Chapter_11 - Chapter 11 Rolling Torque and Angular Momentum...

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Chapter 11 Rolling, Torque, and Angular Momentum In this chapter we will cover the following topics: - Rolling of circular objects -Redefinition of torque to describe rotational problems that are more complicated than the rotation of a rigid body about a fixed axis - Angular momentum of single particle and a system of particles - Conservation of angular momentum --Applications of the conservation of angular momentum (11-1)
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t 1 = 0 t 2 = T Consider a circular object that rolls along a surface without slipping. We can simplify its study by treating it as a combination of translation of COM and Rolling as Translation and Rotation Combined rotation about COM Consider two snapshots of a rolling bicycle wheel. An observer on the ground sees the COM (point O) of the wheel move forward with a speed . During the time interval between the two snapshots p com v T oint O covers a distance (eqs.1) During time interval T, the bicycle rider sees the wheel rotate by an angle about O, so (eqs.2). If we equation 1 with equation 2, we g = = = com s v T s R TR combine θ θ ϖ et the condition for rolling without slipping. com v R ϖ = (11-2)
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Rolling is a combination of a pure translation with speed and a pure rotation about COM with angular velocity . The velocity of each point is the vector sum of the velocities of the two m = com com v v R ϖ otions. For the translational motion, the velocity is the same for every point ( ). The rotational velocity varies from point to point. Its magnitude is equal to where is the distance from com v r r ϖ r P T O. Its direction is tangent to the circular orbit. The net velocity is the vector sum of these two terms. For example, v = 0 v = 0 = 2 . - = + = + = com com O com com com com com v v v v v v v v r r r r r r r r r r com v R ϖ = (11-3)
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A B v A v B v T v O We consider rolling as a pure rotation about an axis that passes through the contact point P between the wheel and the road. The angular velocity of the rot Alternative view : Rolling as Pure Rotation ation is The rotation axis at P changes with time = Note : com v R ϖ Let's find the velocity vector v for each point on the wheel. The direction of v points along the tangent to the circular orbit. For example, at point A, is perpendicular to the line AP. The A v r r r magnitude of v is , where is the distance between a given point and the contact point P. For example, At point T 2 2 2 . At point O . At point P 0 = = = = = = = = T com O com P v r r r R v R v r R v R v r v ϖ ϖ ϖ r 0 = (11-4)
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For a rolling object (mass and radius ) we calculate its kinetic energy by considering rolling as a pure rotation about the contact point P. The Kinetic Energy of Rolling M R ( 29 2 2 2 2 2 2 1 The kinetic energy is . Here is the rotational inertia of the rolling 2 body about point P.
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