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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
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The Principle of Conservation of Mechanical Energy (PCME)
Section 6.5
Total Mechanical Energy
Review
KE = ½mv2
PE = mgh
Total Mechanical Energy Defined
E = KE + PE
E = ½mv2 + mgh
Work from Nonconservative Forces
Wnc = ΔKE + ΔPE
= KEf – KE0 + PEf – PE0
= KEf + PEf – KE0 – PE0
= (KEf + PEf) – (KE0 + PE0)
Ef E0
Wnc = Ef – E0
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.
PCME
Work-EnergyTheorem
Newton’sSecond Law
Work
Kinetic Energy
IsWnc=0?
PCMEAppliesY
es
Wk = Wc + Wnc
PCME
No
PCME Applies
Wnc = 0 = Ef – E0
Ef = E0
Total mechanical energy remains constant
Kinetic energy and potential energy can be interchanged
Ex. 8: A Daredevil Motorcyclist
v0 = 38.0 m/s
h0 = 70.0 m
hf = 35.0 m
Find vf
Ignore air resistance
Ex. 8: Reasoning
Only gravity acts on cycle
Since air resistance is ignored
Wnc = 0
PCME applies
Ex. 8: A Daredevil Motorcyclist
Ef = E0
½mvf2 + mghf = ½mv0
2 + mgh0
½vf2 + ghf = ½v0
2 + gh0
½vf2 = ½v0
2 + gh0 – ghf
½vf2 = ½v0
2 + g(h0 – hf)
vf2 = v0
2 + 2g(h0 – hf)
vf = (v02 + 2g(h0 – hf))
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
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:
WeightTension in the ropeForce due to air resist.
Initial & final heightsare known.Can the PCME be used tofind his speed vf when helets go of the rope?
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
Ex. 10: The Magnum XL-200
One of the fastest roller coasters in the worldVertical drop of 59.4 m
Assume vtop ≈ 0Neglect frictionFind speed at bottom of hill
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)
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))