Chapter 7Chapter 7Potential Energy and Energy Potential Energy and Energy

ConservationConservation

HomeworkHomeworkRead 241 Read 241 -- 2602609, 11, 13, 17, 199, 11, 13, 17, 19Read 260 Read 260 –– 27227235, 37, 43, 47, 49, 6335, 37, 43, 47, 49, 63

Goals for Chapter 7Goals for Chapter 7––To study gravitational and elastic To study gravitational and elastic

potential energypotential energy––To determine when total mechanical To determine when total mechanical

energy is conservedenergy is conserved––To examine situations when total To examine situations when total

mechanical energy is not conservedmechanical energy is not conserved––To examine conservative forces, To examine conservative forces,

nonconservative forces, and the law of nonconservative forces, and the law of energy conservationenergy conservation

––To determine force from potential To determine force from potential energyenergy

OverviewOverviewThe diver on page 241 hits the water fast, with The diver on page 241 hits the water fast, with a lot of kinetic energy. His kinetic energy a lot of kinetic energy. His kinetic energy increases by an amount equal to the work done increases by an amount equal to the work done on him.on him.There is a useful alternate way to think about There is a useful alternate way to think about work and kinetic energy.work and kinetic energy.This new approach uses ideas of potential This new approach uses ideas of potential energy, energy due to condition or position.energy, energy due to condition or position.The diver had stored energy, potential energy, The diver had stored energy, potential energy, before he left the board. That stored energy before he left the board. That stored energy was transferred into kinetic energy.was transferred into kinetic energy.WeWe’’ll investigate some cases where the sum of ll investigate some cases where the sum of a systema system’’s kinetic and potential energies is s kinetic and potential energies is constant during the systems motion.constant during the systems motion.

ObjectivesObjectivesSC.AP1.50.03SC.AP1.50.03 The student will The student will apply the concept of energy to the apply the concept of energy to the workwork--energy theorem, conservative energy theorem, conservative force fields, potential energy, force fields, potential energy, conservation of energy and power. conservation of energy and power. (CS 5.12.1, CLG 5.1.4*)(CS 5.12.1, CLG 5.1.4*) TATABURR.AP1.01BURR.AP1.01 The student will The student will score a 3 or better on the AP exam.score a 3 or better on the AP exam.

7.1 Gravitational Potential Energy7.1 Gravitational Potential EnergyWe learned in chapter 6 that during any We learned in chapter 6 that during any interaction the change in kinetic energy is equal interaction the change in kinetic energy is equal to the total work done on the particle by the to the total work done on the particle by the forces acting on it.forces acting on it.In many situations, forces acting on a system In many situations, forces acting on a system add stored energy, like depositing money into a add stored energy, like depositing money into a bank.bank.This stored energy is called potential energy This stored energy is called potential energy because it has the potential to do work.because it has the potential to do work.If a body falls without air resistance its kinetic If a body falls without air resistance its kinetic energy increases as its gravitational potential energy increases as its gravitational potential energy decreases. The workenergy decreases. The work--energy theorem energy theorem says that this is due to the weight doing work.says that this is due to the weight doing work.LetLet’’s use the works use the work--energy theorem to show energy theorem to show that these two descriptions are equivalent.that these two descriptions are equivalent.

Figure 7.1aFigure 7.1aConsider a body of mass Consider a body of mass mm that that moves along the vertical moves along the vertical yy--axis. axis. The forces acting on it are its The forces acting on it are its weight and possible other forces weight and possible other forces with the vector sum with the vector sum FFotherother..Find the work done by the weight Find the work done by the weight when the body drops from when the body drops from yy11 to to yy22. The weight and displacement . The weight and displacement are in the same direction, so the are in the same direction, so the work work WWgravgrav done on the body is done on the body is positive. The following positive. The following expression shows how we must expression shows how we must express the work in order for express the work in order for WWgravgrav to be positive.to be positive.

2121 )( mgymgyyywFsWgrav −=−==

Figure 7.1bFigure 7.1bThis expression also gives the This expression also gives the correct work if we increase correct work if we increase height from height from yy11 to to yy22..In that case the quantity (In that case the quantity (yy11 --yy22) is negative and ) is negative and WWgravgrav is is negative because force and negative because force and displacement are in opposite displacement are in opposite directions.directions.We can express the work done We can express the work done by the weight in terms of the by the weight in terms of the quantity quantity mgymgy at the beginning at the beginning and end of the displacement.and end of the displacement.

