Chapter 7Chapter 7

EnergyEnergyandandEnergy TransferEnergy Transfer

February 22, 2006February 22, 2006

Subject to much change

HAPPY BIRTHDAY HAPPY BIRTHDAY GEORGE!GEORGE!

Kalendar

Today we start the new TOPIC OF ENERGY No Quiz on Friday, but there MAY be one on

Monday The BAD NEWS:

EXAM #2 will be on March 3 (Friday)

ENERGY

We use energy to walk, run or even sleep We use energy when we lift a weight We use energy when we drive a car We even use energy to THINK! BUT …..

Introduction to EnergyIntroduction to Energy

The concept of energy is one of The concept of energy is one of the most important topics in the most important topics in sciencescience

Every physical process that Every physical process that occurs in the Universe involves occurs in the Universe involves energy and energy transfers or energy and energy transfers or transformationstransformations

Energy is not easily definedEnergy is not easily defined

Systems

A system is a small portion of the Universe We identify a number of particles or objects and

draw a sphere around them There are no forces acting on anything inside the

sphere from outside the sphere We will ignore the details outside of the sphere.

A critical skill is to identify the system

Valid System

A valid system may be a single object or particle be a collection of objects or particles be a region of space vary in size and shape

Environment

There is a system boundary around the system The boundary is an imaginary surface It does not necessarily correspond to a physical

boundary The boundary divides the system from the

environment The environment is “the rest of the Universe”

Work

The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude, F, of the force, the magnitude r of the displacement of the point of application of the force, and cos where is the angle between the force and the displacement vectors

Working, Working, Working

WORK = Component of the applied force x the displacement=

Fcos() x r

Work, cont.

W = F r cos

The displacement is that of the point of application of the force

A force does no work on the object if the force does not move through a displacement

The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application

rF (Later for the dot!)

Work Example

The normal force, n, and the gravitational force, m g, do no work on the object cos = cos 90° = 0

The force F does do work on the object

Same amount as in the previous overhead

More About Work

The system and the environment must be determined when dealing with work The environment does work on the system

Work by the environment on the system The sign of the work depends on the direction of F

relative to r Work is positive when projection of F onto r is in the same

direction as the displacement Work is negative when the projection is in the opposite

direction

Units of Work

Work is a scalar quantity The unit of work is a joule (J)

1 joule = 1 newton . 1 meter J = N · m

A block of mass 2.50 kg is pushed 2.20 m along a frictionless horizontal table by a constant 16.0-N force directed 25.0 below the horizontal. Determine the work done on the block by (a) the applied force, (b) the normal force exerted by the table, and (c) the gravitational force. (d) Determine the total work done on the block.

A raindrop of mass 3.35 10–5 kg falls vertically at constant speed under the influence of gravity and air resistance. Model the drop as a particle. As it falls 100 m, what is the work done on the raindrop (a) by the gravitational force and (b) by air resistance?

Work Is An Energy Transfer

This is important for a system approach to solving a problem

If the work is done on a system and it is positive, energy is transferred to the system

If the work done on the system is negative, energy is transferred from the system

Work Is An Energy Transfer, cont If a system interacts with its environment, this

interaction can be described as a transfer of energy across the system boundary This will result in a change in the amount of

energy stored in the system

ShejuleShejule

Continue to work on energy.Continue to work on energy. Exam on March 3Exam on March 3rdrd.. Material … as far as we get by Material … as far as we get by

March 1March 1stst.. Mucho WebAssign StuffMucho WebAssign Stuff

LAST TIMELAST TIME

We defined the “Dot Product”We defined the “Dot Product” We defined WORKWe defined WORK We discussed the “system” and We discussed the “system” and

the “environment”the “environment” System + Environment = Entire System + Environment = Entire

Universe (Dumb concept!)Universe (Dumb concept!)

WORK

)cos(rFW rF

F is the NET force acting on the block.

