red in order to be more clear about what I'm saying. Okay, So what are the measures off this element off area? As you can see, you have a height here on that height is D y. And you have a weaved about here on that we've as you can guess, is just x So the moment off inertia is given by the integral off Y squared D eight, which happens to be X. The why So now you have two variables. How can you get read off one off those two variables? It happens that you have two choices. You can solve this equation for X and then substitute that expression for X here. Or you can just plugging. Why squared here and then work with an interval over X. You can do whatever you want. You have these two choices how you go. We've serving his equation for acts and then plugging here. So so in that equation for X is nothing outside of these words. All you have to do is the following You send one to the other side. So why squared? Minus one is minus 0.5 times acts and then you send minor zero points five to the other side. So these results in the following X is equal Stroup Why squared Divided by minor 0.5 plus one divided by 0.5 And this is miners Cheuk Why squared plus two Okay, so X is minus true y squared plus two. Let us plugging these expression here inside are integral. So the moment off inertia is then given by the integral off white squared times miners Chu y squared plus two leasing Ken Rosen integral over the Y axis. And then we can plaguing the limits off integration which go from zero o que won because our figure here is 1 m high. Okay, No, All they have to do is solve this integral. Let's do it. The moment off in your share is given by the integral from 0 to 1 off minus two times wiped the fourth. I'm just multiplying by y squared here plus two times y squared Do you wipe on the zis? Minus two times the integral from 0 to 1 off like the fourth B y plus three times the integral from 0 to 1 off. Why squared The white danger grow off white? The fourth is very, very easy. If you remember, it is just 1/5 time. Why? To the fifth Wendy's goes from 0 to 1 and then you have the integral off. Why squared, which follows the same idea. It is white, littered, divided by tree and it goes from 0 to 1. Then the moment of ownership is given by minus True divided by five times one to the fifth, which is one minus zero, which is zero plus cheer, divided by three times. Want to the third, which is one minus zero, which is zero a land. These results in the following minus cheer over five plus chew over tree and then you have to some distractions. To do that, you have to make that the nominee just people so we can multiply the first fraction by tree. Sure, we have minor six divided by 15 on the second fraction by five, so we have 10 divided by 15 and the result off this calculation is four divided by 15. Therefore, your moment off inertia is given by four divided by 15 and the units are meters. The fourth. You know that the units are meters to the fourth because you have units off meters squared coming from his wife squared and then you have another meters squared coming from the element off area. And this is the answer to this question.

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