Potential energy and conservation of Potential energy and conservation of energyenergy

Chapter8Chapter8

I.I. Potential energy Potential energy Energy of configuration Energy of configuration

II.II. Work and potential energyWork and potential energy

III.III. Conservative / Non-conservative forcesConservative / Non-conservative forces

IV.IV. Determining potential energy values:Determining potential energy values:

- Gravitational potential energy- Gravitational potential energy - Elastic potential energy- Elastic potential energy

V. V. Conservation of mechanical energyConservation of mechanical energy

VI.VI. External work and thermal energyExternal work and thermal energy

VII.VII. External forces and internal energy changesExternal forces and internal energy changes

I. Potential energyPotential energy

Energy associated with the Energy associated with the arrangementarrangement of a system of of a system of objects that exert forces on one another. objects that exert forces on one another.

Units: Units: 1J1J

Examples:Examples:

- Gravitational potential energy:Gravitational potential energy: associated with the state of associated with the state of separation between objects which can attract one another separation between objects which can attract one another by the gravitational force.by the gravitational force.

- Elastic potential energy:Elastic potential energy: associated with the state of associated with the state of compression/extension of an elastic object. compression/extension of an elastic object.

II. Work and potential energyWork and potential energy

If tomato If tomato risesrises gravitational force gravitational force transfers energy “transfers energy “fromfrom” tomato’s kinetic ” tomato’s kinetic energy “energy “toto” the gravitational potential ” the gravitational potential energy of the tomato-Earth system.energy of the tomato-Earth system.

If tomato If tomato falls downfalls down gravitational force transfers gravitational force transfers energy “energy “fromfrom” the gravitational potential energy “” the gravitational potential energy “toto” the ” the tomato’s kinetic energy.tomato’s kinetic energy.

WU Also valid for elastic potential energyAlso valid for elastic potential energy

Spring forceSpring force does –W on block does –W on block energy transfer from kinetic energy transfer from kinetic energy of the block to potential energy of the block to potential elastic energy of the spring.elastic energy of the spring.

Spring forceSpring force does +W on block does +W on block energy transfer from potential energy transfer from potential energy of the spring to kinetic energy of the spring to kinetic energy of the block.energy of the block.

fs

fs

Spring compressionSpring compression

Spring extensionSpring extension

General:General:- System of two or more objects.System of two or more objects.

- A force acts between a particle in the system and the rest of the A force acts between a particle in the system and the rest of the system.system.

- - When system configuration changes When system configuration changes force does work on the force does work on the

object (Wobject (W11) transferring energy between KE of the object and ) transferring energy between KE of the object and

some other form of energy of the system.some other form of energy of the system.

- When the configuration change is reversed When the configuration change is reversed force reverses the force reverses the

energy transfer, doing Wenergy transfer, doing W22..

III. Conservative / Nonconservative forcesIII. Conservative / Nonconservative forces

- If WIf W11=W=W22 always always conservative force. conservative force.

Examples:Examples: Gravitational force and spring force Gravitational force and spring force associated associated potential energiespotential energies..

- If WIf W11≠≠WW22 non-conservative force. non-conservative force.

Examples:Examples: Drag force, frictional force Drag force, frictional force KE transferred into KE transferred into thermal energy. Non-reversible processthermal energy. Non-reversible process.

- Thermal energy:- Thermal energy: Energy associated with the random Energy associated with the random movement of atoms and molecules. This is not a potential movement of atoms and molecules. This is not a potential energy.energy.

- Conservative force:Conservative force: The net work it does on a particle moving The net work it does on a particle moving around every closed path, from an initial point and then back to around every closed path, from an initial point and then back to that point is zero.that point is zero.

Conservative forceConservative force W Wab,1ab,1= W= Wab,2ab,2

WWab,1ab,1+ W+ Wba,2ba,2=0 =0 W Wab,1ab,1= -W= -Wba,2ba,2

WWab,2ab,2= - W= - Wba,2ba,2

- The net work it does on a particle moving between two The net work it does on a particle moving between two points does not depend on the particle’s path.points does not depend on the particle’s path.