Scalar or DOT Product of Two Vectors The scalar product of

two vectors is written as A . B It is also called the dot

product A . B = A B cos

is the angle between A and B

Scalar Product Properties

The scalar product is commutative A . B = B . A

The scalar product obeys the distributive law of multiplication

A . (B + C) = A . B + A . C

Dot Products of Unit Vectors

Using component form with A and B:

0kjkiji

1kkjjii

zzyyxx

zyx

zyx

BABABABA

kBjBiBB

kAjAiAA

A force acts on a particle that undergoes a displacement .

Find (a) the work done by the force on the particle and (b) the angle between F and r.

F 6ˆ i – 2ˆ j N

r 3ˆ i ˆ j m

Work Done by a Varying Force

dxxFWf

i

x

x

x )(

f

i

x

xx

W F x

11. The force acting on a particle varies as in Figure P7.11. Find the work done by the force on the particle as it moves (a) from x = 0 to x = 8.00 m, (b) from x = 8.00 m to x = 10.0 m, and (c) from x = 0 to x = 10.0 m.

Work Done By Multiple Forces AGAINAGAIN … If more than one force acts on a

system and the system can be modeled as a particle, the total work done on the system is the work done by the net force

f

i

x

net xxW W F dx

Hooke’s Law

The force exerted by the spring is

Fs = - kx x is the position of the block with respect to the equilibrium position (x = 0) k is called the spring constant or force constant and measures the

stiffness of the spring This is called Hooke’s Law

Hooke’s Law, cont.

When x is positive (spring is stretched), F is negative

When x is 0 (at the equilibrium position), F is 0

When x is negative (spring is compressed), F is positive

Work Done by a Spring Identify the block as the

system Calculate the work as the

block moves from xi = - xmax to xf = 0

The total work done as the block moves from

–xmax to xmax is zero

max

0 2max

1

2

f

i

x

s xx xW F dx kx dx kx

ENERGY IS STORED IN THE SPRING AND THEN RECOVERED AND THEN STORED AND THEN RECOVERED AND THEN STORED AND THEN RECOVERED AND THEN STORED AND THEN RECOVERED AND THEN ….

Spring with an Applied Force

Suppose an external agent, Fapp, stretches the spring

The applied force is equal and opposite to the spring force

Fapp = -Fs = -(-kx) = kx

Work done by Fapp is equal to ½ kx2

max

19. If it takes 4.00 J of work to stretch a Hooke's-law spring 10.0 cm from its unstressed length, determine the extra work required to stretch it an additional 10.0 cm.

21. A light spring with spring constant 1 200 N/m is hung from an elevated support. From its lower end a second light spring is hung, which has spring constant 1 800 N/m. An object of mass 1.50 kg is hung at rest from the lower end of the second spring. (a) Find the total extension distance of the pair of springs. (b) Find the effective spring constant of the pair of springs as a system. We describe these springs as in series.

Consider the following:

F

22

2

1

2

1if

v

v

x

x

x

x

x

x

x

x

mvmvW

mvdvdt

dxmdvW

dxdt

dvmmadxW

FdxW

f

i

f

i

f

i

f

i

f

i

Kinetic Energy

Kinetic Energy is the energy of a particle due to its motion K = ½ mv2

K is the kinetic energy m is the mass of the

particle v is the speed of the

particle A change in kinetic energy

is one possible result of doing work to transfer energy into a system

22

2

1

2

1if mvmvW

With no friction or other strange forces,the work done by a force on a particle (or system of particles) = the change inthe particles kinetic energy.

Work-Kinetic Energy Theorem The Work-Kinetic Energy Principle states W = Kf –

Ki = K In the case in which work is done on a system and

the only change in the system is in its speed, the work done by the net force equals the change in kinetic energy of the system.

We can also define the kinetic energy K = ½ mv2

Nonisolated System

A nonisolated system is one that interacts with or is influenced by its environment An isolated system would not interact with its

environment The Work-Kinetic Energy Theorem can be

applied to nonisolated systems

BREAK POINTBREAK POINT

February 27, 2006February 27, 2006

Stuff HappensStuff Happens

Today and WednesdayToday and Wednesday– More on potential energyMore on potential energy– some material from the next chaptersome material from the next chapter

FridayFriday– EXAMINATION #2EXAMINATION #2– All material since last exam.All material since last exam.– You should be studying by now.You should be studying by now.

