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(II) When discussing moments of inertia, especially for unusual or irregularly shaped objects, it is sometimes convenient to work with the radius of gyration, $k$ . This radius is defined so that if all the mass of the object were concentrated at this distance from the axis, the moment of inertia would be the same as that of the original object. Thus, the moment of inertia of any object can be written in terms of its mass $M$ and the radius of gyration as $I=M k^{2} .$ Determine the radius of gyration for each of the objects (hoop, cylinder, sphere, etc. $.$ shown in Fig. $20 .$

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a. $R_{0}$b. $\sqrt{\frac{1}{2} R_{0}^{2}+\frac{1}{12} w^{2}}$c. $\sqrt{\frac{1}{2}} R_{0}$d. $\sqrt{\frac{1}{2}\left(R_{1}^{2}+R_{2}^{2}\right)}$e. $\sqrt{\frac{2}{5}} r_{0}$f. $\sqrt{\frac{1}{12}} \ell$g. $\sqrt{\frac{1}{3}} \ell$h. $\sqrt{\frac{1}{12}\left(\ell^{2}+w^{2}\right)}$

Physics 101 Mechanics

Chapter 10

Rotational Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

University of Michigan - Ann Arbor

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Lectures

02:21

In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

02:34

In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

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party. We have a thin hoop with the radius are. And we can say that the moment of inertia, I would be equaling m k squared. And this is equaling m r squared. And so here the radius of gyration with Kay would be eagling are the radius. Here you have the moment of inertia equaling again m k squared. This would be equaling to 1/2 um, r squared plus 1/12. Um, W squared. And so here we can say that Kay, the radius of gyration would be equaling the square root of one over two r squared plus one over 12. Ah, with squared. And so this would this would be the radius of gyration with a thin hoop everythin hoop with a radius and a with. Now we have a solid cylinder where here, the moment of inertia is equal in again M k squared. And then specifically for a solid cylinder, we have 1/2 mm r squared. So here the radius of gyration is equaling. Are the radius multiplied by the square root of 1/2 for a hollow cylinder? We have I equaling m k squared, and this would be equaling 1/2 times and sometimes the smaller radius plus squared plus the larger radius squared. And so here, the radius of gyration kay would be equaling the square root of 1/2 times our sub one squared. Let's are set to square four part E a uniform sphere. We have the moment of inertia equaling again M k squared and then specifically for a uniform sphere, we have 2/5 um, r squared. And so the radius of gyration for ah uniforms speak uniforms. Sphere would be the radius times the square root of two over five for a long rod through the center, eyes equaling again at K squared, This would be like one over 12 times. AM l squared where the radius of gyration K is now equaling oh, multiplied by one over 12 and then finally four part G and H Gene, we have a long rod through the end. This would be I equaling, encased, squared and this would be equaling 1/3 m l squared. And so the radius of gyration K is equaling l times the square root of one over three. And now we have new rectangular thin plate. This would be the moment of inertia equaling M K squared. Uh, this would be equaling 1/12 Um, times l squared, plus the length squared plus the with squared and so Okay, The radius of gyration would be equal to the square root of 1 12 l squared plus w squared. That is the end of the solution. Thank you. For what?

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