work energy and power.pdf

PHYSICS 3 (GENERAL PHYSICS I)

Chapter Objectives

 At the end of this chapter, you should be able to:

1. Define Energy, Work, and Power;

2. Apply the Work-Kinetic Energy Theorem in some

dynamics problem;

3. Define and apply the conservation law of mechanical energy in some dynamics situations and problems; and 4. Relate Newton’_{s Laws and the Conservation of}

Introduction



Forms of energy:

 Mechanical

 chemical

 electromagnetic

 nuclear



Energy can be transformed from one form to

another

 Essential to the study of physics, chemistry, biology, geology, astronomy



Can be used in place of Newton

’

s laws to

solve certain problems more simply

Work



Provides a link between force and energy



The work,

W, done by a constant force on

an object

is defined as the product of the component of the force

along the direction of displacement and the magnitude of

the displacement

x

F

W



(

cos



)



 (F cos θ) is the component of the force in the direction of the displacement

WORK

 accomplished action of a

force when it acts on an object as the object moves through a distance.

 SI UNIT: Joule (J)

Physical Properties:

Dimensions: ML

/T

Type: Derived, Scalar

Units :

Joule = 1N-m

1 J = 0.7376 ft-lb erg = 1 dyne-cm

 A force of 15 N is exerted on a box at an angle of θ =45o

Work is positive Work is negative

Work is zero

when:



there is

no displacement

(holding a bucket)



force and displacement are

perpendicular to each

other, as cos 90

°

= 0

(if

PROBLEM

Eric decided to clean his

dorm room with his

vacuum cleaner. While

doing so, he pulls the

canister of the vacuum

cleaner with a force of

magnitude F=50.0 N at an

angle 30.0

°

. He moves

Problem:

cleaning the dorm room

Given:

angle: a=30° force: F=55.0 N

Find:

Work W_F=? W_n=? W_mg=?

1. Introduce coordinate frame: Oy: y is directed up

Ox: x is directed right

2. Note: horizontal displacement only, Work: W=(F cos ) s

J m N m N s F

W_F ( cos)  (55.0 )(cos30.0)(3.00 ) 130  130

J m

n s

n

W_n ( cos)  ( )(cos90.0)(3.00 )  0

J m

n s

mg

W_mg  ( cos)  ( )(cos(90.0))(3.00 )  0

No work as force is perpendicular to the displacement

Example 1:

lifting a cement block…

then moving it horizontally



Work done by the person:



is

positive

when lifting the box



is

negative

when lowering the box



Work done by gravity:



is

negative

when lifting the box



is

positive

when lowering the box

Work for multiple forces

n

W

W

W

W

W

work

Total

:







....



work at

different instances

, the total work is:

W

= F

Δx

+ F

Δx

+ F

Δx

+...

work on a particle all at the same time, the total work is:

W_total = F_1xΔx₁ + F_2xΔx₂ + F_3xΔx₃ +... = (F_1x+ F_2x + F_3x +...) Δx



Example 1: lifting a cement block…



Work done by the person:

 is positive when lifting the box

 is negative when lowering the box



Example 2: … then moving it

horizontally



Work done by gravity:

 is negative when lifting the box

 is positive when lowering the box

 is zero when moving it horizontally

0 0

:W W₁ W₂ W₃  mgh mgh  work

Total



A farmer hitches his tractor to a

sled loaded with firewood and

pulls it a distance of 20 m along

level ground The total weight

of sled and load is 14,700 N.

The tractor exerts a constant

5OO0-N force at an angle of

36.9

°

above the horizontal.

There is a 35OO-N friction force

opposing the sled's motion.

SOLUTION:



Given:



distance of 20 m



total weight of sled and

load is 14,700 N.



tractor exerts a constant

5OO0-N force at an angle of

36.9

°

above the

Graphical Representation of Work

i

i

F

x

W



(

cos



)



into many small displacements x

 For each small displacement:

 Thus, total work is:

which is total area under the F(x) curve!













i

i x

i

i

tot

W

F

x

 Can be carried out using graphical methods!

WORK AND POWER

272 kg

272 kg

4.6 m

Although each man

did the

same amount

of work

,

the power

calculation of the man

on the lift motor will

be higher because

he

accomplished his

POWER

 WORK is accomplished action of a force when it acts on

an object as the object moves through a distance.

 POWER supplied by a force is the rate at which the

force does work.