Internal Energy

The energy associated with an object’s temperature is called its internal energy, Eint

In this example, the surface is the system

The friction does work and increases the internal energy of the surface

FRICTIONFRICTION

Potential Energy

Potential energy is energy related to the configuration of a system in which the components of the system interact by forces

Examples include: elastic potential energy – stored in a spring gravitational potential energy electrical potential energy

Conservation of EnergyConservation of Energy

Energy is conserved !!!! This means that energy cannot be created or

destroyed If the total amount of energy in a system changes,

it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer

A 2 100-kg pile driver is used to drive a steel I-beam into the ground. The pile driver falls 5.00 m before coming into contact with the top of the beam, and it drives the beam 12.0 cm farther into the ground before coming to rest. Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest.

31. A 40.0-kg box initially at rest is pushed 5.00 m along a rough, horizontal floor with a constant applied horizontal force of 130 N. If the coefficient of friction between box and floor is 0.300, find (a) the work done by the applied force, (b) the increase in internal energy in the box-floor system due to friction, (c) the work done by the normal force, (d) the work done by the gravitational force, (e) the change in kinetic energy of the box, and (f) the final speed of the box.

63. The ball launcher in a pinball machine has a spring that has a force constant of 1.20 N/cm (Fig. P7.63). The surface on which the ball moves is inclined 10.0° with respect to the horizontal. If the spring is initially compressed 5.00 cm, find the launching speed of a 100-g ball when the plunger is released. Friction and the mass of the plunger are negligible.

32. A 2.00-kg block is attached to a spring of force constant 500 N/m as in Figure 7.10. The block is pulled 5.00 cm to the right of equilibrium and released from rest. Find the speed of the block as it passes through equilibrium if (a) the horizontal surface is frictionless and (b) the coefficient of friction between block and surface is 0.350.

35. A sled of mass m is given a kick on a frozen pond. The kick imparts to it an initial speed of 2.00 m/s. The coefficient of kinetic friction between sled and ice is 0.100. Use energy considerations to find the distance the sled moves before it stops.

Power

The time rate of energy transfer is called power

The average power is given by

when the method of energy transfer is work

WP

t

Instantaneous Power

The instantaneous power is the limiting value of the average power as t approaches zero

This can also be written as

lim0t

W dWP

t dt

dW drP F F v

dt dt

Power Generalized

Power can be related to any type of energy transfer In general, power can be expressed as

dE/dt is the rate rate at which energy is crossing the boundary of the system for a given transfer mechanism

dEP

dt

Units of Power

The SI unit of power is called the watt 1 watt = 1 joule / second = 1 kg . m2 / s2

A unit of power in the US Customary system is horsepower 1 hp = 746 W

Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x106 J

49. A 4.00-kg particle moves along the x axis. Its position varies with time according to x = t + 2.0t3, where x is in meters and t is in seconds. Find (a) the kinetic energy at any time t, (b) the acceleration of the particle and the force acting on it at time t, (c) the power being delivered to the particle at time t, and (d) the work done on the particle in the interval t = 0 to t = 2.00 s.

BREAKBREAK

EXAM #2 This Friday!

Let’s Review with some

Problems

24. A 5.00-kg object placed on a frictionless, horizontal table is connected to a cable that passes over a pulley and then is fastened to a hanging 9.00-kg object, as in Figure P5.24. Draw free-body diagrams of both objects. Find the acceleration of the two objects and the tension in the string.

In Figure P5.29, the man and the platform together weigh 950 N. The pulley can be modeled as frictionless. Determine how hard the man has to pull on the rope to lift himself steadily upward above the ground. (Or is it impossible? If so, explain why.)

