23 work, energy and power 20130909

7/29/2019 23 Work, Energy and Power 20130909

1/32

2.3 Work, Energy and Power

Kari Eloranta2013

Jyvskyln Lyseon lukio

International Baccalaureate

September 11, 2013

Kari Eloranta 2013 (Jyvskyln Lyseon lukio International Baccalaureate)

2.3 Work, Energy and Power September 11, 2013 1 / 29
http://find/


2/32

2.3.1 Introduction to Work Done by Force on Object

When an object moves in the direction of a force acting on the object,the force transfers energy to the object.

Work W is a quantity that gives the amount of energy transferred bythe force

#

F. The SI unit of work is the unit of energy, 1 J (joule).

When the work done by a force on an object is positive (W> 0), theforce transfers energy to the object.

When the work done by a force on an object is negative (W< 0), theforce transfers energy away from the object.

For example, the gravitational force accelerates a falling apple. Thework done by the gravitational force on the apple is positive. The airresistance opposes the motion. The work done by the air resistance on

the apple is negative. Kari Eloranta 2011 (Jyvskyln Lyseon lukio2.3 Work, Energy and Power

September 11, 2013 2 / 29
http://find/


3/32

2.3.1 Definition of Work

Definition of WorkWork is the product of a force on an object, and the displacement of theobject in the direction of the force.

WorkWork W done by a constant force

#

F acting on an object is

W= F scos (1)

where scos is the displacement of the object in the direction of the force.

In Equation 1 the symbol (greek theta) is the angle between thedisplacement vector #s and force

#

F.

Kari Eloranta 2011 (Jyvskyln Lyseon lukio2.3 Work, Energy and Power

September 11, 2013 3 / 29
http://find/


4/32

2.3.4 Introduction to Kinetic Energy

Assume that an object moves to the right along a horizontal surface suchthat its speed increases uniformly at constant acceleration. According toNewtons Second Law of Motion, the net force

#

F that causes theaccelerated motion points in the direction of acceleration. If we denote thedisplacement of the object by #s, the work done by the net force on theobject is

W=F scos0 =mas1=mv

ts. (2)

However, in uniformly accelerated motion the distance travelled (magnitudeof displacement) is s= vavt=12 (v+v0)t. Substituting this expression

and the change in velocity v= vv0 into Equation 2 we have

Kari Eloranta 2011 (Jyvskyln Lyseon lukio2.3 Work, Energy and Power September 11, 2013 4 / 29
http://goforward/http://find/http://goback/


5/32

2.3.4 Introduction to Kinetic Energy

mvv0

t

1

2(v+v0)t=

1

2m(vv0)(v+v0)=

1

2m(v2v20 ).

The quantity 12m(v2v20)=

12mv

2

12mv

20 is defined as the change in

kinetic energy of the object.

We conclude by stating an important principle in physics that is not a partof IB syllabus, but is often useful in problem solving.

Definition of Work-Energy Theorem (NOT IBO)The work done by the net force on the object is equal to the change in thekinetic energy of the object.

http://find/


6/32

2.3.4 Definition of Kinetic Energy EK

Kinetic energy is energy that an object has due to its motion.

Definition of Kinetic Energy (NOT IBO)

The kinetic energy of an object is the energy the object has due to itsmotion in a reference frame with respect to the speed of the object ismeasured.

Kinetic Energy EK

If an object moves by the speed of v, its kinetic energy is

EK=1

2

mv2 (3)

where m is the mass of the object.

The unit of kinetic energy is [EK]= [m][v]2= kg m2 s2 = 1J (joule).

Kinetic energy is a scalar quantity. Kari Eloranta 2011 (Jyvskyln Lyseon lukio2.3 Work, Energy and Power September 11, 2013 6 / 29
http://find/


7/32

Example on Work Done by Gravity

Example

Your physics book falls from rest down to the floor from a 82 cm high table.a) Calculate the work done by the gravitational force on the book.b) State the change in the kinetic energy of the object.c) Where does the energy come from?

The mass of the book is 1450 g. You may ignore the effect of air resistance.

http://find/http://goback/


8/32


Example

Your physics book falls from rest down to the floor from a 82 cm high table.a) Calculate the work done by the gravitational force on the book.b) State the change in the kinetic energy of the object.c) Where does the energy come from?

The mass of the book is 1450 g. You may ignore the effect of air resistance.

a) Because the gravitational force on the book is G=mg, the work doneby the gravity on the book is

W= F scos0 =Gh1

=mgh= 1.450kg9.81m s20.82m 12J



9/32


Example

Your physics book falls from rest down to the floor from a 82 cm high table.a) Calculate the work done by the gravitational on the book.b) State the change in the kinetic energy of the object.c) Where does the energy come from?The mass of the book is 1450 g. You may ignore the effect of air resistance.