 POWER (P) is the rate of energy generation (or

absorption) over time:

POWER

 SI unit : Watt (representing the generation or

absorption of energy at the rate of 1 Joule/sec)

 English system: horsepower (1 HP = 735.7 Watts)

PHYSICAL PROPETIES:

 Dimensions [F*L/T] = [F*V]

 Type: Derived, Scalar

›

*

W is Watt

, after James

Watt, famous for his

POWER and WORK

Electric companies charge for

energy not power, usually by the kilowatt-hour (kW•h)

A kilowatt-hour of energy is

1 kW•h = (10

W)(3600s) = 3.6x10

W•s

Power: Example 1



A person pulls a block with a force of 25N at

an angle of 30

with the horizontal. If the

Power: Example 2

 A small motor is used to operate a

lift that raises a load of bricks

weighing 300 N to a height of 20 m in 25s.

 What is the minimum power the

SOLUTION

Given:

weight= 300 N

height= 20 m

time= 25 seconds

Find:

1. Minimum power

Solution:

Based on the problem, we can draw :

Assuming the bricks are lifted without acceleration, the upward force exerted by the motor equals the weight of the bricks (F=1200N).

The speed of the bricks is

WORK AND ENERGY



An object

’

s kinetic

energy can also be

thought of as the amount

of work the moving

object could do in

coming to rest

 The moving hammer has kinetic energy and can do work on the nail

•

Mechanical work

is the amount of ENERGY

ENERGY

 A quantifiable attribute of a

physical system

 Not an object nor any

substance

 Defined as the amount of

work one system CAN do on another.

Physical Properties:

Dimensions: ML

/T

Type: Derived, Scalar

SI UNIT: Joule (J)

Symbols:

E

and many other

depending on type.

Energy



Cannot be created nor destroyed



Energy can be transformed from one form to another



Can be used in place of Newton

’

s laws to solve

Energy Work

Definition Ability to do work -ENERGY transferred by a FORCE

-product of force and displacement

Dimension ML2_/T2 _ML2_/T2

Unit Joule (J) Joule (J)

Type Derived, scalar Derived, scalar

Symbol E W

Equation (depends on the

type) W=Fx∆x=Fcosø∆x

*Note: 1Joule = 1 Newton-meter

Types of Energy

 Gravitational potential - Stored energy in raised

objects.

 Kinetic - The energy in moving objects. Also called

movement energy.

 Heat - Also called thermal energy.  Light - Also called radiant energy.

 Chemical - Stored energy in fuels, foods and batteries.  Sound - Energy released by vibrating objects.

 Electrical - Energy in moving or static electric charges.

 Elastic potential - Stored energy in stretched or squashed

objects.

Work and Kinetic Energy



An object

’

s kinetic

energy can also be

thought of as the amount

of work the moving

object could do in

coming to rest

 The moving hammer has kinetic energy and can do work on the nail

•

Mechanical work

is the amount of ENERGY

Kinetic Energy

 Energy associated with the motion of an object  Scalar quantity with the same units as work

 Work is related to kinetic energy  Let F be a constant force:

. 2 1 2 1 2 : . 2 , 2 : , ) ( 2 0 2 2 0 2 2 0 2 2 0 2 mv mv v v m W Thus v v s a or s a v v but s ma Fs W net net                   2 2 1 mv KE 

Work-Kinetic Energy Theorem.

 There is an important relation between the total work

done on a particle and the initial and final speeds of the particle.

 The total work done on a particle is equal to the

change in its kinetic energy:

W_total = ΔK = ½ mv_f2 _{- ½ mv}

i2

 This result is known as the work-kinetic energy

Work-Kinetic Energy Theorem

 When work is done by a net force on an object and

the only change in the object is its speed, the work done is equal to the change in the object’s kinetic

energy

 Speed will increase if work is positive  Speed will decrease if work is negative

KE

KE

KE

Work and Kinetic Energy: Example

A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31kN on the truck. This force, which is just strong enough to get the truck started upward, is applied over a distance of 2m. Find

a) Work done by the crane b) Work done by gravity

Potential Energy : TYPES

Potential energy

is the work done by a force (

such

as gravitational force or spring force

) when the relative

positions of particles are changed within a physical

system.

•

It is the work done when an object is

moved from

one point to another.

•

Symbol: U or PE[subscript refers to type]

•

Types:

•

1. Gravitational Potential Energy (U

OR PE

)

Potential Energy



Energy

associated with the

position

of the object

within some system



Scalar quantity

with the

same units as work



Potential energy is a property of the

system

, not the

object



A

system

is a collection of objects or particles

interacting via forces or processes that are internal to

the system



Units

of

Potential Energy

are the same as those of

Work

and

Kinetic Energy [J]

mgh

•

Gravitational Potential Energy

is the energy associated with

the relative position of an

object in space near the

Earth

’

s surface

– Objects interact with the earth through the gravitational force

– The potential energy of the earth-object system

Work and Gravitational Potential Energy

 Consider block of mass m at initial height y_i

 Work done by the gravitational force

f i

gravity

PE

PE

W













This quantity is called potential energy:

mgy PE 

 Note: _{Important: work is related to the}

Potential Energy Stored in a Spring



Involves the

spring constant

(or

force constant),

k



Hooke

’

s Law gives the force

F_s = - k x

 F_s is the restoring force

 F is in the opposite direction of x

 k depends on how the spring was formed, the

Potential Energy in a Spring



Elastic Potential Energy

 related to the work required to compress a spring from its equilibrium position to some final, arbitrary, position x

2

2 1

kx PE_s 





Spring/Elastic Potential Energy

 Potential Energy associated with hookean spring forces.

