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))