34. An object of mass m1 on a frictionless horizontal table is connected to an object of mass m2 through a very light pulley P1 and a light fixed pulley P2 as shown in Figure P5.34. (a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation between these accelerations? Express (b) the tensions in the strings and (c) the accelerations a1 and a2 in terms of the masses m1, and m2, and g.

44. Three objects are connected on the table as shown in Figure P5.44. The table is rough and has a coefficient of kinetic friction of 0.350. The objects have masses 4.00 kg, 1.00 kg and 2.00 kg, as shown, and the pulleys are frictionless. Draw free-body diagrams of each of the objects. (a) Determine the acceleration of each object and their directions. (b) Determine the tensions in the two cords.

68. Two blocks of mass 3.50 kg and 8.00 kg are connected by a massless string that passes over a frictionless pulley (Fig. P5.68). The inclines are frictionless. Find (a) the magnitude of the acceleration of each block and (b) the tension in the string.

19. A roller coaster car (Fig. P6.19) has a mass of 500 kg when fully loaded with passengers. (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at B and still remain on the track?

25. A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later is stopping, the scale reading is 391 N. Assume the magnitude of the acceleration is the same during starting and stopping, and determine (a) the weight of the person, (b) the person's mass, and (c) the acceleration of the elevator.

7. A crate of eggs is located in the middle of the flat bed of a pickup truck as the truck negotiates an unbanked curve in the road. The curve may be regarded as an arc of a circle of radius 35.0 m. If the coefficient of static friction between crate and truck is 0.600, how fast can the truck be moving without the crate sliding?

8. Find the scalar product of the vectors in Figure P7.8.

15. When a 4.00-kg object is hung vertically on a certain light spring that obeys Hooke's law, the spring stretches 2.50 cm. If the 4.00-kg object is removed, (a) how far will the spring stretch if a 1.50-kg block is hung on it, and (b) how much work must an external agent do to stretch the same spring 4.00 cm from its unstretched position?

20. A small particle of mass m is pulled to the top of a frictionless half-cylinder (of radius R) by a cord that passes over the top of the cylinder, as illustrated in Figure P7.20. (a) If the particle moves at a constant speed, show that F = mgcos. (Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times.) (b) By directly integrating W = Fdr, find the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder.

49. A 4.00-kg particle moves along the x axis. Its position varies with time according to x = t + 2.0t3, where x is in meters and t is in seconds. Find (a) the kinetic energy at any time t, (b) the acceleration of the particle and the force acting on it at time t, (c) the power being delivered to the particle at time t, and (d) the work done on the particle in the interval t = 0 to t = 2.00 s.

Potential EnergyPotential Energy

~March 7, 2006~March 7, 2006

Potential (Stored) Energy

Potential energy is the energy associated with the configuration of a system of objects that exert forces on each other This can be used only with conservative forces

Conservative forces are NOT Republicans When conservative forces act within an isolated system,

the kinetic energy gained (or lost) by the system as its members change their relative positions is balanced by an equal loss (or gain) in potential energy. This is Conservation of Mechanical Energy

Types of Potential Energy

There are many forms of potential energy, including: Gravitational Electromagnetic Chemical Nuclear

One form of energy in a system can be converted into another Nuclear heat easy Heat Nuclear probably impossible!

Conversion from one type to another type of energy is not always reversible.

Systems with Multiple Particles We can extend our definition of a system to

include multiple objects The force can be internal to the system The kinetic energy of the system is the

algebraic sum of the kinetic energies of the individual objects Sometimes, the kinetic energy of one of the

objects may be negligible

System Example This system consists of Earth and a

book Do work on the system by lifting the

book through y The work done by you is mg(yb – ya) At the top it is at rest. The amount of work that you did is

called the potential energy of the system with respect to the ground.

The PE’s initial value is mgya

The FINAL value is mgyb

The difference is the work done.

Let’s drop the book from yb and see what it is doing at ya.