Proof:Proof:

WWab,2ab,2= W= Wab,1ab,1

IV. Determining potential energy valuesIV. Determining potential energy values

f

i

x

xUdxxFW )( Force F is conservativeForce F is conservative

Gravitational potential energy:Gravitational potential energy:

Change in the gravitational potential energy of the particle-Earth system.

f

i

f

i

y

y ifyy ymgyymgymgdymgU )()(

mgyyUyU ii )(0,0

The gravitational potential energy associated with particle-The gravitational potential energy associated with particle-Earth system depends Earth system depends onlyonly on particle’s vertical position “y” on particle’s vertical position “y” relative to the reference position y=0, not on the horizontal relative to the reference position y=0, not on the horizontal position.position.

Reference configurationReference configuration

Elastic potential energy:Elastic potential energy:

Change in the elastic potential energy of the spring-block Change in the elastic potential energy of the spring-block system.system.

222

2

1

2

1

2)( i

x

x f

x

x kxkxxk

dxkxU f

i

f

i

2

2

1)(0,0 kxxUxU ii

Reference configuration Reference configuration when the spring is at its relaxed when the spring is at its relaxed length and the block is at xlength and the block is at xii=0.=0.

Remember!Remember! Potential energy is always associated with a Potential energy is always associated with a system.system.

EEmecmec= U + K= U + K

Assumptions:Assumptions: Only conservative forces cause energy transfer Only conservative forces cause energy transfer within the system.within the system.

UW

KW

iiffifif UKUKUUKKUK 0)()(0

ΔΔEEmecmec= = ΔΔK + K + ΔΔU = 0U = 0

The system is isolated from its environment The system is isolated from its environment No external No external force from an object outside the system causes energy force from an object outside the system causes energy changes inside the system.changes inside the system.

V. Conservation of mechanical energyV. Conservation of mechanical energyMechanical energy Mechanical energy of a systemof a system: : Sum of the its potential (U) Sum of the its potential (U) and kinetic (K) energies.and kinetic (K) energies.

- In an In an isolated systemisolated system where where only conservative only conservative forcesforces cause energy changes, the kinetic energy and cause energy changes, the kinetic energy and potential energy can change, but their sum, the potential energy can change, but their sum, the mechanical energymechanical energy of the system of the system cannot changecannot change..

- When the When the mechanical energymechanical energy of a system is of a system is conservedconserved, we can relate the sum of kinetic energy , we can relate the sum of kinetic energy and potential energy at one instant to that at another and potential energy at one instant to that at another instant without considering the intermediate motion instant without considering the intermediate motion and without finding the work done by the forces and without finding the work done by the forces involved.involved.

EEmecmec= = constantconstant

iiff

mec

UKUK

UKE

0

Potential energy curvesPotential energy curves

x

y

Finding the force analytically:Finding the force analytically:

)1()(

)()()( motionDdx

xdUxFxxFWxU

- The force is the negative of the slope of the curve U(x) The force is the negative of the slope of the curve U(x) versus x.versus x.

- The particle’s kinetic energy is: The particle’s kinetic energy is: K(x) = EK(x) = Emecmec – U(x) – U(x)

Turning point: Turning point: a point a point x at which the particle x at which the particle reverses its motion reverses its motion (K=0).(K=0).

Equilibrium pointsEquilibrium points: : where the slope of the U(x) curve is zero where the slope of the U(x) curve is zero F(x)=0 F(x)=0

ΔU = -F(x) dx ΔU = -F(x) dx ΔU/dx = -F(x) ΔU/dx = -F(x)

K always K always ≥≥0 0 (K=0.5mv(K=0.5mv2 2 ≥0≥0 ))

Examples:Examples:

x= xx= x11 EEmecmec= 5J=5J+K = 5J=5J+K

K=0K=0

x<xx<x11 E Emecmec= 5J= >5J+K= 5J= >5J+K

K<0 K<0 impossibleimpossible

Example: Example: x x ≥≥ x x55 E Emec,1mec,1= 4J=4J+K = 4J=4J+K K=0 and also F=0 K=0 and also F=0 xx55

neutral equilibriumneutral equilibrium

Emec,1

Emec,2

Emec,3

ΔU(x)/dx = -F(x) ΔU(x)/dx = -F(x) -F(x) -F(x) Slope Slope

Equilibrium pointsEquilibrium points

xx44 E Emec,3mec,3=1J=1J+K =1J=1J+K K=0, F=0, it cannot move to x>x K=0, F=0, it cannot move to x>x44 or x<x or x<x44, ,

since then K<0 since then K<0 Stable equilibriumStable equilibrium

xx22<x<x<x<x11, x, x55<x<x<x<x44 E Emec,2mec,2= 3J= 3J+K = 3J= 3J+K K=0 K=0 Turning pointsTurning points

xx33 K=0, F=0 K=0, F=0 particle stationary particle stationary Unstable equilibriumUnstable equilibrium