10/32


Example

Your physics book falls from rest down to the floor from a 82 cm high table.a) Calculate the work done by the gravitational on the book.b) State the change in the kinetic energy of the object.c) Where does the energy come from?The mass of the book is 1450 g. You may ignore the effect of air resistance.

b) 12 J. (By the work energy theorem the work done by the net force

equals the change in kinetic energy. Neglecting the air resistance, the netforce equals the gravitational force. Because the book starts from rest, thechange in kinetic energy equals final the kinetic energy.)

http://find/


11/32


Example

Your physics book falls from rest down to the floor from a 82 cm high table.a) Calculate the work done by the gravitational on the book.

b) State the change in the kinetic energy of the object.c) Where does the energy come from?The mass of the book is 1450 g. You may ignore the effect of air resistance.

http://find/


12/32


Example

Your physics book falls from rest down to the floor from a 82 cm high table.a) Calculate the work done by the gravitational on the book.

b) State the change in the kinetic energy of the object.c) Where does the energy come from?The mass of the book is 1450 g. You may ignore the effect of air resistance.

c) The energy comes from the gravitational field. When the book falls and

gains kinetic energy, the work done by the gravitational force equals theamount by which the energy of the gravitational field decreases.

http://find/


13/32

Kinetic Energy Ek in Terms of Linear Momentum p

Next we derive the expression for kinetic energy in terms of magnitude of

linear momentum p=mv.

The kinetic energy is

Ek=1

2mv2 =

mv2

2=

(mv)2

2m(4)

where the term in the numerator (mv)2 is the square of the magnitude oflinear momentum p2 = (mv)2.

Kinetic Energy Ek

If the magnitude of the linear momentum of an object is p=mv (mass mtimes speed v) the kinetic energy of the object is

Ek=p2

2m. (5)

http://find/


14/32

6 Field of Force

Mass creates a gravitational field around it, and electric charge an electricfield.

Fields carry energy in them. When you change the field, the energy of the

field changes.Distant interaction can be understood in terms of fields. The types ofdistant interaction at macroscopic level are gravitational interaction,electric interaction and magnetic interaction.

The interacting objects form a system, in which the distant forcescontribute to the total potential energy of the system.

http://find/


15/32

2.3.5 Introduction to Potential Energy

Definition of Potential Energy (NOT IBO)

Potential energy is energy stored in a system as a result of internal forcesthat depend on the position of interacting objects in the system.

The types of potential energy include gravitational potential energy, elasticpotential energy, electrical potential energy, nuclear energy, and chemicalenergy.

Each type of potential energy relates to a different type of force acting in asystem.

http://find/


16/32

2.3.5 Gravitational Potential Energy

The Earth creates a gravitational field around it. When an object is placedin Earths gravitational field, the object and the Earth form a system that isbound together by gravity.

Gravitational potential energy is potential energy associated with thegravitational force acting inside a system.

Definition of Gravitational Potential Energy (NOT IBO)

Gravitational potential energy is energy an object has due to its position ingravitational field, and the gravitational force acting on the object.

The gravitational potential energy is a property of the gravitational fieldformed by the object and the Earth.

Instead of considering the actual value of gravitational potential energy, weconsider the changes in it.

http://find/


17/32

2.3.5 Change in Gravitational Potential Energy

When an object moves in a gravitational field, the work done by the field

on the object depends only on the initial and final position of the object.

Change in Gravitational Potential Energy

When the vertical position of an object placed in Earths gravitational fieldchanges by h, the change in the gravitational potential energy of the

object isEP =mgh, (6)

where m is the mass of the object, and g= 9.81m s2 is the accelerationdue gravity.

The gravitational potential energy is a property of the gravitational fieldformed by the object and the Earth.

Instead of considering the actual value of gravitational potential energy, we

consider the changes in it. Kari Eloranta 2011 (Jyvskyln Lyseon lukio2.3 Work, Energy and Power September 11, 2013 14 / 29
http://find/


18/32

2.3.6 Conservation of Energy

Definition of an Isolated System

An isolated system is a system that cannot exchange energy or matter withits surroundings.

The law of conservation of energy states that the energy of an isolatedsystem remains constant over time.

Definition of Law of Conservation of Energy

The total energy of an isolated system is conserved.

In any process, energy is neither created or destroyed.

http://find/


19/32

2.3.6 Mechanical Energy

Definition of Mechanical Energy

The mechanical energy of a system is the sum of the gravitational potentialand kinetic energy in the system.

When an object moves in a gravitational field, the work done by the

gravitational force on the object depends only on the initial and finalposition of the object. It does not depend on the path the object travelled.

Conservative Force (NOT IBO)

A force is conservative, if the work done by the force on the object, as the

object moves from point A to point B, depends only on the points A andB, not on the path along which the object travelled from A to B.

Gravitational and electric force are examples of conservative forces. Kineticfriction and air resistance are examples of nonconservative forces.

http://find/


20/32


Conservation Law of Mechanical EnergyWhen only conservative forces are acting in a system, the mechanicalenergy of the system is conserved.

When an object is in the gravitational field of the Earth, the Earth does notmove with respect to the object. As a result, the kinetic energy of thesystem formed by the object and Earth equals the kinetic energy of theobject in the reference frame attached to the Earth.

Likewise, when the position of the object changes in the Earthsgravitational field, we usually say that the gravitational potential energy ofthe object changes, instead of saying that the potential energy of thesystem changes.



