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CH 6 Lecture 2 Conservation of Energy

CH 6 Lecture 2 Conservation of Energy. Elastic Potential Energy Pushing/Pulling on a spring Stretching or compressing an object from a preferred state requires work The force opposing is elastic if the object snaps back into shape We give the object Elastic Potential Energy Springs

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CH 6 Lecture 2 Conservation of Energy

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  1. CH 6 Lecture 2 Conservation of Energy • Elastic Potential Energy • Pushing/Pulling on a spring • Stretching or compressing an object from a preferred state requires work • The force opposing is elastic if the object snaps back into shape • We give the object Elastic Potential Energy • Springs • Spring constant = k (Tells us how stiff the spring is) • Stiffer the spring, larger k • Force of spring is proportional to distance it is stretched/compressed • Hooke’s Law F = -kx (x = distance stretched or compressed) • -k means spring force is always opposing its motion • Push on the spring, it pushes back • Pull on the spring, it pulls back • Elastic Potential Energy = ½ comes from avg force being half max. force k = 2 kg/s, x = 5 m, F?, W?, PE?

  2. Average Force

  3. Conservation of Energy • Total Energy of a system remains the same unless it is acted on by an external force KE + PE = constant • Pendulum • Side1: KE = 0, Total E = PE • Bottom: PE = 0, KE = Total E • Side2: KE = 0, Total E = PE • Work to get this started, after that W = 0 • Input E into the system • ET = KE + PE = constant • Sides: Initial Work gives us PE • Bottom: Gravity moves bob down (KE) • F = tension = centripetal force; perpendicular to motion, W = Fd = 0 • Friction (air resistance) does small work, eventually stopping bob • In a vacuum, the pendulum would keep swinging forever • Could we describe the pendulum with Newton’s laws? • Velocity is continually changing • Calculations would be hard F

  4. Use Conservation of Energy to Solve hard Mechanics problems • Pendulum, m = 0.5 kg h = 12 cm v at bottom? At top, ET = PE = mgh = (0.5kg)(9.8m/s)(0.12m) = 0.588 J At bottom, ET = KE = ½ mv2 = (0.5)(0.5kg)v2 = 0.588 J v2 = 2.35 m2/s2 v = 1.53 m/s E) Energy on a mountain

  5. KE = ½ mv2 If KE decreases, v decreases

  6. Springs and Harmonic Motion • Simple Harmonic Motion: repetitive motion with constant conversion KE/PE • Pendulum • Mass at the end of a spring • Add E with Initial amount of Work • Plotting Position vs. Time • “Harmonic Function” (sin or cos) • One complete cycle = T, period • Frequency = # cycles per second (Hz = s-1) • Amplitude = max. distance from starting pt. • Frequency for • Loose/Tight spring? • Large/Small mass? KE = 0 PE = 0 KE = 0

  7. Restoring Forces lead to Harmonic Motion • Wants to bring mass back to starting position • If F is proportional to d, get harmonic motion (F = -kx) • What is restoring force for a simple pendulum? • Case of a vertical spring/mass system • Force of Gravity pulling down is constant • Restoring force pulling up varies FT = FR + FG • Equilibrium point is lower than without gravity • Harmonic Motion just like horizontal setup • FT is still proportional to x • PE = gravity + elastic potential energy Animations

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