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The Principle of Conservation of Mechanical Energy (PCME)

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

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The Principle of Conservation of Mechanical Energy (PCME)

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  1. The Principle of Conservation of Mechanical Energy (PCME) Section 6.5

  2. Total Mechanical Energy • Review • KE = ½mv2 • PE = mgh • Total Mechanical Energy Defined • E = KE + PE • E = ½mv2 + mgh

  3. Work from Nonconservative Forces • Wnc = ΔKE + ΔPE = KEf – KE0 + PEf – PE0 = KEf+ PEf – KE0 – PE0 = (KEf + PEf) – (KE0 + PE0) Ef E0 • Wnc = Ef – E0

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

  5. Newton’s Second Law PCME Applies Y e s Is Wnc=0? N o PCME Work Kinetic Energy PCME Work-Energy Theorem Wk = Wc + Wnc

  6. PCME Applies • Wnc = 0 = Ef– E0 • Ef = E0 • Total mechanical energy remains constant • Kinetic energy and potential energy can be interchanged

  7. Ex. 8: A Daredevil Motorcyclist • v0 = 38.0 m/s • h0 = 70.0 m • hf = 35.0 m • Find vf • Ignore air resistance

  8. Ex. 8: Reasoning • Only gravity acts on cycle • Since air resistance is ignored • Wnc = 0 • PCME applies

  9. Ex. 8: A Daredevil Motorcyclist • Ef = E0 • ½mvf2 + mghf = ½mv02 + mgh0 • ½vf2 + ghf = ½v02 + gh0 • ½vf2 = ½v02 + gh0 – ghf • ½vf2 = ½v02 + g(h0 – hf) • vf2 = v02 + 2g(h0 – hf) • vf = (v02 + 2g(h0 – hf))

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

  11. Ex. 9: The Favorite Swimming Hole • A 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: • Weight • Tension in the rope • Force due to air resist. • Initial & final heightsare known. • Can the PCME be used tofind his speed vfwhen helets go of the rope?

  12. Ex. 9: Rationale • Tension is nonconservative force • Tension on rope is  to motion • Tension does no work • Force due to air resistance is opposite motion • Wair =(Faircos180)s • Wair ≠ 0 •  Wnc ≠ 0 • Cannot use PCME

  13. Ex. 10: The Magnum XL-200 • One of the fastest roller coasters in the world • Vertical drop of 59.4 m • Assume vtop≈ 0 • Neglect friction • Find speed at bottom of hill

  14. Ex. 10: The Magnum XL-200 • Neglecting friction • Normal force  to motion • Neither contributes to work • Wnc = 0 • vf = (v02 + 2g(h0 – hf)) • vf = (02 + 2(9.8)(59.4)) • vf = 34.1 m/s (about 76 mph)

  15. Reasoning Strategy • Identify external conserv & nconserv forces acting on object • To apply PCME, Wnc must equal 0 • Either ignored (approximation) • Or  to motion • Choose location where PE = 0 • Arbitrary → Let h0 = 0 or hf = 0 • vf = (v02 + 2g(h0 – hf))

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