2121 )( mgymgyyywFsWgrav −=−==

Figure 7.1bFigure 7.1bThis quantity, the product of the weight This quantity, the product of the weight mgmg and and height height yy, is called the , is called the gravitational potential gravitational potential energyenergy, , U = mgyU = mgyThe change in The change in UU is its final minus its initial is its final minus its initial value, or value, or ΔΔU = UU = U22 –– UU11. We can express the . We can express the work done by the gravitational force during the work done by the gravitational force during the displacement from displacement from yy11 to to yy22 asas

The negative sign in front of The negative sign in front of ΔΔU is U is essentialessential..

UUUUUWgrav Δ−=−−=−= )( 1221

UUUUUWgrav Δ−=−−=−= )( 1221When the body moves up, y increases, the When the body moves up, y increases, the work is negative, and the gravitational potential work is negative, and the gravitational potential energy increases.energy increases.When the body moves down, y decreases, the When the body moves down, y decreases, the work is positive, and the gravitational potential work is positive, and the gravitational potential energy decreases.energy decreases.Decreasing Decreasing UU is like a withdrawal from the is like a withdrawal from the energy reservesenergy reserves, while increasing is a deposit., while increasing is a deposit.The units are the same as work, Joules.The units are the same as work, Joules.

2121 )( mgymgyyywFsWgrav −=−==

Conservation of Mechanical Energy Conservation of Mechanical Energy (Gravitational Forces Only)(Gravitational Forces Only)

Suppose the only force acting on the body in Suppose the only force acting on the body in 7.1 on page 242 is its weight, so 7.1 on page 242 is its weight, so FFotherother = 0. It = 0. It is freely falling with no air resistance.is freely falling with no air resistance.Let its speed at Let its speed at yy11 be be vv11 and let its speed at and let its speed at yy22bebe vv22..By the workBy the work--energy theorem, the total work energy theorem, the total work done on the body is equal to the change in done on the body is equal to the change in kinetic energy kinetic energy WWtottot = = ΔΔKK = = KK22 -- KK11

If gravity is the only force that acts on it then If gravity is the only force that acts on it then WWtottot = = WWgravgrav = = -- ΔΔUU = = UU11 –– UU22

Putting these together yields:Putting these together yields:ΔΔK = K = --ΔΔUU or or KK22 –– KK11 = U= U11 –– UU22

If only gravity does work:If only gravity does work:We can rewrite the equation as We can rewrite the equation as

KK11 + + UU11 = = KK22 + + UU22 or as or as ½½mvmv11

22 + + mgymgy11 = = ½½mvmv2222 + + mgymgy22

We define the sum of both potential We define the sum of both potential and kinetic energy to be the and kinetic energy to be the mechanical energy of the systemmechanical energy of the system, , EE. .

The mechanical energy of the system does The mechanical energy of the system does not change if only gravity does work in the not change if only gravity does work in the system: system: E = KE = K + + UUWe say that the mechanical energy is We say that the mechanical energy is conserved because the quantity is constant.conserved because the quantity is constant.Lets try to apply this to a common example:Lets try to apply this to a common example:

Height of a baseball from energy conservationHeight of a baseball from energy conservationYou throw a 0.145You throw a 0.145--kg baseball straight kg baseball straight up in the air, giving it an initial upward up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. velocity of magnitude 20.0 m/s. Use conservation of energy to find how Use conservation of energy to find how high it goes, ignoring air resistance.high it goes, ignoring air resistance.What is What is yy11? ? yy22??What is What is vv11? ? vv22??

