chapter 4 work energy power

7/27/2019 Chapter 4 Work Energy Power

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4. Work, Energy and Power

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Animations of roller coasters,downhill skiers and pendulaare used to illustrate therelationship between theconcepts of work and energy .

4.1 Work and Energy4.1 Work and Energy

Work dW = F.ds

W = dW

W = F . ds

WORK

Thework W done by an agent exerting a constant force, F and causing a displacement sequals the magnitude of the displacement, s, times the component of F along the directions.


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If an object undergoes a displacement of s, the work done by the forceF is (F cos )s

Notes : If s = 0 W = 0 ( no work is done when holding a heavy box, or pushing against wall). W = 0 if ( no work is done by carrying a bucket of water horizontally). Work is ascalar The SI units of work areJoules (J).

( 1 Joule = 1 Newton meter )

Energy

A body which is capable of doing work is said to possess energy. The amount of energythat a body has is equal to the amount of work that it can do.

Mechanical energy :Potential energy - positionKinetic energy - motion

4.2 Potential Energy

Potential Energy

An object canstore energy as the result of its position. Thisstored energy of position isreferred to aspotential energy .

Potential energy is the stored energy of position possessed by an object.


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Gravitational potential energy (PEgrav)

- is the energy stored in an object as the result of its vertical position.(i.e height)PEgrav = mass * g * height


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Gravitational potential energy (PEgrav)

Elastic potential energy

Elastic potential energy

Energy stored in elastic materials as the result of their stretching or compressing.The more stretch, the more stored energy.

Springs are a special instance of device which can store elastic potential energy dueither compression or stretching.


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Elastic potential energy (Spring)

A force required to stretch a spring.For certain springs, the amount of force required is directly proportional to the

amount of stretch or compression (x); the constant of proportionality, spring constant (k).Such spring are said to followHookes Law


Work required to stretch a spring from its equilibrium position to some final arbitra position, x is,



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4.3 Kinetic Energy, K

The kinetic energy, K , of a particle of massm moving with speedv is defined as

K = mv2

Kinetic energy is a scalar quantity and has the same units as work (Joule).

Kinetic energy is energy associated with the motion of a body.It is often convenient to write equation in the form :

Wnet = K f K i = K work-energy theorem

The net work done by the force F isWnet= Fs = (ma)s

Relationships are valid when a particle undergoes constant acceleration :

s = ( vi + vf )t ;t

vva i f

=

where vi is the speed at t = 0 and vf is the speed at time t. Substituting these expressionsinto equation gives

Work done on a particle equals the change in its kinetic energy.

Example 1:


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6.0 kg block initially at rest is pulled to the right along a horizontal, frictionless surface a constant, horizontal force of 12N as in figure 1.

Find the speed of the block after is has moved 3.0 m.

Solution 1:W = Fs = (12N)(3.0m) = 36 Nm = 36 JUsing the work-energy theorem and noting that the initial kinetic energy is zero, w

get

Example 2 :

a) How much work is done to move a 1840 kg Jaguar XJ6 automobile from rest to27.0 m s-1(60mi/h) on a level road?

b) If this takes place over a distance of 117m, what is the average net force?

Solution 2 :

a) We can use the work-energy theorem to find the work ;

We setv1 equal to zero,v2 =27.0 ms-1 and m = 1840kg. Then the work is

b) The average net force can be found from


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4.4 Conservation of energy

The total mechanical energy, E is defined as the sum of the kinetic and potentialenergies, we can write :

E = K + U total mechanical energy

Therefore, we can apply conservation of energy in the form E i = E f ,, or K i + U i = K f + U f the mechanical energy of

an isolated systemremains constant

The principle of conservation of mechanical energy for the system as Ei = Ef

K i + Ui = K f + Uf

Example :

Suppose your hand moves up 0.50 m while you are throwing the ball, which leaves yhand with an upward velocity of 20.0ms-1. Again ignore air resistance.

a) Assuming that your hand exerts a constant upward force on the ball, find the magnitudthat force.

Find the speed of the ball at a point 15.0m above the point where it leaves your hand

Solution :

y1 = -0.50m ; y2 = 0m ;K 1= 0,

U1 = mgy1 =(0.145)(9.80)(-0.50) -0.71 JK 2 = 1/2mv22 = (0.145)(9.80)(0) = 0U2 = mgy2 = (0.145)(9.80)(0) = 0K 1 + U1 + Wother = K 2 + U2Wother = (K 2 K 1) + (U2 U1)= (29.0 0) + (0 (-0.71) )= 29.7 J

a) Assuming that the upward force F that your hand applies is constant, we have


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Since 233

2

1mv K = , the velocity at point 3 is

4.5 Power

Power is defined as the time rate of energy transfer

The time rate of doing work The instantaneous power is the limiting value of the average power as t approaczero

The instantaneous power can be written

The unit of power in SI system:J/s = kg m2s-3

= W


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Example :

1.A 1000kg elevator carries a maximum load of 800 kg. A constant frictional force of 40retards its motion upward. What minimum power , in kilowatts, must the motor delivelift the fully loaded elevator at a constant speed of 3.00 m/s.

Solution :The motor must supply the force, F that pulls the elevator upward. From Newto

second law and from the fact a=0 because v is constant.

Where M is the total mass (elevator plus load) equal to 1800 kg. Therefore,

From Fv P = and the fact that T is in the same direction as v, we have


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Example :

2. A 70 kg person runs up a staircase 3.0m high in 3.5 s. How much power does he devin climbing the steps?

Solution :

In this case, the work done is the change in gravitational potential energy, mgh, so the power is

Example :

3.An astronaut with space suit has mass of 110kg. Climbing up a hill 7.3m high in 7requires the astronaut to expend a power of 200W. Is the astronaut on the earth?

Solution :m=110kgh=7.3mt=7.2sP=200W

From,


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The astronaut not on the earthMECHANICAL EFFICIENCY

Definition

How well a machine transfers the work put into work coming out of it is called efficiency of the machine.

Mathematical formula

efficiency, ek = %100input

output

work

work

The main factor affecting efficiency is friction. If friction could be eliminated, ywould have a machines efficiency.

Example :

1. A steam requires 60 J of work to run it. It is capable of giving a work output of 3What is the machines efficiency?

Solution :


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Example :

2. A force of 545N is exerted on the use of a pulley system and, the rope is pulled 10m. Tforce causes an object with weight of 2520N to be raised 1.5m. What is the efficiency

the pully system?Solution :

chapter 4 work energy power

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