7 Conservation of Energy

• Potential Energy

• The Conservation of Mechanical Energy

• The Conservation of Energy

• Mass and Energy

• Hk: 23, 27, 39, 47, 55, 65, 69, 71

Potential Energy

• Potential Energy is stored energy

• Potential Energy is position dependent (KE is speed dependent)

• Ex. object at higher height has more PE

• Types of PE: gravitational, elastic, electric, magnetic, chemical, nuclear.

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Conservative Forces

• When the work done by a force moving from position 1 to 2 is independent of the path, the force is Conservative.

• The work done by a Conservative Force is zero for any closed path.

• Conservative Forces have associated Potential Energies

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Non Conservative Forces

• Produce thermal energy, e.g. friction

• Work done by Non Conservative Forces is path dependent, e.g. longer path, more work required

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Potential Energy Functions

work produce todecreasemust PE

2

1

UdFW

FunctionEnergy Potential Definition

2

112 dFUUU

Elastic Potential Energy

dxkxFdxdFdU )(

kxdxdU

oUkxkxdxU 221

Energy Potential Elastic Definition

221 kxU

Ex. Elastic Potential Energy

• 100N/m spring is compressed 0.2m.

• F = -kx = -(100N/m)(0.2m) = -20N

• U = ½kx2 = ½(100N/m)(0.2m)2 = 2J

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Gravitational Potential Energy

dymgFdxdFdU )(

mgdydU

oUmgymgdyU

Energy Potential nalGravitatio Definition

mgyU

Ex. Gravitational Potential Energy

• Ex: A 2kg object experiences weight (2kg)(9.8N/kg) = 19.6N.

• At 3m above the floor it has a stored energy of mgy:

• (2kg)(9.8N/kg)(3m) = 48.8Nm = 48.8J.

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Conservation of Energy

• Individual energy levels change.

• Sum of all individual energies is constant.

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Conservation of Mechanical Energy

cncexttotal WWWW

ctotalncext WWWW

)( syssyssyssysncext UKEUKEWW

0 & when Conserved

Energy Mechanical of Definition

ncext

syssysmech

WW

UKEE

Ex. Conservation of Mechanical Energy: Object dropped from height h above floor.

12

2212

21

2

221

1

)0(

)0(

MEME

mvmvmgME

mghmmghME

ghv

mghmv

2

221

Energy E1 E2 E3

Kinetic 0 ½mv22 0

PE-g 0 0 mgh

PE-spring

½kx2 0 0

Totals

½kx2 ½mv22 mgh

Energy E(h) E(y)

Kinetic 0 ½mv2

PE-g mgh mgy

Totals mgh ½mv2 + mgy

Energies and speeds are same at height y

Accelerations at y are not same

Work Energy with Friction

Frictionh Energy witWork relkmechthermmechext sfEEEW

conserved is system isolatedan ofEnergy Total

constant

0

thermmech

thermmech

EEE

EE

mechncext EWW mechthermext EEW )(

Energy Ei Ef

Kinetic ½mvi2 0

PE-g 0 0

Thermal 0 fks

Totals ½mvi2 fks

Example: The smaller the frictional force fk, the larger the distance, s, it will travel before stopping.

s

A 2.00kg ball is dropped from rest from a height of 1.0m above the floor. The ball rebounds to a height of 0.500m. A movie-frame type diagram of the motion is shown below.

Type E1 E2 E3 E4 E5

gravita-tional

mg(1) 0 0 0 mg(1/2)

kinetic 0 ½ m(v2)2 0 ½ m(v4)2 0

elastic 0 0 PE-elastic 0 0

thermal 0 0 E-thermal E-thermal E-thermal

By energy conservation, the sum of all energies in each column is the same, = E1 = mg(1) = 19.6J

Calculate v2: (use 1st and 2nd columns)mg(1) = ½ m(v2)2.

g = ½ (v2)2.v2 = 4.43m/s

Calculate PE-thermal: (use 1st and 5th columns)mg(1) = mg(1/2) + PE-thermal

mg(1/2) = PE-thermalPE-thermal = 9.8J

Calculate PE-elastic: (use 1st and 3rd columns)PE-elastic + PE-thermal = mg(1)

PE-elastic + 9.8 = 19.6PE-elastic = 9.8J

Calculate v4: (use 1st and 4th columns)½ m(v4)2 + PE-thermal = mg(1)

½ m(v4)2 + 9.8 = 19.6½ m(v4)2 = 9.8 (v4)2 = 2(9.8)/2

v4 = 3.13m/s

Potential Energy & Force

dx

dUF

dxFdFdU

x

x

kxkxdx

d

dx

dUF

kxU

x

)(

Ex.

221

221

Equilibrium

• Stable: small displacement in any direction results in a restoring force toward Equilibrium Point

• Unstable: small displacement in any direction results in a force away from Equilibrium Point

• Neutral: small displacement in any direction results in zero force

Mass and Energy

smcmcE /100.3 82 2mcE

J

mcE

kgg

kgg

10

286-2

6-3-

109

m/s)10kg(310

101000

1

1

101

milligram 1 ofEnergy

Efficiency & Thermodynamics

efficiency 100% impliesequation This

UW

engine gasoline e.g. heat,Q

:efficient Less

out

sysoutout UQW

outoutsys QWU

micsThermodyna of LawFirst

inonsys QWU

Summary

• Potential Energy function & force

• The Conservation of Mechanical Energy

• The Conservation of Energy

• Mass and Energy

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