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Review from Last Lecture

Review from Last Lecture. Elastic & Inelastic Collisions Momentum conserved Energy conserved in elastic but not in inelastic Rocket Propulsion No “pushing against,” but conservation of momentum Rotational Kinematics Analogous to linear kinematics. Rotational Kinetic Energy.

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Review from Last Lecture

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  1. Review from Last Lecture • Elastic & Inelastic Collisions • Momentum conserved • Energy conserved in elastic but not in inelastic • Rocket Propulsion • No “pushing against,” but conservation of momentum • Rotational Kinematics • Analogous to linear kinematics

  2. Rotational Kinetic Energy • What is the kinetic energy of a spinning object? • Just the sum of it’s parts!

  3. Moment of Inertia • What is the moment of inertia of an extended object • Break it up into little pieces

  4. Integrals • Recall: All the differential calculus you need (in 5 minutes)

  5. Integrals • Now, all the integral calculus you need in 10

  6. Integrals • A definite integral is just a sum • The “area under the curve” • Consider • Sum of rectangles • Width: dx • Height: kx • Area of triangle • Half base times height • A = ½ a ka • From integration formula ka kx x a

  7. Moments of Inertia • Moment of inertia of a hoop or a thin cylinder • All the mass is at the same R!

  8. Moments of Inertia • Moment of inertia of a disk or solid cylinder • Consider ring at r, with volume 2prLdr

  9. Moments of Inertia • Moment of inertia of a thin rod about CM

  10. Moments of Inertia

  11. Moments of Inertia • What about an arbitrary axis? • Use “Parallel Axis Theorem” I = ICM + MD2 • Moment of inertia about any axis is just moment of inertia about center of mass plus moment of inertia of “CM” about the axis

  12. Moments of Inertia • What about an arbitrary axis? • “Parallel Axis Theorem”

  13. Torque • If angular acceleration (a) is analogous to acceleration (a), what is analogous to force (F)? • Since a = rac, use t = rFc =rF sinf • Call it “Torque” (like a it too is a vector)

  14. Torque and Angular Acceleration • Consider a particle at position r • What are the kimematics? • Analogous to F=ma (linear) we have t=Ia (angular)

  15. Torque and Angular Acceleration • What about for extended object? • Net torque gives rise to angular acceleration

  16. Work and Power • Work and power for a rotating object

  17. Example: Trapdoor • What is acceleration of tip when horizontal? • What is speed of tip when vertical?

  18. Rolling Motion • For a rolling object the point in contact with the surface in momentarily stationary • Thus, center (also CM) moves thru a distance s=rq, at a velocity v=rw • Top moves at speed 2vCM • Kinetic energy is rotation plus translation

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