Effect of Other ForcesEffect of Other ForcesWhen other forces act on a body, then When other forces act on a body, then FFotherother is is notnot zero.zero.Examples of possible other forces doing work on the Examples of possible other forces doing work on the object are friction and air resistance.object are friction and air resistance.The work that these forces do, WThe work that these forces do, Wotherother, is added to the , is added to the work done by the weight work done by the weight

WWtottot = W= Wgravgrav + W+ WotherotherEquating this to the change in kinetic energy (workEquating this to the change in kinetic energy (work--energy theorem) energy theorem)

WWtottot = = WWotherother + W+ Wgravgrav = K= K22 –– KK11We know that WWe know that Wgravgrav = U= U11 –– UU22

WWotherother + U+ U11 –– UU22 = K= K22 –– KK11KK11 + U+ U11 + W+ Wotherother = K= K22 +U+U22

The work done by all forces other than the gravitational The work done by all forces other than the gravitational force equals the change in the total mechanical energy force equals the change in the total mechanical energy of the system.of the system.If WIf Wotherother is positive, energy increases, if Wis positive, energy increases, if Wotherother is is negative, energy decreases. If no external forces act negative, energy decreases. If no external forces act on the object, energy does not change.on the object, energy does not change.

PSS: Problems Using Mechanical EnergyPSS: Problems Using Mechanical EnergyIdentifyIdentify

Decide whether the problem should be Decide whether the problem should be solved using energy methods, Newtonsolved using energy methods, Newton’’s s second law, or a combination.second law, or a combination.If the problem involves time, the energy If the problem involves time, the energy approach is approach is NOTNOT the best choice.the best choice.

Set UpSet UpChoose initial and final states (position Choose initial and final states (position and velocity) of the system.and velocity) of the system.Define your coordinate system (where y Define your coordinate system (where y = 0)= 0)

PSS: Problems Using Mechanical EnergyPSS: Problems Using Mechanical EnergyIdentify all nongravitational forces that Identify all nongravitational forces that do work.do work.List the known and unknown quantities.List the known and unknown quantities.

ExecuteExecuteWrite expressions for the initial and final Write expressions for the initial and final energies.energies.Relate these to WRelate these to Wotherother..

EvaluateEvaluateDoes your answer make physical sense?Does your answer make physical sense?Gravitational work is in Gravitational work is in ––ΔΔU, not WU, not Wotherother..

Work and energy throwing a baseballWork and energy throwing a baseballIn the previous example, suppose your In the previous example, suppose your hand moves up 0.50 m while you are hand moves up 0.50 m while you are throwing the ball, which leaves your throwing the ball, which leaves your hand with an upward velocity of 20.0 hand with an upward velocity of 20.0 m/s. Again ignore air resistance.m/s. Again ignore air resistance.

a)a) Assuming that your hand exerts a Assuming that your hand exerts a constant upward force on the ball, find constant upward force on the ball, find the magnitude of that force.the magnitude of that force.

b)b) Find the speed of the ball at a point Find the speed of the ball at a point 15.0 m above the point where it leaves 15.0 m above the point where it leaves your hand.your hand.

Gravitational Potential Energy for Gravitational Potential Energy for Motion Along a Curved PathMotion Along a Curved Path

Because the work done Because the work done by the force of gravity by the force of gravity can only be done along can only be done along the vertical, any the vertical, any horizontal displacements horizontal displacements are not considered.are not considered.Work done by other Work done by other forces must take the forces must take the horizontal displacements horizontal displacements into account.into account.

U for Motion Along a Curved PathU for Motion Along a Curved PathIn short:In short:–– The work is unaffected by any horizontal The work is unaffected by any horizontal

motion that may occur. motion that may occur. –– So we can use the same expression for So we can use the same expression for

gravitational potential energy whether gravitational potential energy whether the bodythe body’’s path is curved or straight.s path is curved or straight.

Energy in Projectile MotionEnergy in Projectile MotionA batter hits two identical baseballs A batter hits two identical baseballs with the same initial speed and with the same initial speed and height but different initial angles. height but different initial angles. Prove that at a given height Prove that at a given height hh, both , both balls have the same speed if air balls have the same speed if air resistance can be neglected.resistance can be neglected.

