Translation and Rotation 8.01 W11D2Next Reading Assignment: W11D3 Quiz 8: Rotational Dynamics 2Worked Example: Cylinder on Inclined Plane, Torque Method A hollow cylinder of outer radius R and mass m with moment of inertia I cm about the center of mass starts from rest and moves down an incline tilted at an angle θ from the horizontal. The center of mass of the cylinder has dropped a vertical distance h when it reaches the bottom of the incline. Let g denote the gravitational constant. The coefficient of static friction between the cylinder and the surface is µs. The cylinder rolls without slipping down the incline. Using torque about a fixed point lying along the line of contact between the cylinder and the surface, calculate the acceleration of the center of mass of the cylinder when it reaches the bottom of the incline.Rules to Live By: Angular Momentum and Torque 1) About any fixed point S 2) Independent of the CM motion, even if and are not parallel ,S about cm of cm about cm s cm total cmr m v= + = + ×L L L L τS=τS ,iext=dLSdti∑about cmabout cmddtτ=Labout cmLωTable Problem: Cylinder on Inclined Plane, Torque About Center of Mass A hollow cylinder of outer radius R and mass m with moment of inertia I cm about the center of mass starts from rest and moves down an incline tilted at an angle θ from the horizontal. The center of mass of the cylinder has dropped a vertical distance h when it reaches the bottom of the incline. Let g denote the gravitational constant. The coefficient of static friction between the cylinder and the surface is µs. The cylinder rolls without slipping down the incline. Using the torque method about the center of mass, calculate the velocity of the center of mass of the cylinder when it reaches the bottom of the incline.Rules to Live By: Kinetic Energy of Rotation and Translation Change in kinetic energy of rotation about center-of-mass Change in rotational and translational kinetic energy ΔK = ΔKtrans+ ΔKrot ΔKrot≡ Krot, f− Krot,i=12Icmωcm, f2−12Icmωcm,i2 ΔK = ΔKtrans+ ΔKrot=12mvcm, f2−12mvcm,i2⎛⎝⎜⎞⎠⎟+12Icmωcm, f2−12Icmωcm,i2⎛⎝⎜⎞⎠⎟Table Problem: Cylinder on Inclined Plane, Energy Method A hollow cylinder of outer radius R and mass m with moment of inertia I cm about the center of mass starts from rest and moves down an incline tilted at an angle θ from the horizontal. The center of mass of the cylinder has dropped a vertical distance h when it reaches the bottom of the incline. Let g denote the gravitational constant. The coefficient of static friction between the cylinder and the surface is µs. The cylinder rolls without slipping down the incline. Using energy techniques calculate the velocity of the center of mass of the cylinder when it reaches the bottom of the incline.Demo B 113: Rolling CylindersDemo B117: Stationary Cylinder A cylinder weighted off-center seems to defy gravity by remaining stationary on an inclined plane.Table Problem: Descending and Ascending Yo-Yo A Yo-Yo of mass m has an axle of radius b and a spool of radius R. It’s moment of inertia about the center of mass can be taken to be I = (1/2)mR2 and the thickness of the string can be neglected. The Yo-Yo is released from rest. What is the acceleration of the Yo-Yo as it descends. 10Demo B107: Descending and Ascending Yo-Yo wheel+axle435 gM =outer6.3 cmR ≅inner4.9 cmR ≅( )2 2cm outer inner4 2121.385 10 g cmI M R R≅ += × ⋅Concept Question: Rotation and TranslationConcept Question: Rotation and TranslationMini-Experiment: Pulling Spool 1. Which way does the yo-yo roll when you pull it horizontal? 2. Is there some angle at which you can pull the string in which the yo-yo doesn’t roll forward or
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