Work, Energy And Power

Chapter 4:

WORK, ENERGY AND POWER

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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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Objectives : 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. 3

4.1 Work Definition of work done by a constant force : product of the magnitude of the force and the displacement of the body in the direction of the force. or scalar (dot) product between force and displacement of the body. F

Equation of work :

  W = F•s W  Fs cos 

)θ

F cos θ

s

where , F : magnitude of force s : displacement of the body   θ : the angle between F and s • Scalar quantity • Dimension : [W] = ML2T-2 • SI unit : kg m2 s-2 or joule (J) or N m 1 kg m2 s-2 = 1 N m = 1 J One joule is the work done by a force of 1 N which results in a displacement of 1 m in the direction of the4 force.

Work done by a varying force : F/N

s2

W   Fds s1

0 si

sf

s/m

W  area under the force-displacement graph

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4.2 Applications of work equation for constant force. Case 1 :  Work done by a horizontal force, F on an object: W  Fsθcos ; where  =0o

 W  Fs

 F

 s

Case 2 :  Work done by a horizontal forces, F1 and  F2 on an object:

F1

 F2

 s W2 = F2 s cos 0 

W1 = F1 s cos 0  and

∑W =W

1

+ W2 = ( F1 s + F2 s )

∑W =( F + F ) s  W   Fnet  s 1



2

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

F

θ = 90



 s

W = Fs cos θ  W 0J

Case 4 :  Work done by a force, F and the friction  force, f on an object:

 f W = ( Fnet ) s

F

)θ

 s

W = ( F cos θ − f ) s

 W  mas

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• Notes :

Work done on an object is zero if : (i) F = 0

(ii) s = 0

(iii) θ = 90o

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4.2 Applications of work equation for constant force. Sign convention :

W = Fs cos θ 0°<θ <90° (acute angle) 90°<θ <180° (obtuse angle)

W > 0 (positive)

work is being done on the system ( by the external force)

energy is transferred to the system.

W < 0 (negative)

work is being done by the system

energy is transferred from the system.

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4.1 & 4.2

Conclusion :  Work done by constant force:

W = Fs cos θ  Work done by a varying force:

W 



s2

s1

Fds

 Force-displacement graph :

W = area under the graph

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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 Chemical Electrical

Sound Mechanical a. Kinetic

Description Energy released when chemical bonds between atoms and molecules are broken. Energy that is associated with the flow of electrical charge. Energy transmitted through the propagation of a series of compression and refaction in solid, liquid or gas.

Energy associated with the motion of a body. b. Gravitational Energy associated with the position of a body in a gravitational field. potential Energy stored in a compressed or 12 c. Elastic stretched spring. potential

Forms of Energy Heat Internal Nuclear

Mass

Radiant Heat

Description Energy that flows from one place to another as a result of a temperature difference. Sum of kinetic and potential energy of atoms or molecules within a body. Energy released by the splitting of heavy nuclei. 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 Energy associated with infra-red radiation.

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4.3.1 Kinetic Energy • Definition: energy of a body due to its motion. • Equation : 1 K = mv 2 where; K = kinetic energy 2 m = mass of a body v = speed of a body

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

W = ∆K Prove :

m

 F  F  mma 

v  2s

2

 F  s

v

2 0



v 2  vo2 ; where a  2s

1 1 2 2 Fs = mv − mv0 2 2  K f - K i  W  K

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4.3.2 Gravitational Potential Energy, U Definition : – energy stored in a body or system because of its position. Equation :

U = mgh where; U = gravitational potential energy m = mass of a body g = acceleration due to gravity h = height of a body from the initial position 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”

W  U

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• Derivation of W = ∆U :

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

h= h1 – h2 h1 h2

Work done by the gravitational force,

Wg  mgh  mg  h1  h2   mgh1  mgh2  U f Ui

 W  U

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4.3.3 Elastic Potential Energy, Us Definition : energy stored in an elastic materials as the result of their stretching or compressing. Equation :

1 2 1 U s = kx = Fs x 2 2

where; Us = elastic potential energy k = spring constant x = extension or compression of the spring Fs = restoring force of spring • Dimension of the spring constant, k :

[ Fs ] [ k ] = = MT −2 [ x]

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

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Consider a spring is stretched by a force F :

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”

Fs ∝ − x

where ;

 Fs   kx

Fx = the restoring force of spring x = extension or compression (xf - xi) k = spring constant or force constant -ve sign : direction of Fs is always opposite to the direction of the amount of 18 extension or compression, x.

A graph of F against x : F (x)

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

1 W  Fx 2 1  (kx) x 2 1 2 W  kx 2

Work-elastic potential energy theorem : “The work done to overcome the elasticity of the spring equals the change in elastic potential”

W  U s

 U sf  U si 1 2 1 2  kx f  kxi 2 2

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• Determining the spring constant, k : Case 1 :  The spring is hung vertically and it is stretched by a suspended object of mass m :

Initial position

 Fs

x

Final position

  W = mg The spring is in equilibrium ;

  Fs  W  mg  kx

mg  k x 20

Case 2 :  The spring is attached to an object and it is stretched and compressed by a force F: Fs is negative, x is positive

 Fs

 F

x x=0 Fs = 0 x=0 (Equilibrium position) x=0

 Fs

 F

Fs is positive, x is negative

x x=0

The spring is in equilibrium ;

  Fs  F  kx F  k x

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Conclusion : Formula for energy :  Kinetic energy

1 K = mv 2 2

 Gravitational potential energy

U = mgh  Elastic potential energy

1 2 1 U s = kx = Fs x 2 2  Work-energy theorem:

W = ∆K

and

W  U 22

4.4 Principle of Conservation of Energy State ; “ The total energy in an isolated (closed) system is conserved (constant)”

∑E = ∑E i

Total of initial energy

=

f

Total of final energy

Conservation of mechanical energy (without friction) :

E  K  U  constant

or

Ki + U i = K f + U f

Conservation of mechanical energy (with friction) :

K i + U i + Wother = K f + U f work done by the frictional force or losses of energy.

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4.5 Power Definition : rate of doing work or rate at which energy is transferred • Average power , Pave ;

Pave

W E   t t

• Instantaneous power, P ;

∆W dW P = limit = ∆t →0 ∆t dt

Scalar quantity. Dimension :

[ ∆W ] ML2T −2 [ P] = = [ ∆t ] T

= ML T 2

−3

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

Other unit : horsepower (hp) 24 1 hp = 550 ft.lb s-1 = 746 watts

Relation of P, F and v ; Consider the net force F applied to an object and its velocity v :

dW P dt F cos   ds   dt

 Fv cos 

   P  F v

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4.6 Mechanical efficiency (ek or η ) Definition : – ratio of the useful work done, Wout to the energy input, Ein

Wout ek   100% Ein or – ratio of the useful power output, Pout to

the power input, Pin

Pout ek   100% Pin

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4.4, 4.5 & 4.6

Conclusion : Conservation of energy :

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

 Ei   E f

in solving problems regarding conservation of energy

 Power :

and

W E Pav   t t

P  Fv cos 

Mechanical efficiency :

Wout ek   100% Ein

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THE END… Next Chapter… CHAPTER 5 : Static

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