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The Principle of Conservation of Mechanical Energy (PCME). Section 6.5. Total Mechanical Energy. Review KE = ½mv 2 PE = mgh Total Mechanical Energy Defined E = KE + PE E = ½mv 2 + mgh. Work from Nonconservative Forces. W nc = Δ KE+ Δ PE - PowerPoint PPT Presentation

### Transcript of The Principle of Conservation of Mechanical Energy (PCME)

• The Principle of Conservation of Mechanical Energy (PCME)Section 6.5

• Total Mechanical EnergyReviewKE = mv2PE = mghTotal Mechanical Energy DefinedE = KE + PE E = mv2 + mgh

• Work from Nonconservative ForcesWnc= KE+ PE = KEf KE0+ PEf PE0= KEf + PEf KE0 PE0= (KEf + PEf) (KE0 + PE0)

Ef E0 Wnc= Ef E0

• Work from Nonconservative ForcesThe net work done by external nonconservative forces changes the total mechanical energy from an initial value E0 to a final value Ef.

• PCMEWork-EnergyTheoremWk = Wc + Wnc

• PCME AppliesWnc = 0 = Ef E0Ef = E0Total mechanical energy remains constantKinetic energy and potential energy can be interchanged

• Ex. 8: A Daredevil Motorcyclistv0 = 38.0 m/sh0 = 70.0 mhf = 35.0 mFind vfIgnore air resistance

• Ex. 8: ReasoningOnly gravity acts on cycleSince air resistance is ignoredWnc = 0PCME applies

• Ex. 8: A Daredevil MotorcyclistEf = E0mvf2 + mghf = mv02 + mgh0 vf2 + ghf = v02 + gh0vf2 = v02 + gh0 ghfvf2 = v02 + g(h0 hf)vf2 = v02 + 2g(h0 hf)vf = (v02 + 2g(h0 hf))

• Ex. 8: A Daredevil Motorcyclistvf = (v02 + 2g(h0 hf))vf = (38.02 + 2(9.80)(70.0 35.0))vf = 46.2 m/s

• Ex. 9: The Favorite Swimming HoleA rope is tied to a tree limb and used by a swimmer to swing into the water below. The person starts from rest with the rope held in the horizontal position, then lets go of the rope.3 forces act on him:WeightTension in the ropeForce due to air resist.Initial & final heights are known.Can the PCME be used to find his speed vf when he lets go of the rope?

• Ex. 9: RationaleTension is nonconservative forceTension on rope is to motionTension does no workForce due to air resistance is opposite motionWair =(Faircos180)s Wair 0 Wnc 0Cannot use PCME

• Ex. 10: The Magnum XL-200One of the fastest roller coasters in the worldVertical drop of 59.4 mAssume vtop 0Neglect frictionFind speed at bottom of hill

• Ex. 10: The Magnum XL-200Neglecting frictionNormal force to motionNeither contributes to workWnc = 0vf = (v02 + 2g(h0 hf)) vf = (02 + 2(9.8)(59.4)) vf = 34.1 m/s (about 76 mph)

• Reasoning StrategyIdentify external conserv & nconserv forces acting on objectTo apply PCME, Wnc must equal 0Either ignored (approximation)Or to motion Choose location where PE = 0Arbitrary Let h0 = 0 or hf = 0 vf = (v02 + 2g(h0 hf))

Conservation of Mechanical Energy