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WORK, ENERGY AND POWER

Chapter 4:

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Chapter outline :

4.1 WORK

4.2 APPLICATIONS OF WORK EQUATION FOR CONSTANT FORCE

4.3 ENERGY

4.4 PRINCIPLE OF CONSERVATION OF ENERGY

4.5 POWER

4.6 MECHANICAL EFFICIENCY

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Define work done by a force, dW=Fds and use the force – displacement graph (straight line case).Define energy, kinetic energy and potential energy and use the formula for kinetic energy ½mv2, gravitational potential energy mgh and elastic potential energy for spring ½kx2

Understand the work-energy theorem and use the related equation. State and use the principle of Conservation of Energy, and solve problems regarding conversion between kinetic and potential energy.To define power, P=W/t , derive and use the formula P=F.v To understand the concept of mechanical efficiency and the concequences of dissipated heat, ek=Woutput/Winput.

Objectives :Objectives :

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Definition of work done by a constant force :

product of the magnitude of the force and the product of the magnitude of the force and the displacement of the body in the direction of the displacement of the body in the direction of the force.force.

or

scalar (dot) product between force and scalar (dot) product between force and displacement of the body.displacement of the body.

4.1 Work4.1 Work

sFW

sFθ

and between angle the:

where ,

body theofnt displaceme : sforce of magnitude:F

Equation of work :

cosW Fs

Scalar quantity Dimension : [W] = ML2T-2

SI unit : kg m2 s-2 or joule (J) or N m

One joule is the work done by a force of 1 N which One joule is the work done by a force of 1 N which results in a displacement of 1 m in the direction of the results in a displacement of 1 m in the direction of the force.force.

1 kg m2 s-2 = 1 N m = 1 J

F

)θF cos θ

s

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Work done by a varying force :

2

1

ds

sW F s

area under the force-displacement graphW

F / N

s / m0 si sf

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4.2 4.2 Applications of work equation for Applications of work equation for constant force.constant force.Case 1 :

Work done by a horizontal force, F on an object:

ocos ; where =0W Fs θ

F

s

W Fs

Case 2 : Work done by a horizontal forces, F1 and

F2 on an object:

s

2F1F

cos 0sFW 11 cos 0sFW 22

sFsFWWW 2121

sFFW 21 netW F s

and

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θFsW cos 0 W J

F

s

90θ

Case 3 : Work done by a vertical force, F on an

object:

Case 4 : Work done by a force, F and the friction

force, f on an object:

f

s

F

)

sFW net sfFW cos

W mas

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Notes :Work done on an object is zero if :

(ii) s = 0

(iii) = 90o

(i) F = 0

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cosFsW

4.2 Applications of work equation for 4.2 Applications of work equation for constant force. constant force.

Sign convention :

0< <90 (acute angle)

W > 0 (positive)

work is being done on the system ( by the

external force)

90< <180 (obtuse angle)

W < 0 (negative)

work is being done by the

system

energy is transferred to the

system.

energy is transferred from the system.

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Conclusion :Conclusion :

Work done by constant force:

θ FsW cos

Work done by a varying force:

2

1

ds

sW F s

Force-displacement graph :

4.1 & 4.24.1 & 4.2

W = area under the graph

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4.3 Energy 4.3 Energy Definition : system’s ability to do work.

dimension : [energy] = [work] = ML2T-2

SI unit : kg m2 s-2 or joule (J)

Scalar quantity.

Forms of Energy

Description

ChemicalEnergy released when chemical bonds between atoms and molecules are broken.

ElectricalEnergy that is associated with the flow of electrical charge.

Sound

Energy transmitted through the propagation of a series of compression and refaction in solid, liquid or gas.

Mechanical

a. Kinetic

b. Gravitational

potential

c. Elastic potential

Energy associated with the motion of a body.

Energy associated with the position of a body in a gravitational field.

Energy stored in a compressed or stretched spring.

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Forms of Energy

Description

HeatEnergy that flows from one place to another as a result of a temperature difference.

InternalSum of kinetic and potential energy of atoms or molecules within a body.

NuclearEnergy released by the splitting of heavy nuclei.

Mass

Energy released when there is a loss of small amount of mass in a nuclear process. The amount of energy can be calculated from Einstein’s mass-energy equation, E = mc2

Radiant Heat

Energy associated with infra-red radiation.

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Definition: energy of a body due to its motion.energy of a body due to its motion.

Equation : 2mv2

1K

KW

4.3.1 Kinetic Energy4.3.1 Kinetic Energy

Work-kinetic energy theorem : “The work done by the net force on a body equals the change in the kinetic energy of the body’

where;

K = kinetic energy

m = mass of a body

v = speed of a body

s

F

mProve :

F F m a 2 2

0 2

mv v

s

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2 mv2

1mv

2

1Fs

f iK - K W K

2 2o; where

2

v va

s

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Definition :

energy stored in a body or system because of its position.

