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HW #8 due tonight, 11:59 p.m. HW #9 available, due Tuesday Nov 16, 11:59 p.m.

HW #8 due tonight, 11:59 p.m. HW #9 available, due Tuesday Nov 16, 11:59 p.m. Last Time : Moment of Inertia, Torque and Angular Acceleration Today : Rotational Kinetic Energy + the “Race of the Shapes ” !!. Define: Translational & Rotational KE.

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HW #8 due tonight, 11:59 p.m. HW #9 available, due Tuesday Nov 16, 11:59 p.m.

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  1. HW #8 due tonight, 11:59 p.m. • HW #9 available, due Tuesday Nov 16, 11:59 p.m. • Last Time: Moment of Inertia, Torque and Angular Acceleration • Today: Rotational Kinetic Energy + the “Race of the Shapes” !!

  2. Define: Translational & Rotational KE Recall, we defined the kinetic energy of an object of mass m moving with a speed v to be : m v We will call this the translational kinetic energy. An object that is rotating about some axis with an angular speed ω has rotational kinetic energy I : moment of inertia [kg-m2] ω : angular speed [rad/s] KEr : [kg-m2/s2 ] = [Joule]

  3. ω Rotational KE r2 Where does this formula come from ? r1 r3 Think of this wheel as being made up of many small masses m1, m2, m3, … The (tangential) velocity of each of these masses is: v = rω. So the KE due to their rotational motion is : Remember: ω is the same everywhere on the wheel !! moment of inertia

  4. Now: Three Types of Energy ! Consider this ball rolling down an incline. ω v (speed of center of gravity) h There are three different types of energy that are relevant here : • Gravitational potential energy: PE = mgh • Translational kinetic energy: KE = ½ mv2 • Rotational kinetic energy: KEr = ½ Iω2

  5. Now: Three Types of Energy ! ωi If there is no friction, the total mechanical energy is still conserved. ωf vi But now also have to account for KEr ! hi vf hf

  6. Example 10 kg, radius R A 10.0-kg solid sphere rolls without slipping. When its center of gravity has a speed of 10.0 m/s, determine : 10 m/s (a) Its translational kinetic energy (b) Its rotational kinetic energy (c) Its total kinetic energy Note : “without slipping” means that we can use v = Rω .

  7. Example 1.0 m A ball of mass 1.0-kg and radius 1.0-m starts on a ramp from rest at a height of 1.0-m. What is the speed of the ball when it reaches the bottom? Assume it rolls without slipping. How does this compare to the speed of a 1.0-kg block that starts from rest at a height of 1.0-m, and then slides without friction down the ramp?

  8. Demo: The Race of the Shapes !! cylindrical shell solid sphere solid cylinder Which one will reach the bottom of the ramp first ?? [Assuming they roll without slipping.] Or will they all reach the bottom at the same time ??

  9. Why ?? Moment of Inertia KEr [v = Rω, no slip] KE + KEr [total KE] Energy Conservation: mgh = KE + KEr

  10. Conceptual Question Two uniform, solid spheres are released from rest simultaneously and roll (without slipping) down the same hill. • One is a large, more massive sphere; • The other is a small, less massive sphere. Which one reaches the bottom of the hill first ? (a) Large, more massive sphere (b) Small, less massive sphere (c) Both reach the bottom at the same time (d) Depends on their exact masses and radii, which aren’t given

  11. Next Two Classes • 8.7 : Angular Momentum

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