lecture20

lecture20 - Translational Kinetic Energy: KETran = 1 mv 2 2...

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Translational Kinetic Energy: 2 2 1 KE mv Tran = Rotational Kinetic Energy: 2 2 1 KE ω I Rot = 2 2 1 KE I Rot = So, for an object with both translational motion and rotational motion, the total KE is given by: Rot Tran tot KE KE KE + = 2 2 1 2 2 1 I mv + = In principle now, we have another term which can contribute to the total mechanical energy of an object: Grav Rot Tran tot PE KE KE E + + = mgh I mv + + = 2 2 1 2 2 1 As before, if W NC = 0, then E tot is a constant of the motion.
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Rolling Motion Release a solid sphere from rest at the top of an incline and let it roll without slipping to the bottom. h The total mechanical energy at the top of the ramp must be equal to the total mechanical energy at the bottom of the ramp: bot top E E = E E E E E E top E bot E bot Rot Trans top Rot Trans PE KE KE PE KE KE bot bot top top + + = + + bot bot Rot Trans top KE KE PE + = 2 2 1 2 2 1 bot bot I mv mgh ω + = Notice, for the sphere which is released from rest and rolls without slipping, the initial gravitational PE goes into both the translational KE and the rotational KE at the bottom of the ramp.
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Example : Last time we showed that a solid disk of radius r and mass m will roll faster down an incline than a ring of radius r and mass m, since the moment of inertia of the disk is smaller than that of the ring. Use conservation of energy to calculate the translational speed of the ring and disk at the bottom of the ramp and show that it is faster for the disk. h Since gravity is the only conservative force that does work on the objects, by conservation of energy, the total mechanical energy at the top of the ramp for each object must equal the total mechanical energy at the bottom of the ramp. top E ot p E E = bot E bot top bot Rot Trans top Rot Trans PE KE KE PE KE KE bot bot top top + + = + + bot bot Rot Trans top KE KE PE + = 2 2 1 2 2 1 bot bot I mv mgh ω + = r v bot bot = 2 2 1 2 2 1 + = r v I mv mgh bot bot Plug this in above: Solve this for v bot :
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2 2 bot
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lecture20 - Translational Kinetic Energy: KETran = 1 mv 2 2...

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