2012_1061_Lecture_Ch_10

E we ignore friction about a hinge or pin attached to

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Unformatted text preview: r is: •  The work principle holds for rotation of a rigid body about a fixed axis Rotating Rod A rotating rod of mass M and length l can pivot freely (i.e. we ignore friction) about a hinge or pin attached to the case of a large machine as in the figure. The rod is held horizontally and then released. At the moment of release (when you are no longer exerting a force to hold it up), determine (a) the angular acceleration of the rod and (b) the linear acceleration of the tip of the rod. Assume the force of gravity acts as the center of mass of the rod, as shown. The only torque is that of the force of gravity F = Mg which act with a lever arm l/2 at the moment of release. There is also a force at the hinge but its moment is zero since the hinge is the axis of rotation. Rotational Plus Translational Motion; Rolling • Rolling without slipping •  (a) A wheel rolling to the right. Its center C moves with velocity . Point P is at rest at this instant. • (b) The same wheel as seen from a reference frame in which the axle C is at rest- that is, we are moving to the right with velocity relative to the ground. Point P, which was at rest in (a) , here in (b) is moving with velocity Instantaneous Axis A rotating wheel rotates about the instantaneous axis (perpendicular to the page) passing through the point of contact with the ground, P. The arrows represent the instantaneous velocity of each point (b) Photograph of a rolling wheel. The spokes are more blurred when the speed is greater Total Kinetic Energy = KCM+KROT 1 2 KTranslation = M VCM 2 KRotation/CM 1 = ICM ω 2 2 With respect to point P all is rotation KRotation/P KRotation/P KT OT 1 = IP ω 2 2 1 1 2 = ICM ω + M R2 ω 2 2 2 1 1 2 2 = ICM ω + M VCM 2 2 Sphere rolling down an Incline Evaluate the speed of the center of mass of a solid sphere of Mass M and radius r0 when it reaches the bottom if it starts from rest at a vertical height H and rolls without slipping. Compare you result with an object sliding down a frictionless incline The total mechanical energy at any point along the incline is Figure 10.34 Figure 10.35 Analysis of a sphere on an incline using forces...
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