VI. Work done on a system by an external forceVI. Work done on a system by an external force

No Friction:No Friction:

Work is energy transfer Work is energy transfer “to”“to” or or “from”“from” a system by means of an a system by means of an external force acting on that system.external force acting on that system.

When more than one force acts on When more than one force acts on a system their net work is the energy a system their net work is the energy transferred transferred toto or or fromfrom the system. the system.

W = W = ΔΔEEmecmec= = ΔΔK+ K+ ΔΔU U Ext. force Ext. force

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mafF k

/)(5.02 20

220

2

Friction:Friction:

Remember!Remember! ΔΔEEmecmec= = ΔΔK+ K+ ΔΔU = 0 U = 0 only when:only when:

- System isolated.System isolated. - No external forces act on a system.No external forces act on a system.- All internal forces are conservative.All internal forces are conservative.

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General:General: Example:Example: Block sliding up a ramp. Block sliding up a ramp.

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/)(5.02 20

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dfE kth Thermal energy:Thermal energy:

thmec EEFdW

Work done on a system by an external force, friction involvedWork done on a system by an external force, friction involved

Friction due to cold welding Friction due to cold welding between two surfaces. As between two surfaces. As the block slides over the the block slides over the floor, the sliding causes floor, the sliding causes tearing and reforming of the tearing and reforming of the welds between the block and welds between the block and the floor, which makes the the floor, which makes the block-floor warmer.block-floor warmer.

VI. Conservation of energyVI. Conservation of energy

- The total energy of a system can only change by amounts The total energy of a system can only change by amounts of energy transferred of energy transferred “from”“from” or or “to”“to” the system. the system.

intEEEW thmec Experimental lawExperimental law

-The total energy of an isolated system cannot change. The total energy of an isolated system cannot change. (There cannot be energy transfers to or from it).(There cannot be energy transfers to or from it).

Total energy of a system = ETotal energy of a system = Emechanicalmechanical + E + Ethermalthermal + E + Einternalinternal

Isolated system:Isolated system:

0int EEE thmec

In an isolated system we can relate the total energy at one In an isolated system we can relate the total energy at one instant to the total energy at another instant without instant to the total energy at another instant without considering the energies at intermediate states.considering the energies at intermediate states.

VII. External forces and internal energy changesVII. External forces and internal energy changes

Example:Example: skater pushes herself away from a railing. There is a skater pushes herself away from a railing. There is a force force FF on her from the railing that increases her kinetic energy. on her from the railing that increases her kinetic energy.

i)i) One part of an object (skater’s arm) One part of an object (skater’s arm) does not move like the rest of body.does not move like the rest of body.

ii) Internal energy transfer (from one part ii) Internal energy transfer (from one part of the system to another) using the of the system to another) using the external force external force FF. Biochemical energy . Biochemical energy from muscles transferred to kinetic from muscles transferred to kinetic energy of the body.energy of the body.

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)(cos

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Change in system’s mechanical energy by an external forceChange in system’s mechanical energy by an external force

cos

0

int

int

FdEE

EE

mec

mec

Change in system’s Change in system’s

internal energy by an internal energy by an external forceexternal force

Proof:Proof:

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1

2

1

)(2

20

2

20

2

FdK

dMaMvMv

Mdavv

x

x

A massless rigid rod of length A massless rigid rod of length LL has a ball of mass has a ball of mass m m attached to one end. attached to one end. The other end is pivoted in such a way that the ball will move in a vertical The other end is pivoted in such a way that the ball will move in a vertical circle. First, assume that there is no friction at the pivot. The system is circle. First, assume that there is no friction at the pivot. The system is launched downward from the horizontal position launched downward from the horizontal position AA with initial speed with initial speed vv00. .