Maximum height of a projectile, using energy methodsMaximum height of a projectile, using energy methods

In Example 3.10 (page 95) we derived an In Example 3.10 (page 95) we derived an expression for the maximum height expression for the maximum height hh of a of a projectile launched with initial speed projectile launched with initial speed vvoo at angle at angle ααoo::

Derive this expression using energy Derive this expression using energy considerations.considerations.

gvh oo

2sin22 α

=

Revenge of the ThrockmortonRevenge of the ThrockmortonYour cousin Throckmorton skateboards down a curved Your cousin Throckmorton skateboards down a curved playground ramp. Treating Throcky and his skateboard as a playground ramp. Treating Throcky and his skateboard as a particle, he moves through a quarterparticle, he moves through a quarter--circle with radius circle with radius RR. The . The total mass of Throcky and his skateboard is 25.0total mass of Throcky and his skateboard is 25.0--kg. He starts kg. He starts from rest and there is no friction.from rest and there is no friction.Find his speed at the bottom of the ramp.Find his speed at the bottom of the ramp.Find the normal force that acts on him at the bottom of the curvFind the normal force that acts on him at the bottom of the curve.e.

Throckmorton with FrictionThrockmorton with FrictionIn the previous example, suppose that the ramp is not In the previous example, suppose that the ramp is not frictionless and that Throckyfrictionless and that Throcky’’s speed at the bottom is only s speed at the bottom is only 6.00 m/s.6.00 m/s.What work was done by the friction force acting on him? Use What work was done by the friction force acting on him? Use RR = 3.00 m.= 3.00 m.

An inclined plane with frictionAn inclined plane with frictionWe want to load a 12We want to load a 12--kg crate into a truck by kg crate into a truck by sliding it up a ramp 2.5 m long, inclined at sliding it up a ramp 2.5 m long, inclined at 3030oo. A worker, giving no thought to friction, . A worker, giving no thought to friction, calculates that he can get the crate up the calculates that he can get the crate up the ramp by giving it an initial speed of 5.0 m/s at ramp by giving it an initial speed of 5.0 m/s at the bottom and letting it go. But friction is the bottom and letting it go. But friction is notnotnegligible; the crate slides 1.6 m up the ramp, negligible; the crate slides 1.6 m up the ramp, stops, and slides back down.stops, and slides back down.

a)a) Assuming that the friction force acting on the Assuming that the friction force acting on the crate is constant, find its magnitude.crate is constant, find its magnitude.

b)b) How fast is the crate moving when it reaches How fast is the crate moving when it reaches the bottom of the ramp?the bottom of the ramp?

HomeworkHomeworkRead 241 Read 241 -- 2602609, 11, 13, 17, 199, 11, 13, 17, 19Read 260 Read 260 –– 27227235, 37, 43, 47, 49, 6335, 37, 43, 47, 49, 63

7.2: Elastic Potential Energy7.2: Elastic Potential EnergyWhen a spring or rubber band is When a spring or rubber band is deformed, work is done to deform the deformed, work is done to deform the object.object.This work stores energy in the object This work stores energy in the object which is released later. which is released later. When energy is stored in a deformable When energy is stored in a deformable object we call that energy object we call that energy elastic elastic potential energypotential energy..If an object returns to its original shape If an object returns to its original shape and size, it is called elastic.and size, it is called elastic.Ideal Spring: Ideal Spring: F = kxF = kx

Work Done on an Ideal SpringWork Done on an Ideal SpringIf the force required to keep an ideal spring If the force required to keep an ideal spring deformed a distance deformed a distance xx is is F = kxF = kx then the work then the work done on that spring to deform it from position done on that spring to deform it from position xx11 to position to position xx22 is:is:

From NewtonFrom Newton’’s third law, the two quantities of s third law, the two quantities of work are just negatives of each other (equal work are just negatives of each other (equal and opposite forces, same displacement) so the and opposite forces, same displacement) so the work done by the spring is:work done by the spring is:

212

1222

1 kxkxW −=

222

1212

1 kxkxWel −=

In figure 7.13, if In figure 7.13, if xx11 and and xx22 are positive and are positive and xx22>>xx11 the spring does the spring does negative work on the negative work on the block which moves in block which moves in the +the +xx--direction while direction while the spring pulls in the the spring pulls in the ––xx--direction. The block direction. The block slows down.slows down.If If xx22<<xx11 the spring the spring does positive work as it does positive work as it relaxes and the block relaxes and the block speeds up.speeds up.When the spring does When the spring does positive work, the block positive work, the block speeds up. speeds up. When the spring does When the spring does negative work, the negative work, the block slows down.block slows down.