0

)(220

20

2

v

yygvv abv

Multiply by m and divide by 2:

)(2

1 2abf yymgmv

The decrease in Potential Energy equals the increase in Kinetic Energy

Potential Energy

The energy storage mechanism is called potential energy

A potential energy can only be associated with specific types of forces (conservative)

Potential energy is always associated with a system of two or more interacting objects

Gravitational Potential Energy Gravitational Potential Energy is associated

with an object at a given distance above Earth’s surface

Assume the object is in equilibrium and moving at constant velocity

The work done on the object is done by Fapp and the upward displacement is

ˆy r j

Gravitational Potential Energy, cont

The quantity mgy is identified as the gravitational potential energy, Ug Ug = mgy

Units are joules (J)

app

ˆ ˆ( ) b a

b a

W

W mg y y

W mgy mgy

F r

j j

Energy Problems

Draw a diagram of the situation. ESTABLISH AN ORIGIN. THE POTENTIAL ENERGY OF A PARTICAL

OF MASS M IS ALWAYS MEASURED WITH RESPECT TO THIS ORIGIN.

The potential energy of a particle is defined as being ZERO when it is at the origin.

At some height above the origin, the value of the PE is mgh.

Gravitational Potential Energy, final

The gravitational potential energy depends only on the vertical height of the object above Earth’s surface

In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero The choice is arbitrary because you normally need

the difference in potential energy, which is independent of the choice of reference configuration

Conservation of Mechanical Energy The mechanical energy of a system is the algebraic

sum of the kinetic and potential energies in the system Emech = K + Ug

The statement of Conservation of Mechanical Energy for an isolated system is Kf + Uf = Ki+ Ui An isolated system is one for which there are no energy

transfers across the boundary

A bead slides without friction around a loop-the-loop (Fig. P8.5). The bead is released from a height h = 3.50R. (a) What is its speed at point A? (b) How large is the normal force on it if its mass is 5.00 g?

Let’s look at the more general case.

y=0(origin)

y=h

m

We do work tomove mass toy=h. W=mgh

F

m

m

W=mgW=mg

W=0

ANY PATH

Can be broken up into a series of very small vertical moves and horizontal moves.

The horizontal moves require no work. The force is at right angles to the motion. Dot product is

zero. The vertical moves are

mghFhFFW

Conservation of Mechanical Energy, example

Look at the work done by the book as it falls from some height to a lower height

Won book = Kbook

Also, W = mgyb – mgya

So, K = -ug

rK +ug=0

Elastic Potential Energy

Elastic Potential Energy is associated with a spring

The force the spring exerts (on a block, for example) is Fs = - kx

The work done by an external applied force on a spring-block system is W = ½ kxf

2 – ½ kxi2

The work is equal to the difference between the initial and final values of an expression related to the configuration of the system

Elastic Potential Energy, cont This expression is the

elastic potential energy: Us = ½ kx2

The elastic potential energy can be thought of as the energy stored in the deformed spring

The stored potential energy can be converted into kinetic energy

Elastic Potential Energy, final The elastic potential energy stored in a spring is

zero whenever the spring is not deformed (U = 0 when x = 0) The energy is stored in the spring only when the spring is

stretched or compressed The elastic potential energy is a maximum when the

spring has reached its maximum extension or compression

The elastic potential energy is always positive x2 will always be positive

The Bindell Conservation of Energy Equation

The sum of the Kinetic Energy andthe potential energy of the system

before a sequence of events

Is Equal to

The sum of the Kinetic Energy,the potential energy of the systemand the Energy Lost to friction orwitchcraft.

An object of mass m starts from rest and slides a distance d down a frictionless incline of angle . While sliding, it contacts an unstressed spring of negligible mass as shown in Figure P8.10. The object slides an additional distance x as it is brought momentarily to rest by compression of the spring (of force constant k). Find the initial separation d between object and spring.

CAN WE ADD FRICTION???

