lecture-ch11 - Contents 11 Rotational Motion—II 1 11.1...

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Unformatted text preview: Contents 11 Rotational Motion—II 1 11.1 Rolling as Translation and Rotation Combined . . . . . . . . . 1 11.2 The Kinetic Energy of Rolling . . . . . . . . . . . . . . . . . . 2 11.3 Torque by Uniform gravitational Force . . . . . . . . . . . . . 3 11.4 The Force of Rolling . . . . . . . . . . . . . . . . . . . . . . . 4 11.4.1 Rolling Down a Ramp . . . . . . . . . . . . . . . . . . 4 11.4.2 Rolling without Slipping . . . . . . . . . . . . . . . . . 5 11.4.3 Rolling with Slipping . . . . . . . . . . . . . . . . . . . 6 11.5 The Yo-Yo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 11.6 Torque and Angular Momentum . . . . . . . . . . . . . . . . . 8 11.7 Angular Momentum for a Rigid Body about a Fixed Axis . . . 9 11.8 Conservation of Angular Momentum . . . . . . . . . . . . . . 11 11.9 Precession of Gyroscope . . . . . . . . . . . . . . . . . . . . . 14 11 Rotational Motion—II 11.1 Rolling as Translation and Rotation Combined vector R θ s = θR vectorx vectorx − vector R vector R Let vectorv ′ be the velocity for a point vectorx on the rotating body in the center of mass frame. The velocity observed in the laboratory system is vectorv = vectorv cm + vectorv ′ = vectorv cm + vectorω × vectorx 1 Let ˆ n be the unit vector pointing in the direction of the rotation axis. The angular velocity is related to the angle of rotation θ by vectorω = ω ˆ n = dθ dt ˆ n. The center of mass is moving in the direction of vector R R × ˆ n with the magnitude | vectorv cm | = vextendsingle vextendsingle vextendsingle vextendsingle R dθ dt vextendsingle vextendsingle vextendsingle vextendsingle = | Rω | Thus vectorv cm = Rω parenleftBigg vector R R × ˆ n parenrightBigg = vector R × vectorω and vectorv = vectorv cm + vectorω × vectorx = vectorω × parenleftBig vectorx − vector R parenrightBig In particular, vectorx = vector R at the bottom of the wheel, vectorv = 0. Furthermore, parenleftBig vectorx − vector R parenrightBig is the displacement relative to the bottom of the wheel. The rolling motion can be also treated as a rotation around the axis passing at the bottom of the wheel. 11.2 The Kinetic Energy of Rolling First let us prove an identity that relates the total kinetic energy observed in the laboratory system to that in the center of mass frame. In the laboratory system, K = 1 2 summationdisplay i m i vectorv i · vectorv i = 1 2 summationdisplay i m i ( vectorv cm + vectorv ′ i ) · ( vectorv cm + vectorv ′ i ) = 1 2 summationdisplay i m i vectorv cm · vectorv cm + 1 2 summationdisplay i m i vectorv ′ i · vectorv ′ i + parenleftBigg summationdisplay i m i vectorv ′ i parenrightBigg · vectorv cm = 1 2 M | vectorv cm | 2 + K ′ + Mvectorv ′ cm · vectorv cm where M = ∑ i m i is the total mass of the system, K ′ is the total kinetic energy observed in the center of mass frame, and vectorv ′ cm = È i m i vectorv ′ i M the velocity of the center of mass observed in the center of mass frame. By definition,of the center of mass observed in the center of mass frame....
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This note was uploaded on 05/14/2011 for the course ECON 101 taught by Professor Asdaf during the Spring '11 term at Universidad de San Buenaventura Bogota.

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lecture-ch11 - Contents 11 Rotational Motion—II 1 11.1...

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