Mechanical Energy

• Kinetic energy• Potential Gravitational Energy• Potential Elastic Energy

DEFINITION OF GRAVITATIONAL POTENTIAL ENERGY

The gravitational potential energy PE is the energy that anobject of mass m has by virtue of its position relative to thesurface of the earth. That position is measured by the heighth of the object relative to an arbitrary zero level:

mghPE

J joule 1 mN 1

1h

3h

Compare gravitational energies.a) Can they all be equal?b) Can the navy ball have more PE than the green one?

A 5.0-kg object is sliding down (staring at rest) a 5.0 m incline that is making a 30.0o angle with the horizontal. Find the final velocity of the block. Find the final kinetic energy of the block. Compare it to the initial PE of the block.

Elastic Potential EnergyElastic potential energy – energy in a

stretched or compressed elastic objectk – spring constantx – distance of deformation

A Gymnast on a Trampoline

What kinds of mechanical energy are involved in this activity??

A Gymnast on a Trampoline

The 65.0kg gymnast leaves the trampoline at an initial height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast? How much potential elastic energy did the trampoline store? If the trampoline was stretched down by 30.0 cm, and the mass of the what is the k for the material?

Conservation of energy

• Mechanical energy is conserved in the absence of non-conservative forces.

DEFINITION OF A CONSERVATIVE FORCE

Version 1 A force is conservative when the work it doeson a moving object is independent of the path between theobject’s initial and final positions.

Version 2 A force is conservative when it does no work on an object moving around a closed path, starting andfinishing at the same point.

6.4 Conservative Versus Nonconservative Forces

Version 1 A force is conservative when the work it doeson a moving object is independent of the path between theobject’s initial and final positions.

fo hhmgW gravity

Version 2 A force is conservative when it does no work on an object moving around a closed path, starting andfinishing at the same point.

fo hh fo hhmgW gravity

In normal situations both conservative and nonconservativeforces act simultaneously on an object, so the work done bythe net external force can be written as

ncc WWW

KEKEKE of W

PEPEPE fogravity foc mghmghWW

ncc WWW

ncW PEKE

THE WORK-ENERGY THEOREM

PEKE ncW

ofof PEPEKEKEPEKE ncW

ooff PEKEPEKE ncW

of EE ncW

If the net work on an object by nonconservative forcesis zero, then its energy does not change:

of EE

THE PRINCIPLE OF CONSERVATION OF MECHANICAL ENERGY

The total mechanical energy (E = KE + PE) of an objectremains constant as the object moves, provided that the network done by external nononservative forces is zero.

6.5 The Conservation of Mechanical Energy

Example 8 A Daredevil Motorcyclist

A motorcyclist is trying to leap across the canyon by driving horizontally off a cliff 38.0 m/s. Ignoring air resistance, findthe speed with which the cycle strikes the ground on the otherside.

of EE

2212

21

ooff mvmghmvmgh

2212

21

ooff vghvgh

6.5 The Conservation of Mechanical Energy

2212

21

ooff vghvgh

22 ofof vhhgv

sm2.46sm0.38m0.35sm8.92 22 fv

Conceptual Example 9 The Favorite Swimming Hole

The person starts from rest, with the ropeheld in the horizontal position,swings downward, and then letsgo of the rope. Three forces act on him: his weight, thetension in the rope, and theforce of air resistance.

Can the principle of conservation of energybe used to calculate hisfinal speed?

Example 11 Fireworks

Assuming that the nonconservative forcegenerated by the burning propellant does425 J of work, what is the final speedof the rocket. Ignore air resistance.

2

21

221

oo

ffnc

mvmgh

mvmghW

sm61fv

DEFINITION OF AVERAGE POWER

Average power is the rate at which work is done, and itis obtained by dividing the work by the time required to perform the work.

tWP

TimeWork

(W)watt sjoule

Timeenergyin Change

P

watts745.7 secondpoundsfoot 550 horsepower 1

vFP

sFW cos

Constant Force

Variable Force

2211 coscos sFsFW