Conservation of Mechanical Conservation of Mechanical EnergyEnergy

6.56.5

Mechanical EnergyMechanical Energy

KE and PE = total mechanical energyKE and PE = total mechanical energy

E = KE + PEE = KE + PE

Actually there are other forms of energy to include, Actually there are other forms of energy to include, but we haven’t met them yet in the bookbut we haven’t met them yet in the book

Work Energy Theorem (again)Work Energy Theorem (again)

WWncnc = (KE = (KEff – KE – KE00) + (PE) + (PEff – PE – PE00))RearrangeRearrange

WWncnc = (KE = (KEff + PE + PEff) – (KE) – (KE00 + PE + PE00) )

WWncnc = E = Eff – E – E00

Nonconservative forces change the total Nonconservative forces change the total mechanical energy of the systemmechanical energy of the system

WWncnc = 0 = 0

WWncnc = E = Eff – E – E00

If WIf Wncnc = 0 like if friction or air resistance is = 0 like if friction or air resistance is negligiblenegligible

0 = E0 = Eff – E – E00

EEf f = E = E00

If all forces are conservative, then the total If all forces are conservative, then the total mechanical energy does not changemechanical energy does not change

Conservation of Mechanical EnergyConservation of Mechanical Energy

If there is no work done by If there is no work done by nonconservative forcesnonconservative forces

Total mechanical energy is constantTotal mechanical energy is constant

KEKE00 + PE + PE00 = KE = KEff + PE + PEff

Example 1Example 1

A cyclist approaches the bottom of a A cyclist approaches the bottom of a gradual hill at a speed of 11 m/s. The hill gradual hill at a speed of 11 m/s. The hill is 5.0 m high, and the cyclist estimates is 5.0 m high, and the cyclist estimates that she is going fast enough to coast up that she is going fast enough to coast up and over it without pedaling. Ignoring air and over it without pedaling. Ignoring air resistance and friction, find the speed at resistance and friction, find the speed at which the cyclist crests the hill.which the cyclist crests the hill.

v = 4.8 m/sv = 4.8 m/s

Example 2Example 2

A slingshot fires a pebble from the top of a A slingshot fires a pebble from the top of a building at a speed of 14.0 m/s. The building at a speed of 14.0 m/s. The building is 31.0 m tall. Ignoring air building is 31.0 m tall. Ignoring air resistance, find the speed with which the resistance, find the speed with which the pebble strikes the ground when the pebble pebble strikes the ground when the pebble is fired (a) horizontally, (b) vertically is fired (a) horizontally, (b) vertically straight up, and (c) vertically straight straight up, and (c) vertically straight down.down.

v = 28.3 m/sv = 28.3 m/s

Conceptual ExampleConceptual Example

A rope is tied to a tree limb and used by a A rope is tied to a tree limb and used by a swimmer to swing into the water below. swimmer to swing into the water below. The person starts from rest with the rope The person starts from rest with the rope held in the horizontal position, swings held in the horizontal position, swings down and then lets go of the rope. Three down and then lets go of the rope. Three forces act on him; W, T, and air forces act on him; W, T, and air resistance. Can conservation of resistance. Can conservation of mechanical energy be used to find the mechanical energy be used to find the speed when he lets go of the rope?speed when he lets go of the rope?

Conceptual ExampleConceptual Example

In order to use conservation of mechanical In order to use conservation of mechanical energy, Wenergy, Wncnc = 0 = 0

Nonconservative force in exampleNonconservative force in exampleTension Tension perpendicular to the motion so W = 0 perpendicular to the motion so W = 0Air Resistance Air Resistance opposite and parallel to motion opposite and parallel to motion

– Work does not equal 0Work does not equal 0

Cannot use conservation of MECannot use conservation of ME

Reasoning Strategy for Reasoning Strategy for Conservation of MEConservation of ME

Identify the external conservative and Identify the external conservative and nonconservative forces that act on the nonconservative forces that act on the object. For this principle to apply, the total object. For this principle to apply, the total work done by nonconservative forces must work done by nonconservative forces must be zero.be zero.Choose the location where the PE is taken Choose the location where the PE is taken to be zero.to be zero.Set the final total ME equal to the initial Set the final total ME equal to the initial total ME.total ME.

Practice WorkPractice Work

Don’t conserve energy to solve these Don’t conserve energy to solve these problems!problems!

174 CQ 12 – 14, P 34 – 36, 38, 40, 41174 CQ 12 – 14, P 34 – 36, 38, 40, 41

Total 9 problemsTotal 9 problems