Equation :

mghU

W U

4.3.2 Gravitational Potential Energy, 4.3.2 Gravitational Potential Energy, UU

Work-gravitational potential energy theorem :

“The work done by the net force on a body equals the change in the gravitational potential energy of the body”

where;

U = gravitational potential energy

m = mass of a body

g = acceleration due to gravity

h = height of a body from the initial

position

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1 2gW mgh mg h h

1 2mgh mgh

f iU U

Derivation of W = U :

h1

h2

h= h1 – h2

W U

Consider a body of mass m being lifted from a height h2 to a height h1 :

Work done by the gravitational force,

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Definition :

energy stored in an elastic materials energy stored in an elastic materials as the result of their stretching or as the result of their stretching or compressing.compressing.

Equation :

4.3.3 Elastic Potential Energy, 4.3.3 Elastic Potential Energy, UUss

xF2

1kx

2

1U s

2s

where;

Us = elastic potential energy

k = spring constant

x = extension or compression of the spring

Fs = restoring force of spring

2s MTx

Fk

SI unit : kg sSI unit : kg s-2-2 or N m or N m-1-1

Dimension of the spring constant, k :

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xFs sF kx

-ve sign : direction of Fs is always opposite to the direction of the amount of extension or compression, x.

where ;

Fx = the restoring force of spring

x = extension or compression (xf - xi)

k = spring constant or force constant

Hooke’s Law : “The restoring force, Fs of spring is directly

proportional to the amount of extension or compression, x if the limit of

proportionality is not exceeded”

Consider a spring is stretched by a force F :

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A graph of F against x :

F (x)

xx

The work done by the force F is equal to the area under the straight-line graph ;

1

2W Fx

1( )

2kx x

21

2W kx

Work-elastic potential energy theorem :

“The work done to overcome the elasticity of the spring equals the change in elastic potential”

sW U

sf siU U 2 21 1

2 2f ikx kx

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Case 1 : The spring is hung vertically and it is

stretched by a suspended object of mass m :

xInitial position

Final position

sF W mg kx

sF

gmW

The spring is in equilibrium ;

mg

kx

Determining the spring constant, k :

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F

F

sF

sF

(Equilibrium position)

Case 2 : The spring is attached to an object and it is

stretched and compressed by a force F:

sF F kx The spring is in equilibrium ;

F

kx

Fs is positive, x is negative

x = 0

x = 0

x = 0x

x

Fs is negative, x is positive

Fs = 0 x = 0

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Conclusion :Conclusion :

2mv2

1K

mghU

xF2

1kx

2

1U s

2s

KW W U

Formula for energy :

Work-energy theorem:

Gravitational potential energy

Kinetic energy

and

Elastic potential energy

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fi EE

4.44.4 Principle of Conservation of Principle of Conservation of EnergyEnergy

State ; “ The total energy in an isolated (closed)

system is conserved (constant)”

Total of initial energy

Total of final energy=

Conservation of mechanical energy (without friction) :

constantE K U

ffii UKUK or

Conservation of mechanical energy (with friction) :

ffotherii UKWUK

work done by the frictional force or losses of energy.

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ave

W EP

t t

dt

dW

t

WP

0tlimit

4.5 Power4.5 Power Definition :

rate of doing work or rate at which rate of doing work or rate at which energy is transferredenergy is transferred

Average power , Pave ;

Instantaneous power, P ;

Scalar quantity.

Dimension :

3222

TMLT

TML

t

WP

SI unit : kg m2 s-3 or J s-1 or watt (W)

Other unit : horsepower (hp)

1 hp = 550 ft.lb s-1 = 746 watts

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Relation of P, F and v ;

d

dcos d

d

WP

tF s

t

cosFv

P F v

Consider the net force F applied to an object and its velocity v :

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outk

in

100%W

eE

or

outk

in

100%P

eP

4.6 Mechanical efficiency (4.6 Mechanical efficiency (eekk or or ))

Definition :

ratio of the useful work done, ratio of the useful work done, WWoutout to the to the

energy input, energy input, EEinin

ratio of the useful power output, ratio of the useful power output, PPoutout to to

the power input, Pthe power input, Pinin

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Conclusion :Conclusion :

Conservation of energy :

In an isolated (closed) system, the total In an isolated (closed) system, the total energy of that system is constant.energy of that system is constant.

i fE E in solving problems regarding conservation of energy

Power :

av

W EP

t t

cosP Fv

Mechanical efficiency :

outk

in

100%W

eE

4.4, 4.5 & 4.6 4.4, 4.5 & 4.6

Use ,

and

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THE END…

Next Chapter…CHAPTER 5 :

Static