Problem Solving Strategy – Conservation of Mechanical Energy

Define the isolated system and the initial and final configuration of the system The system may include two or more

interacting particles The system may also include springs or other

structures in which elastic potential energy can be stored

Also include all components of the system that exert forces on each other

Problem-Solving Strategy, 2

Identify the configuration for zero potential energy Include both gravitational and elastic potential

energies If more than one force is acting within the system,

write an expression for the potential energy associated with each force

Problem-Solving Strategy, 3

If friction or air resistance is present, mechanical energy of the system is not conserved

Use energy with non-conservative forces instead

Problem-Solving Strategy, 4

If the mechanical energy of the system is conserved, write the total energy as Ei = Ki + Ui for the initial configuration

Ef = Kf + Uf for the final configuration

Since mechanical energy is conserved, Ei = Ef and you can solve for the unknown quantity

Two objects are connected by a light string passing over a light frictionless pulley as in Figure P8.13. The object of mass m1 is released from rest at height h. Using the principle of conservation of energy, (a) determine the speed of m2 just as m1 hits the ground. (b) Find the maximum height to which m2 rises.

A particle of mass m = 5.00 kg is released from point A and slides on the frictionless track shown in Figure P8.24. Determine (a) the particle's speed at points B and C and (b) the net work done by the gravitational force in moving the particle from A to C.

A block slides down a curved frictionless track and then up an inclined plane as in Figure P8.48. The coefficient of kinetic friction between block and incline is k. Use energy methods to show that the maximum height reached by the block is

ymax

h1 k cot

A 10.0-kg block is released from point A in Figure P8.57. The track is frictionless except for the portion between points B and C , which has a length of 6.00 m. The block travels down the track, hits a spring of force constant 2 250 N/m, and compresses the spring 0.300 m from its equilibrium position before coming to rest momentarily. Determine the coefficient of kinetic friction between the block and the rough surface between B and C.

Conservation of Energy, (Pendulum) As the pendulum swings, there

is a continuous change between potential and kinetic energies

At A, the energy is potential At B, all of the potential energy

at A is transformed into kinetic energy Let zero potential energy be

at B At C, the kinetic energy has

been transformed back into potential energy

Conservative Forces

The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle

The work done by a conservative force on a particle moving through any closed path is zero A closed path is one in which the beginning and

ending points are the same

Nonconservative Forces

A nonconservative force does not satisfy the conditions of conservative forces

Nonconservative forces acting in a system cause a change in the mechanical energy of the system

Nonconservative Forces

The work done against friction is greater along the red path than along the blue path

Because the work done depends on the path, friction is a nonconservative force

Nonconservative Forces (Connected Blocks)

The system consists of the two blocks, the spring, and Earth

Gravitational and potential energies are involved

The kinetic energy is zero if our initial and final configurations are at rest

Connected Blocks, cont

Block 2 undergoes a change in gravitational potential energy

The spring undergoes a change in elastic potential energy

The coefficient of kinetic energy can be measured

Conservative Forces and Potential Energy Define a potential energy function, U, such

that the work done by a conservative force equals the decrease in the potential energy of the system

The work done by such a force, F, is

U is negative when F and x are in the same direction

f

i

x

C xxW F dx U

Conservative Forces and Potential Energy The conservative force is related to the potential

energy function through

The x component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to x

x

dUF

dx

Conservative Forces and Potential Energy – Check Look at the case of a deformed spring

This is Hooke’s Law

21

2s

s

dU dF kx kx

dx dx

Energy Diagrams and Equilibrium

Motion in a system can be observed in terms of a graph of its position and energy

In a spring-mass system example, the block oscillates between the turning points, x = ±xmax

The block will always accelerate back toward x = 0

Energy Diagrams and Stable Equilibrium The x = 0 position is one of

stable equilibrium Configurations of stable

equilibrium correspond to those for which U(x) is a minimum

x=xmax and x=-xmax are called the turning points

Energy Diagrams and Unstable Equilibrium Fx = 0 at x = 0, so the

particle is in equilibrium For any other value of x, the

particle moves away from the equilibrium position

This is an example of unstable equilibrium

Configurations of unstable equilibrium correspond to those for which U(x) is a maximum