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rotation, torque, and angular acceleration

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Please see attachment. Thank You your help is greatly needed.

12. A top is spinning counterclockwise as shown in the figure. It is also moving to the right with a linear speed v. What is the direction of the angular velocity?

13. A wrench is used to tighten a nut as shown in the figure. A 12-N force is applied 7.0 cm from the axis of rotation. What is the torque due to the applied force.
14. A string is tied to a doorknob 0.79 m from the hinge as shown in the figure. At the instant shown, the force applied to the string is 5.0 N. What is the torque on the door?
15. A uniform 13-kg trap door is oriented horizontally and hinged as shown. What is the magnitude of the torque on the door at the instant that the release is activated and the door can freely rotate?

16. In the drawing shown, the large wheel has a radius of 8.5 m. A rope is wrapped around the edge of the wheel and a 7.6-kg box hangs from the rope. A smaller disk of radius 1.9 m is attached to the wheel. A rope is wrapped around the edge as shown. An axis of rotation passes through the center of the wheel-disk system. What is the value of the mass M that will prevent the wheel from rotating?

17. A massless frame in the shape of a square with 2-m sides has a 1-kg ball at each corner. What is the moment of inertia of the four balls about an axis through the corner marked O and perpendicular to the plane of the paper?

18. A 50 N. m torque acts on a wheel of moment of inertia 150 kg.m2. If the wheel starts from rest, how long will it take the wheel to make on revolution?
19. A string is wrapped around a pulley of radius 0.05 m and moment of inertia 0.2 kg.m2. If the string is pulled with a force F, the resulting angular acceleration of the pulley is 2 rad/s2. Determine the magnitude of the force F.
20. A string is wrapped around a pulley of radius 0.10 m and moment of inertia 0.15 kg.m2. The string is pulled with a force of 12 N. What is the magnitude of the resulting angular acceleration of the pully?

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Rotation, torque, and angular acceleration are investigated. The solution is detailed and was rated '5/5' by the student who originally posted the questions.

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Please see the attached file.

12. The angular velocity

Using the right-hand rule, we can find that the angular velocity is upward.

Answer: b

13. , where θ = 90° is the angular between the force and the moment arm.
So the torque is 12 * ...

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