# ch11 - Chapter 11 Rolling Torque and Angular Momentum In...

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Chapter 11 Rolling, Torque, and Angular Momentum In this chapter we will cover the following topics: -Rolling of circular objects and its relationship with friction -Redefinition of torque as a vector to describe rotational problems that are more complicated than the rotation of a rigid body about a fixed axis -Angular Momentum of single particles and systems or particles -Newton’s second law for rotational motion -Conservation of angular Momentum -Applications of the conservation of angular momentum (11-1)

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t 1 = 0 t 2 = t Consider an object with circular cross section that rolls along a surface without slipping. This motion, though c Rolling as Translation and R ommon, is complicated. We otation Combin can simplif ed y its study by treating it as a combination of translation of the center of mass and rotation of the object about the center of mass Consider the two snapshots of a rolling bicycle wheel shown in the figure. An observer stationary with the ground will see the center of mass O of the wheel move forward with a speed . The point com v P at which the wheel makes contact with the road also moves with the same speed. During the time interval between the two snapshots both O and P cover a distance . (eqs.1) During t com t ds s v t dt = he bicycle rider sees the wheel rotate by an angle about O so that = (eqs.2) If we cambine equation 1 with equation 2 we get the condition for rolling without slipping. ds d s R R dt dt θ ϖ = = com v R = (11-2)
We have seen that rolling is a combination of purely translational motion with speed and a purely rotaional motion about the center of mass with angular velocity . The velocity of each p com com v v R ϖ= oint is the vector sum of the velocities of the two motions. For the translational motion the velocity vector is the same for every point ( ,see fig.b ). The rotational velocity varies from poi com v r nt to point. Its magnitude is equal to where is the distance of the point from O. Its direction is tangent to the circular orbit (see fig.a). The net velocity is the vector sum of these two ter r r ϖ ms. For example the velocity of point P is always zero. The velocity of the center of mass O is ( 0). Finally the velocity of the top point T is wqual to 2 . com com v r v = r r com v R = (11-3)

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A B v A v B v T v O Another way of looking at rolling is shown in the figure We consider rolling as a pure rotation about an axis of rotation that passes through the contact Rollin point g as Pure Rotatio P between th n e wh eel and the road. The angular velocity of the rotation is com v R ϖ= In order to define the velocity vector for each point we must know its magnitude as well as its direction. The direction for each point on the wheel points along the tangent to its circular orbit. For example at point A the velocity vector is perpendicular to the dotted line that connects pont A with point B. The speed of each point is given by: . Here is the distance between a parti
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ch11 - Chapter 11 Rolling Torque and Angular Momentum In...

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