Elastic Potential EnergyElastic Potential EnergyLike gravity, we can express the work done by Like gravity, we can express the work done by the spring as the energy stored in the spring: the spring as the energy stored in the spring: U=U=½½kxkx22

We can express the work done by the spring as We can express the work done by the spring as follows:follows:

When a spring is stretched, When a spring is stretched, WWelel is negative and is negative and UU increases: energy is stored in the spring.increases: energy is stored in the spring.When a spring is released, When a spring is released, WWelel is positive and is positive and UUdecreases: energy is released from the spring.decreases: energy is released from the spring.Negative values of Negative values of xx refer to a compressed refer to a compressed spring.spring.If the spring is deformed from its relaxed If the spring is deformed from its relaxed position, position, UU increases.increases.

UUUkxkxWel Δ−=−=−= 21222

1212

1

Elastic Potential and the WorkElastic Potential and the Work--Energy Energy TheoremTheorem

The workThe work--energy theorem states that the total energy theorem states that the total work done equals the change in kinetic energy, work done equals the change in kinetic energy, no matter what kind of forces are acting.no matter what kind of forces are acting.If the elastic force is the only force acting:If the elastic force is the only force acting:

Using the workUsing the work--energy theorem:energy theorem:

Rearranging:Rearranging:

Substituting:Substituting:

21 UUWW eltot −==

2112 UUWKKW eltot −==−=

2211 UKUK +=+222

1222

1212

1221

1kxmvkxmv +=+

In this case, mechanical energy is conserved as In this case, mechanical energy is conserved as well.well.This only holds for an ideal spring, where the This only holds for an ideal spring, where the force is force is kxkx and the spring is massless.and the spring is massless.If the spring had mass it would also have If the spring had mass it would also have kinetic energy. In the expression above, the kinetic energy. In the expression above, the kinetic energy belongs to the object attached to kinetic energy belongs to the object attached to the spring, not the spring itself.the spring, not the spring itself.We can neglect the kinetic energy of a spring if We can neglect the kinetic energy of a spring if its mass is much less than that of the body its mass is much less than that of the body attached to the spring.attached to the spring.This holds true for most springs we study.This holds true for most springs we study.

222

1222

1212

1221

1kxmvkxmv +=+

Work Done by Other ForcesWork Done by Other ForcesIf forces other than the elastic force do work on If forces other than the elastic force do work on the object we call that work the object we call that work WWotherother as before.as before.WWtottot = W= Welel + + WWotherother and the workand the work--energy energy theorem gives theorem gives WWelel + + WWotherother = K= K22 –– KK11

Work done by the spring is Work done by the spring is WWelel = = UU11 –– UU22 sosoKK11 + + UU11 + + WWotherother = = KK22 + + UU22

Substituting:Substituting:

The work done by all forces other than the The work done by all forces other than the elastic force equals the change in the total elastic force equals the change in the total mechanical energy mechanical energy E = KE = K + + UU of the system of the system where where UU is the elastic potential energy.is the elastic potential energy.

222

1222

1212

1221

1kxmvWkxmv other +=++

Gravitational and Elastic Potential EnergyGravitational and Elastic Potential EnergyThe previous equations are only valid if we only The previous equations are only valid if we only have one potential energy.have one potential energy.What happens if we have both?What happens if we have both?We can keep the same equations, but now U = We can keep the same equations, but now U = UUgravgrav + U+ Uelel..

The work done by all forces other than the The work done by all forces other than the gravitational force or elastic force equals the gravitational force or elastic force equals the change in the total mechanical energy E = K + change in the total mechanical energy E = K + U of the system, where U is the sum of the U of the system, where U is the sum of the gravitational potential energy and the elastic gravitational potential energy and the elastic potential energy.potential energy.

2,2,21,1,1 elgravotherelgrav UUKWUUK ++=+++

Motion with elastic potential energyMotion with elastic potential energyIn Fig 7.16a a glider with mass In Fig 7.16a a glider with mass mm = 0.200 kg sits on a = 0.200 kg sits on a frictionless horizontal air track, connected to a spring with frictionless horizontal air track, connected to a spring with fore constant fore constant kk = 5.00 N/m. You pull on the glider, = 5.00 N/m. You pull on the glider, stretching the spring 0.100m, and then release it with no stretching the spring 0.100m, and then release it with no initial velocity. The glider begins to move back toward its initial velocity. The glider begins to move back toward its equilibrium position.equilibrium position.What is its xWhat is its x--velocity when velocity when xx = 0.080 m?= 0.080 m?

