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Figure 65 illustrates an $\mathrm{H}_{2} \mathrm{O}$ molecule. The $\mathrm{O}-\mathrm{H}$ bond length is 0.096 $\mathrm{nm}$ and the $\mathrm{H}-\mathrm{O}-\mathrm{H}$ bonds make an angle of $104^{\circ} .$ Calculate the moment of inertia for the $\mathrm{H}_{2} \mathrm{O}$ mole cule about an axis passing through the center of the oxygen atom (a) perpendicular to the plane of the molecule, and (b) in the plane of the molecule, bisecting the H- $\mathrm{O}-\mathrm{H}$ bonds.

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(a) $3.1 \times 10^{-45} \mathrm{kg} \cdot \mathrm{m}^{2}$(b) $1.9 \times 10^{-45} \mathrm{kg} \cdot \mathrm{m}^{2}$

Physics 101 Mechanics

Chapter 10

Rotational Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

University of Washington

University of Sheffield

McMaster University

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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Figure 8-59 illustrates an…

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Calculate the moment of in…

03:27

The H-O-H bond angle for $…

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calculate the moment of in…

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The $\mathrm{N}-\mathrm{N}…

09:32

A model of a benzene molec…

12:05

A diatomic molecule consis…

04:50

So let's draw a diagram of the H 20 molecule we have on oxygen molecule here. It has a bent shape. So this would be the hydrogen molecule and this would be the other hydrogen molecule. We could say that this is an angle defined Fada. Ah, we could say that this distance here would be considered out and this distance here would be considered l sub y And we could say that the mass of a hydrogen atom is gonna be equal to 1.11 point 01 atomic mass units. Or we can say 1.66 times 10 to the negative 27th kilograms since the access passes through the oxygen atom Ah, the auction. Adam itself will have no rotational inertia. Therefore, when calculating for the moment of inertia, do not account for the mass of the oxygen atom because this would be the axis of rotation. So we can say that if the access is perpendicular to the plane of the model than each hydrogen atom is a distance, L away from the axis of rotation therefore ah, the a moment of inertia through the perpendicular access, or we can say we simply say perpendicular access. This would be equal to two times the mass of the hydrogen atom times l squared because there are two hydrogen. Madam's Adams. Therefore, uh, I perpendicular would be equal to two times we said what we actually said 1.1 atomic mass units where one atomic mass units is 1.66 times 10 to the negative 27th kilograms per atomic mass units, so this would be multiplied by 1.1 My apologies. So that would be two times 1.1 times 1.66 times 10 to the negative 27th kilograms multiplied by that length of 0.96 times 10 to the negative ninth meters quantity squared. And we have that the moment of inertia through the perpendicular access would be 3.1 times 10 to the negative 30 negative 45th kilogram. Where'd and this would be your answer for part A. For part B, however, the axis of rotation now is in the plane of the molecule. So if you first part a the axis of rotation was through the oxygen molecule Now this would be the axis of rotation for B uh, this essentially going inside the page, This would be the axis of rotation for part a lie their way. You don't account for the object the mass of the oxygen, madam, because the distance between the auction Adam and the axis of rotation for both cases is going to be zero. So you do not account for that in the moment of inertia and for the for part B. Now, the, uh uh, the axis of rotation is bisecting uh, the bar in the ER is bisecting the the plane of the molecule. Therefore, we can say that l said why would be equal to l sign of state? Uh, and this would be equal to 9.6 times 10 to the negative. 10th multiplied by sine of 52 degrees. This is giving us 7.5664 times 10 to the negative 10th meters. So, essentially, when the plane, when this axis rotation is O is in the plane of the Adam, we have to use Elsa by because now the distance between the eight, the hydrogen atom and the axis of rotation decreases a bit, so we can say that the moment of inertia through the plane of the Adam would be equal to two times the mass of the hydrogen atom times outside y squared. And so this would be equal to two times again 1.1 times, 1.66 times 10 to the negative, 27th kilograms multiplied by 7.564 times, 10 to the negative 10th meters quantity squared and we find that the moment of inertia when the axis of rotation is through the plane is in the plane of the H 20 molecule. This would be equal to 1.9 times 10 to the negative 45th kilogram meters squared. So this would be our answer for part B. That is the end of the solution. Thank you for watching.

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