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(II) A person stands on a platform, initially at rest, that can rotate frecly without friction. The moment of inertia of the person plus the platform is $I_{\mathrm{P}}$ . The person holds a spinning bicycle whecl with its axis horizontal. The whecl has moment of inertia $$I_{\mathrm{w}} \text { and angular velocity } \omega_{\mathrm{W}} $$ What will be the angular velocity $$ \omega_{\mathbf{Y}} $$ of the platform if bthe person moves the axis of the wheel so that it points (a) vertically upward, (b) at a $60^{\circ}$ angle to the vertical, $(c)$ vertically downward? (d) What will $\omega_{\mathrm{P}}$ be if the person reaches up and stops the whecl in part $(a) ?$

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a) $\omega_{p}=-\frac{I_{w}}{I_{p}} \omega_{w}$b) $\omega_{p}=-\frac{I_{\mathrm{w}}}{2 I_{\mathrm{p}}} \omega_{\mathrm{w}}$c) $\omega_{p}=\frac{I_{w}}{I_{p}} \omega_{w}$d) $\omega_{p}=0$

Physics 101 Mechanics

Chapter 11

Angular Momentum; General Rotation

Moment, Impulse, and Collisions

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Cornell University

Simon Fraser University

University of Sheffield

University of Winnipeg

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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.

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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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So we're going to four part. I apply the conservation of angular momentum. We know that. But some initial equals at moments and final, and so we can say that essentially zero there's gonna be equaling. The zero would be the initial angular momentum, and this would be equaling the moment of inertia for the wheel multiplied by the angular velocity, uh, of the wheel, plus the moment of inertia for the platform times the angular velocity of the platform. And so, essentially, we can say that the angular velocity of the platform is going to be equaling the negative, angular velocity of the wheel multiplied by the moment of inertia for the wheel, divided by the moment of inertia. For the platform, this would be our answer to part Hey, now to Part B. If the wheel is pointed at a 60 degree angle to the vertical than the component of its angular momentum, that is along the vertical direction would be, um, we could say that a moment of inertia for the wheel along the vertical direction would be the the angular momentum for the wheel. Along the vertical direction would be the moment of inertia for the wheel times the angular velocity, velocity of the wheel, Times Co. Sign of 60 degrees. And so we can then say that, uh, the initial angular momentum equals the final angular momentum. And we can then say that zero equals I, uh, the moment of inertia, of the wheel, the angular velocity of the real Times co sign of 60 degrees, plus the moment of inertia, the platform, time singing times, the angular velocity for the platform. And so again, we can say that the angular velocity Philip platform would be equaling the negative, angular velocity of the wheel, a moment supplied by the moment of inertia of the wheel divided by two times the moment of inertia of the platform. This would be your answer for Part B again. Ah, here. The negative sign means that the platform is going to be rotating in the opposite direction of the wheel. Of course, that applies to part as well, and then for part C. If the wheel is moved, that's so that its angular momentum points downwards than the person, and the platform must get an equal but opposite angular momentum, which will point upwards. So applying the angular, the conservation of angular momentum. We're getting zero, equaling the negative moment of inertia of the wheel, times the angular velocity of the wheel. Plus that moment of inertia for the platform times the angular velocity of the platform. And here the moments of the angular velocity rather of the platform would be equaling the angular velocity of the wheel multiplied by the moment of inertia for the wheel divided by the moment of inertia for the platform. And as you can see here, the wheel and the platform will be rotating in the same direction. So this would be our answer for part C and then four part D. We can say that the scent, because the total angular momentum is gonna be equaling zero. Um, if the wheel has stopped rotating, this means that the platform will also stop. So we can simply say that the angular velocity of the platform is gonna also equal zero radiance for a second. This would be your answer. Four Part D. That is the end of the solution. Thank you for watching

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