Motion with elastic potential energy and Motion with elastic potential energy and work done by other forceswork done by other forces

For the system of the previous example, For the system of the previous example, suppose the glider is initially at rest at x suppose the glider is initially at rest at x = 0, with the spring un= 0, with the spring un--stretched. You stretched. You then apply a constant force then apply a constant force FF in the +xin the +x--direction with magnitude 0.610 N to the direction with magnitude 0.610 N to the glider.glider.What is the gliderWhat is the glider’’s velocity when it has s velocity when it has moved to x = 0.100 m?moved to x = 0.100 m?

Motion with elastic potential energy Motion with elastic potential energy after other forces have ceasedafter other forces have ceased

In Example 7.9, suppose the force In Example 7.9, suppose the force FF is is removed when the glider reaches the removed when the glider reaches the point x = 0.100 m.point x = 0.100 m.How much farther does the glider move How much farther does the glider move before coming to rest?before coming to rest?

Motion with gravitational, elastic, and Motion with gravitational, elastic, and friction forcesfriction forces

In a In a ““worstworst--casecase”” design scenario, a design scenario, a 20002000--kg elevator with broken kg elevator with broken cables is falling at 25 m/s when it cables is falling at 25 m/s when it first contacts a cushioning spring at first contacts a cushioning spring at the bottom of the shaft. The spring the bottom of the shaft. The spring is supposed to stop the elevator, is supposed to stop the elevator, compressing 3.00 m as it does so. compressing 3.00 m as it does so. During the motion a safety clamp During the motion a safety clamp applies a constant 17,000applies a constant 17,000--N N frictional force to the elevator.frictional force to the elevator.As a design consultant, you are As a design consultant, you are asked to determine what the force asked to determine what the force constant of the spring should be.constant of the spring should be.

7.3 Conservative and Nonconservative 7.3 Conservative and Nonconservative ForcesForces

In our discussions we have talked about In our discussions we have talked about ““storingstoring”” kinetic energy by converting it to kinetic energy by converting it to potential.potential.If If WWotherother = 0 then all = 0 then all ““storedstored”” potential can be potential can be converted to kinetic and vice versa.converted to kinetic and vice versa.The mechanical energy The mechanical energy EE, which is the sum of , which is the sum of kinetic and all forms of potential, is constant or kinetic and all forms of potential, is constant or conserved.conserved.We call the force that causes this two way We call the force that causes this two way conversion between kinetic and potential conversion between kinetic and potential energies a energies a conservative forceconservative force..Energy is Energy is ““depositeddeposited”” and can be and can be ““withdrawnwithdrawn””without energy loss.without energy loss.

Conservative ForceConservative ForceOne important aspect of conservative forces One important aspect of conservative forces is that the work done to move a body from is that the work done to move a body from one point to another is independent of the one point to another is independent of the path taken.path taken.The work done by gravity (if we neglect any The work done by gravity (if we neglect any nonconservative force) is the same for the nonconservative force) is the same for the blue, green and red paths.blue, green and red paths.

If a body moves around a If a body moves around a closed path the work closed path the work done by the gravitational done by the gravitational force is zero.force is zero.When the only forces that When the only forces that do work are conservative do work are conservative forces, the total forces, the total mechanical energy E = K mechanical energy E = K + U is constant.+ U is constant.

The work done by a conservative The work done by a conservative force always has these properties:force always has these properties:1.1. It can always be expressed as the It can always be expressed as the

difference between the initial and final difference between the initial and final values of a values of a potential energypotential energy function.function.

2.2. It is reversible.It is reversible.3.3. It is independent of the path of the body It is independent of the path of the body

and depends only on the starting and and depends only on the starting and ending points.ending points.

4.4. When the starting and ending points are When the starting and ending points are the same, the total work is zero.the same, the total work is zero.

