4 Work and Energy Osw

8/6/2019 4 Work and Energy Osw

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Chapter 4Work, Energy andPower


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4.1 Work


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Work done by aconstant force


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1. The work done by a constant force Fwhen the

displacement of its point of application in d is givenby the scalar product of F and d.

W = F d

Unit: Newton-meter (Nm) = joule, J


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A force F pushes a box through a displacement d.

In this case, where the force and displacementare in the same direction, the work done by the forceis

W = Fd


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2. The greater the force, the greaterthe work.

The greater the distance, the greaterthe work.

3. The work W is zero if the distance dis zero.

For example if you push against a solid

wall you do no work on it, even thoughyou may become tired from your efforts.


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Force at angle to

the displacement

s

F

U


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1. If the force does not act in the direction inwhich motion occurs but an angle to it , thenthe work done is defined as

The product of the component of the force inthe direction of motion and the displacementin that direction


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A person pulls a suitcase with a strap at an angle to thedirection of motion. The component of force in the direction of

the motion is F cos and the work done by the person is

W = ( F cos) d


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3. When = 0 , the force F is in thedirection of the displacement , d

Work done, W = F d


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2. If = 90 ,

Work done, W =F d cos 90 = 0


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2. When an object moves with uniform velocity von asmooth horizontal surface, no work is done by theweight W of the object and the normal reactionR.

3. This is because both W and R are perpendicular to

the direction of the displacement


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Negative work

and Total work


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1. The work done by a force may be positiveor negative.


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2. Work depends on the angle between theforce , and the displacement (ordirection of motion).

F d


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3. This dependence gives rise to three distinct

possibilities :


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a) Work is positive if the force has a component inthe direction of motion ( < 90). This occurswhen is acute ( tirus )


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b) Work is zero if the force has no

component in the direction of motion (= 90) or the force is at right anglesto the displacement.


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c) Work is negative if the force has acomponent opposite to the direction ofmotion ( > 90). This occurs when isobtuse (cakah).


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A car of mass m coastsdown a hill inclined at anangle below thehorizontal. The car is actedon by three forces:

(i) the normal force Nexerted by the road,

(ii) a force due to air resistance, Fair, and

(iii) the forc

e of gravity,mg.


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4. For example, a force F being used to stopa moving object.

If the displacement of the object is dbefore it stops, then the work done by

the force F is W =F d cos which isnegative.


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The normal force is perpendicular to the motion of the car, and thus

does no work.

Air resistan

ce points in a direction that opposes the motion, so itdoes negative work.

Gravity has a component ( mg sin) in the direction of motion.

Therefore, its work is positive.


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5. Thus, whenever we calculate work wemust be careful about its sign , and not

just assume it to be positive.

6. The distinction (perbezaan) is importantsince positive work increases speed,whereas negative work decreases speed.

Zero work has no effect on speed.


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7. When more than one force acts on anobject, the total work is the sum of thework done by each force separately.


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8. Thus , if force does work W1 , force

does work W2 , and so on , the totalwork is

1F

itotalWWWWW 7!! .........

32

2F


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9. Equivalently , the total work can be calculatedby first performing a vector sum of all theforces acting on an object to obtain andthen using the basic definition of work

Where is the angle between and the displacement

total

UU coscos dFdFWtotaltotaltotal

!!

t o t a l F d


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Example 1Force F of magnitude 10 N and friction force F

1of magnitude

3.0 N act on an object, which lies on a horizontal rough surface.If the object is displaced 2.0 m , determine

a) The work done by force F , [ +17.3 J ]b) The work done by F1 , [ -6.0 J ]c) The work done by the weight of the object , [ 0 J ]

d) The work done by the normal reaction acting on the object , [ 0 J]e) The total work done by all the external forces [ + 11.3 J]


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An object of mass 2.0 kg is placed on a horizontal

plane. Force F of magnitude 10 N acts on theobject, as shown in figure below. The object

moves in a straight line at constant velocity. After

the object has moved through 3.0 m , determine

a) The work done by force F [ +26.0 J ]

b) The work done by the frictional force [ -26.0 J ]

c) The coefficient of kinetic friction between the

object and the plane. [ 0.59 ]

Example 2


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1. We have calculated work only forconstant forces, yet most forces innature vary with position.

For example, the force exerted by aspring depends on how far the spring isstretched.


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Graphical Representation of the Work Done by aConstant Force


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Work Done by a Non-Constant Force


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2. If the force F varies with the displacement x, thework done can be obtained from a force-displacement graph in which the component of theforce in the direction of the displacement.


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Work Done by a Continuously Varying Force


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Work donebya variableforce

=

= areaundertheforce-displacement

graph

b

a

dxFW


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3. Within the elastic limit of the spring , the

force F is given by Hookes law as

Where k is a constant known as the spring constant.

kxF !

x F


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Work Needed to Stretch a Spring aDistance x

a) The work done is equalto the shaded area,which is a right

triangle.

b) The area of thetriangle is (x)(kx) =

k x2

.


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x

FdxW0

Fx2

1!

2

2

1kx!

= shaded area under the graph


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5. Worktostretch orcompress aspringa

distance x fromequilibrium ,

2

2

1

kxW !