The ball just barely reaches point The ball just barely reaches point DD and then stops. and then stops.

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(a)(a) Derive an expression for Derive an expression for vv00 in terms of in terms of LL, , mm and and gg. .

129. A massless rigid rod of length A massless rigid rod of length LL has a ball of mass has a ball of mass mm attached to one attached to one end. The other end is pivoted in such a way that the ball will move in a end. The other end is pivoted in such a way that the ball will move in a vertical circle. First, assume that there is no friction at the pivot. The vertical circle. First, assume that there is no friction at the pivot. The system is launched downward from the horizontal position system is launched downward from the horizontal position AA with initial with initial speed speed vv00. The ball just barely reaches point . The ball just barely reaches point D D and then stops. and then stops.

D

CA

B

L

v0

y

x

mg

TFc

(b) What is the tension in the rod when the ball passes (b) What is the tension in the rod when the ball passes through through BB? ?

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The difference in heights or in gravitational potential energies between the The difference in heights or in gravitational potential energies between the positions C (reached by the ball when there is friction) and D during the positions C (reached by the ball when there is friction) and D during the frictionless movement Is going to be the loss of mechanical energy which goes frictionless movement Is going to be the loss of mechanical energy which goes into thermal energy. into thermal energy.

In the figure below, a block slides along a path that is In the figure below, a block slides along a path that is without friction until the block reaches the section of length without friction until the block reaches the section of length L=0.75m, which begins at height h=2m. In that section, the L=0.75m, which begins at height h=2m. In that section, the coefficient of kinetic friction is 0.4. The block passes through coefficient of kinetic friction is 0.4. The block passes through point A with a speed of point A with a speed of

8m/s. Does it reach point B 8m/s. Does it reach point B (where the section of (where the section of friction ends)? If so, what is friction ends)? If so, what is the speed there and if not, the speed there and if not, what greatest height above what greatest height above point A does it reach?point A does it reach?

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22

C

In the figure below, a block slides along a path that is without friction until the block reaches the section of length L=0.75m, which begins at height h=2m. In that section, the coefficient of kinetic friction is 0.4. The block passes through

point A with a speed of 8m/s. Does it reach point B (where the section of friction ends)? If so, what is the speed there and if not, what greatest height above point A does it reach?

mg

N

fC

The kinetic energy in C turns into thermal and potential energy Block stops.

BreachesBlockmLd

metersdmddmgmdfmgyK

mmvK

kc

cc

75.0

49.14.3)30sin(4.12

4.125.0 2

61. In the figure below, a block slides along a path that is without friction until the block reaches the section of length L=0.75m, which begins at height h=2m. In that section, the coefficient of kinetic friction is 0.4. The block passes through point A with a speed of

8m/s. Does it reach point B (where the section of friction ends)? If so, what is the speed there and if not, what greatest height above point A does it reach?

mg

N

fC

smvmmmvm

mgLmgLmvmgLyymgmvm

LfUKUKEUKEsystemIsolated

BB

kBkcBB

kBBCCth

/5.35.267.35.04.12

30cos30sin5.030cos)(5.04.12

0

2

22

A 3.2kg sloth hangs 3m above the ground. (a) What is the gravitational potential energy of the sloth-Earth system if we take the reference point y=0 to be at the ground? If the sloth drops to the ground and air drag on it is assumed to be negligible, what are (b) the kinetic energy and (c) the speed of the sloth just before it reaches the ground?

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A metal tool is sharpen by being held against the rim of a wheel on a grinding machine by a force of 180N. The frictional forces between the rim and the tool grind small pieces of the tool. The wheel has a radius of 20cm and rotates at 2.5 rev/s. The coefficient of kinetic friction between the wheel and the tool is 0.32. At what rate is energy being transferred from the motor driving the wheel and the tool to the kinetic energy of the material thrown from the tool?

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A block with a kinetic energy of 30J is about to collide with a spring at its relaxed length. As the block compresses the spring, a frictional force between the block and floor acts on the block. The figure below gives the kinetic energy of the block (K(x)) and the potential energy of the spring (U(x)) as a function of the position x of the block, as the spring is compressed. What is the increase in thermal energy of the block and the floor when (a) the block reaches position 0.1 m and (b) the spring reaches its maximum compression?

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