Nonconservative ForcesNonconservative ForcesThe work done by nonconservative forces The work done by nonconservative forces cannot be expressed by a potential cannot be expressed by a potential energy function.energy function.The kinetic energy lost from these forces The kinetic energy lost from these forces cannot be recovered.cannot be recovered.The energy is not destroyed, merely The energy is not destroyed, merely dissipated. Nonconservative forces are dissipated. Nonconservative forces are sometimes referred to as dissipative sometimes referred to as dissipative forces.forces.Other nonconservative forces Other nonconservative forces increaseincreasekinetic energy, like rocket motors or the kinetic energy, like rocket motors or the food you eat.food you eat.

Frictional work depends on the pathFrictional work depends on the pathYou are rearranging your furniture and wish to move a 40.0 kg You are rearranging your furniture and wish to move a 40.0 kg futon 2.50 m across the room. However, the straightfuton 2.50 m across the room. However, the straight--line path line path is blocked by a heavy coffee table that you donis blocked by a heavy coffee table that you don’’t want to t want to move. Instead, you slide the futon in a dogleg path over the move. Instead, you slide the futon in a dogleg path over the floor; the doglegs are 2.00 m and 1.50 m long. floor; the doglegs are 2.00 m and 1.50 m long. Compared to the straightCompared to the straight--line path, how much more work must line path, how much more work must you do to push the futon in the dogleg path? The coefficient ofyou do to push the futon in the dogleg path? The coefficient ofkinetic friction is 0.200.kinetic friction is 0.200.

Conservative or Conservative or nonconservativenonconservative??In a certain region of space the force on an electron is In a certain region of space the force on an electron is

where C is a positive constant. The electron moves in a where C is a positive constant. The electron moves in a countercounter--clockwise direction around a square loop in the clockwise direction around a square loop in the xyxy--plane. The corners of the square are plane. The corners of the square are

ˆF Cxj=

Calculate the work done on the Calculate the work done on the electron by the force electron by the force FF during during one complete trip around the one complete trip around the square.square.Is this force conservative?Is this force conservative?

( ) ( ) ( ) ( ) ( ), 0,0 , 0, , , ,0x y L L L and L=

HomeworkHomeworkRead 241 Read 241 -- 2602609, 11, 13, 17, 199, 11, 13, 17, 19Read 260 Read 260 –– 27227235, 37, 43, 47, 49, 6335, 37, 43, 47, 49, 63

Internal EnergyInternal EnergyWe cannot express the work done by We cannot express the work done by nonconservative forces (NCF) in terms of nonconservative forces (NCF) in terms of potential energy, but we can express it in potential energy, but we can express it in terms of a different kind of energy.terms of a different kind of energy.When a car skids to a halt both the tires When a car skids to a halt both the tires and road surface heat up. and road surface heat up. This type of energy is called This type of energy is called internal internal energyenergy..From precise measurements it is found From precise measurements it is found that the increase in temperature equals that the increase in temperature equals the work done by friction.the work done by friction.

ΔΔUUintint = = --WWotherother

The Law of Conservation of EnergyThe Law of Conservation of EnergyΔΔUUintint is the change in internal energy. If is the change in internal energy. If we substitute this into our expression for we substitute this into our expression for conservation of energy:conservation of energy:

KK11++UU11 -- ΔΔUUintint = = KK22 + + UU22

OrOrΔΔKK + + ΔΔUU + + ΔΔUUintint = 0= 0

Energy is neither created nor destroyed, Energy is neither created nor destroyed, just transferred from one form to just transferred from one form to another.another.

7.4 Force and Potential Energy7.4 Force and Potential EnergyFor our two conservative forces we have For our two conservative forces we have been able to determine a potential been able to determine a potential energy function from the behavior of the energy function from the behavior of the force.force.For gravity: For gravity: FFyy = = --mgmg UU(y)(y) = mgy= mgyFor ideal spring: For ideal spring: FFxx = = --kxkx UU(x)(x) = = ½½kxkx22

In some cases you will be given an In some cases you will be given an expression of the potential energy and expression of the potential energy and you must find the corresponding force.you must find the corresponding force.