21/32

2.3.5 Zero Level of Gravitational Potential Energy

Only the changes in gravitational potential energy are physically relevant.However, if we choose a reference level of potential energy, and attach avalue ofEP = 0J to that level, we may speak of the potential energy of anobject at vertical distance h from the reference level.

As an example, consider a ball on the table. If the height of the table isdenoted by h, the ball on the table has gravitational potential energyEp =mgh.

Ep =mgh

Ep = 0Jreference level

height h

http://find/


22/32

Definition of Center of Mass (NOT IBO)

The motion of an object under the influence of a net force can be describedin terms of the centre of mass of the object.

Definition of Centre of Mass

The centre of mass of an object is the point where the mass ob the objectcan be thought of residing such that the net force on the object acts onthat point.

For a homogeneous and symmetric object the centre of mass lies at thecentre of the object.

http://find/


23/32

Center of Mass and Change in Gravitational Potential

Energy

When an object falls a distance h, the vertical position of its center ofmass changes by h as well. That explains why the magnitude of the changein gravitational potential energy is given by Ep =mgh.

Ep =mgh

Ep = 0Jreference level

height hposition of centre ofmass changes also by h

http://find/


24/32

Center of Mass Motion

The photograph illustrates the path of a bouncing basketball.

Even though the ball is rotating its centre of mass follows a parabolic path.The shape of the path can be understood in terms of net force acting on

the centre of mass. Kari Eloranta 2011 (Jyvskyln Lyseon lukio2.3 Work, Energy and Power September 11, 2013 21 / 29
http://find/


25/32


When a ball is at rest on a table, its mechanical energy with respect to the

floor isE= Ep+Ek=mgh+

1

2mv21 =mgh, (7)

because the initial speed is v1 = 0, and the kinetic energy is thusEk=

12mv21 =

12m (0m s1)2 = 0J. The ball has only potential energy and

no kinetic energy.Ep =mgh,Ek= 0J

height h



26/32


Because the ball is relatively heavy and the distance fallen is small, we may

neglect the effect of air resistance. As a result, the only force acting on thefalling ball is the gravitational force.

By the work energy theorem, the work done by the gravity on the ballequals the change in the kinetic energy of the ball.

Ep =mgh,Ek= 0J

#

G

Ep =Ek

height h

http://find/


27/32


As the ball reaches the ground, the potential energy has transformed

entirely into kinetic energy. The mechanical energy of the ball is justkinetic energy

E= Ep+Ek= 0J+1

2mv22 =

1

2mv22. (8)

E= Ep+Ek=mgh+0J=mgh

E=Ep+Ek= 0J+12mv2 = 1

2mv2

http://goforward/http://find/http://goback/


28/32


Now we are finally ready to introduce the law of conservation of mechanical

energy in symbolic form.At point 1, the object has gravitational potential energy Ep,1=mgh1, andkinetic energy Ek,1 =

12mv21. At point 2, the potential energy is

Ep,2=mgh2, and kinetic energy Ek,2 =12mv22.

Conservation Law of Mechanical Energy

When an object moves from point 1 to point 2 in Earths gravitationalfield, in the absence of resistive forces the mechanical energy of the objectis conserved so that

Ep,1+Ek,1 =Ep,2+Ek,2 (9)That is,

mgh1+1

2mv21 =mgh2+

1

2mv22. (10)



29/32

Energy Transformation in Free Fall

In the absence of resistive forces, the mechanical energy of an object in

Earths gravitational field is conserved. When the ball falls from rest to thefloor, its gravitational potential energy transforms entirely into kineticenergy.

Ep,1+Ek,1 =mgh

Ep,2+Ek,2 = 12mv2

final velocity is #v

#v

Here we have denoted the initial height h1 by h and final speed v2 by v forsimplicity.



30/32

Energy Transformation in Free Fall

As an equation the conversion of gravitational potential energy into kinetic

energy is expressed asmgh=

1

2mv2. (11)

Solving for the final speed gives

mgh=1

2mv

2 (mass mis cancelled)

v2 = 2gh

v=

2gh

In free fall from rest, the final speed depends only on the distance fallen hand acceleration due gravity g= 9.81m s2, not on mass m.

http://find/


31/32

2.3.9 Definition of Power

When a force does work on an object, energy is exchanged between theobject and its surroundings.

The rate at which energy is exchanged is called power.

Because work relates to energy transfer, the power can be defined in terms

of work done.Definition of Power in Terms of Work

Power is the rate at which work is done on an object.

Alternatively, the power can be defined in terms of energy.Definition of Power in Terms of Energy

Power is the rate at which energy is transferred.


( )
http://find/


32/32

2.3.9 Power (cont.)

For example, if the power of an incandescent lamp is 15W, it transformselectrical energy into thermal energy and light at the rate of15 J in onesecond.

If the power of a car engine is 120 kW, the engine can do work at the rateof120 kJ in one second. Of that power, only a fraction is used to actually

accelerate the car.

Average Power

Average power is

P=W

t

(12)

where W is the work done on the object in time t.

The unit of power is [P]= [W][t]

= J s1 = 1W (watt = joule per second).

http://find/

23 work, energy and power 20130909

Documents