EPE = U_S = ½ k∆x2

Reference Levels for Gravitational

Potential Energy



A location where the gravitational potential energy is

zero must be chosen for each problem

 The choice is arbitrary since the change in the potential energy is the important quantity

 Choose a convenient location for the zero reference height

 often the Earth’s surface

Example problem

U_G1=(1)(9.8)(3) U_G2= (1)(9.8)(5)

Ref 1

Spring and

Gravitational: Example

 Find the potential energy of the basketball player hanging on the rim.

 Find the total potential energy of the system.



Conservation in general



To say a physical quantity is

conserved

is to say that

the numerical value of the quantity remains

constant



In Conservation of Energy, the total mechanical

energy remains constant



In any isolated system of objects that interact only

through

conservative forces,

the total mechanical

energy of the system remains constant.



A force is conservative if the work it does on an

object moving between two points is independent of

the path the objects take between the points

 The work depends only upon the initial and final positions of

the object

 Any conservative force can have a potential energy function

associated with it

Note: a force is conservative if the work it does on an object moving through any closed path is zero.

Examples of Conservative Forces:

 Examples of conservative forces include:

 Gravity

 Spring force

 Electromagnetic forces

 Since work is independent of the path:

 : only initial and final points

f i

c

PE

PE

Conservation of Mechanical Energy



Mechanical energy is the sum of the

kinetic and potential energies in the

system, in a particular state



Other types of energy can be added to

modify this equation

i f

i i f f

E

E

KE

PE

KE

PE



Conservation of Energy including a Spring

 The PE of the spring is added to both sides of the

conservation of energy equation



f s g

i s

g

PE

KE

PE

PE

PE

KE

)

(

)

Problem Solving with Conservation of

Energy

 Define the system

 Select the location of zero gravitational potential energy

 Do not change this location while solving the problem

 Determine whether or not non-conservative forces are

present

 If only conservative forces are present, apply conservation

m=105 kg

PE= 0

V=? H=115m

 A force is nonconservative if the work it does on an

object depends on the path taken by the object between its final and starting points.

 Examples of nonconservative forces

 kinetic friction, air drag, propulsive forces

Example: Friction as a Nonconservative

Force

 The friction force transforms kinetic energy of the object

into a type of energy associated with temperature

 the objects are warmer than they were before the movement  Internal Energy is the term used for the energy associated with

an object’s temperature

 The blue path is shorter

than the red path

 The work required is less

on the blue path than on the red path

 Friction depends on the

path and so is a



If the book moves on an incline that is not frictionless,

a change in the gravitational potential energy of the

book –Earth system also occurs,

and -

fkd

is the amount by which the mechanical

energy of the system changes because of the force of

kinetic friction. In such cases,

WHERE

Non -Conservative Forces and Energy

When nonconservative forces are present, the total

mechanical energy of the system is

not

constant



In equation form:



The energy can either cross a boundary or the

energy is transformed into a form not yet accounted

for.

)

(

)

(

)

(

PE

KE

PE

KE

W

or

PE

PE

KE

KE

W

















Energy

W





Problem Solving with Nonconservative

Forces

 Define the system

 Write expressions for the total initial and final energies  Set the W_nc equal to the difference between the final and

initial total energy

 Follow the general rules for solving Conservation of

Example 1: Conservation of

Energy with spring and NC forces

Problem:

A block with mass of 5.00 kg is

attached to a horizontal spring with

spring constant k = 4.00 x 102 _N/m.

The surface the block rests upon is frictionless. If the block is pulled out

to x_i = 0.0500 m and released, (a)

find the speed of the block at the

equilibrium point, (b) find the speed

when x = 0.0250 m, and (c) repeat

part (a) if friction acts on the block,

Example 1: Conservation of Energy with

spring and NC forces

Given:

Mass of bloc: m = 5 kg

Spring constant: k = 4.00 x 102_N/m

Initial x: x_i = 0.0500 m

At some point: x = 0.0250 m

Coefficient of friction: μ_k = 0.150

Find:

(a) speed of the block at the equi. Position

(b) speed of the block at x = 0.0250 m

(c) repeat (a) if there is friction with μ_k = 0.150

Strategy: In parts (a) and (b) there are no nonconservative forces, so conservation of

energy, can be applied. In part (c), the definition of work and the work–energy theorem are