Finding Force from PotentialFinding Force from PotentialLetLet’’s consider motion along a straight s consider motion along a straight line, with initial position line, with initial position xx..The work done by a conservative force:The work done by a conservative force:

W = W = --ΔΔUUApply this to a small displacement Apply this to a small displacement ΔΔxx. . The work done by the force The work done by the force FFxx((xx) is ) is approximatelyapproximately FFxx((xx))ΔΔxx so:so:FFxx((xx))ΔΔxx = = --ΔΔUU and and FFxx((xx) = ) = --((ΔΔUU//ΔΔxx))If we take the limit of If we take the limit of xx 0 we get an 0 we get an exact relation.exact relation.

( ) ( )dx

xdUxFx −=When the force is in the When the force is in the positive direction, the change in positive direction, the change in potential is negative. When the potential is negative. When the force is in the negative force is in the negative direction, the change in direction, the change in potential is positive.potential is positive.Think about the conservative Think about the conservative forces that we have. Does this forces that we have. Does this make sense?make sense?As a check, calculate the force As a check, calculate the force corresponding to potential of an corresponding to potential of an ideal spring: ideal spring: UU((xx) = ) = ½½kxkx22

An electric force and its potential energyAn electric force and its potential energyAn electrically charged particle is held at An electrically charged particle is held at rest at the point rest at the point xx = 0, while a second = 0, while a second particle with equal charge is free to move particle with equal charge is free to move along the positive xalong the positive x--axis. The potential axis. The potential energy of the system is energy of the system is

UU((xx) = ) = CC//xxwhere C is a positive constant that depends where C is a positive constant that depends on the magnitude of the charges.on the magnitude of the charges.Derive an expression for the Derive an expression for the xx--component component of force acting on the movable charge, as a of force acting on the movable charge, as a function of its position. function of its position.

Force and Potential Energy in 3Force and Potential Energy in 3--DDConservative forces are not limited to one Conservative forces are not limited to one dimension. A particle may move in the dimension. A particle may move in the xx, , yyand and zz directions and forces may act on it with directions and forces may act on it with components components FFxx, , FFyy and and FFzz..Following the same reasoning as with straight Following the same reasoning as with straight line motion, the forces can be expressed in line motion, the forces can be expressed in terms of the potential energy:terms of the potential energy:

xUFx Δ

Δ−=

zUFz Δ

Δ−=

yUFy Δ

Δ−=

If we take the limit as If we take the limit as ΔΔxx, , ΔΔyy and and ΔΔzz 0 then 0 then the ratios become derivatives.the ratios become derivatives.Because Because UU may include all three variables we may include all three variables we need to remember that only one variable need to remember that only one variable changes at a time.changes at a time.We need a We need a partial derivative. partial derivative. Using unit vectors:Using unit vectors:

This indicates the gradient.This indicates the gradient.

Exact relations by limitsExact relations by limits

zUF

yUF

xUF

z

y

x

∂∂

−=

∂∂

−=

∂∂

−=

UF

kzUj

yUi

xUF

∇−=

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

−= ˆˆˆ

Check ItCheck ItFind the force related with the Find the force related with the potential potential UU((yy) = ) = mgymgy by taking the by taking the gradient of the potential.gradient of the potential.I.E. Find I.E. Find FFxx, , FFyy and and FFzz..

Force and potential energy in two Force and potential energy in two dimentionsdimentionsA puck slides on a level, frictionless airA puck slides on a level, frictionless air--hockey hockey table. The coordinates of the puck are table. The coordinates of the puck are xx and and yy. . It is acted on by a conservative force described It is acted on by a conservative force described by the potentialby the potential--energy function:energy function:

Derive an expression for the force acting on the Derive an expression for the force acting on the puck, and find an expression for the magnitude puck, and find an expression for the magnitude of the force as a function of position.of the force as a function of position.

( ) 2221, yxkyxU +=

7.5 Energy Diagrams7.5 Energy DiagramsWe can We can gleanglean a lot of information from the graph of an objecta lot of information from the graph of an object’’s s potential energy.potential energy.

HomeworkHomeworkRead 241 Read 241 -- 2602609, 11, 13, 17, 199, 11, 13, 17, 19Read 260 Read 260 –– 27227235, 37, 43, 47, 49, 6335, 37, 43, 47, 49, 63