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Energy


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1. Energy enables a body to do work.2. There are various forms of energy, such

as mechanical energy, chemical energyand nuclear energy.

3. We will discuss mechanical energy in thistopic

4. The two main forms of mechanical energy

are kinetic energy and potential energy.


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Kinetic Energy and theWork-Energy Theorem


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1. Kinetic energy is the energy of a bodydue to its motion.

2. In general, whenever the total work done

an object is positive, its speed increases.When the total work done on an object isnegative, its speed decreases.


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3. The work done on the body has causedit to move with a velocity of v.

Kinetic energy of the body,

2

2

1mvK !


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4. Kinetic energy is never negative.

Instead, K is always greater than or equalto zero.

Independent of the direction of motionor the direction of any forces.


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5. Work- energy theorem states :

The total work done by a resultantforce on an object is equal to thechange in its kinetic energy of the

object:Work done,

energykineticinchange

mumvKW

!

!(! ,

2

1

2

1 22


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1. The potential energy of a body is theenergy of the body due to its relativeposition or its physical state.


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2. Gravitational Potentialenergy

mghU !

Depends only on the height h, andis independent of horizontal position.


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Gravitational Potential Energy


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3. Elastic Potential energy

When a spring is stretched by a force F, and the extensionproduced is x, the work done in stretching the spring is

The work done on string becomes the springs elastic potentialenergy.

2

2

1kxU !

Since U depends on x2, which is positive even if x is negative


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4. The elastic potential energy stored inthe spring is always greater than orequal to zero when it is stretched orcompressed.

Thus, elastic potential energy increaseswhenever it is displaced from equilibrium.


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1. In Physics, we classify forces accordingto whether they are conservative or nonconservative.

2. The key different is that when aconservative force acts, the work it doesis stored in the form of energy that canbe released at later time.


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7. Hence, the net work done by the gravity

on the ball during the upward journey and

downward journey is zero.

8. The force of gravity is an example ofconservative force.


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Lifting a box against gravity with constant speed takes a workmgh. When the box is released, gravity does the same work on thebox as it falls. Gravity is a conservative force


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10. Figure below shows a body of mass m being raised

from the point A to B through a height of h.

Th W k D b C ti F


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The WorkDone by a Conservative Force

Is Zero on Any Closed Path


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11. Another example of a conservative force isthe elastic force of a spring.


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12. On the other hand, friction is not a

conservative force (nonconservativeforce).

Pushing a box with constant speed against

friction takes a work. When the box isreleased it quickly comes to rest andfriction does no further work.


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Conservative and Nonconservative Forces

Force

Conservativeforces

Gravity

Springforce

Nonconservativeforces

Friction

Tensioninarope,cable

Forcesexertedbyamotor

Forcesexertedbymuscles


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Principle ofConservation of Energy


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1. An important principle in physics is theprinciple of conservation of energy.

2. This principle states that energy is

always conserved.


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3. Although energy can be converted fromone form to another, energy cannot becreated or destroyed

4. According to the principle ofconservation of energy, the total energyof a closed / isolated system is

constant.


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5. For a mechanical system, the mechanical energy E

is the sum of the potential energy and kineticenergy of an object.

Kinetic energy + potential energy = constant

EUK!

ffii UKUK !

fi EE !

(Constant)

E = constant

E is conserved


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Question 1

A force F of magnitude 10N

acts on an object of mass2.0 kgplaced on a rough horizontal plane as shown in figure below.

The object moves in a straight line at constant speed 15 ms-1.The frictional force acting on the object can be assumed to beconstant. At a particular instant, force F becomes zero.Determine

a) The velocity of the object after it has travelled through 5.0m from the moment that force F becomes zero. [ 13.5 ms-1 ]b) The additional distance the object must travel through for it

to come to rest. [ 21.0 m ]

Q i 2


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The figure below shows a block X of mass 4.0 kg on a

smooth inclined plane. It is connected by a string over asmooth pulley to block Y of mass 6.0 kg. Assuming that the

acceleration due to gravity g = 10 ms-2, what is the total

kinetic energy of the system immediately after block Y falls

through a vertical distance of 0.5 m from rest? [20 J]

Question 2


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Refer to figure below. Determine the speed of theobject when it reaches

a) point A , b) point B. [ 14.1 ms-1 , 11.8 ms-1 ]

Question 3


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Question 4A 1.9-kg block slides down a frictionless ramp, as shown in figure

below. The top of the ramp is 1.5 m above the ground; the bottomof the ramp is 0.25 m above the ground. The block leaves theramp moving horizontally, and lands a horizontal distance d away.Find

a) the distance d. [ 1.1 m ]

b) Suppose the ramp is not frictionless. Find the distance d for the

case in which friction on the ramp does -9.7 J of work on theblock before it becomes airborne. [0.85 m ]

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Question 5

Figure below shows a spring-mass system. The springconstant is 80 Nm-1. The object has a mass of 1.5 kg.It oscillates to the left and right about the point ofequilibrium O, which is at x = 0. When the objectmoves through O, its velocity is 1.0 ms-1. Determine

a) The extension of the spring when the velocity of theobject is 0.50 ms-1 ,

b) The maximum extension [ 0.12 m , 0.14 m ]

Q


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Objects P and Q of masses 2.0 kg and 4.0 kg respectivelyare connected by a light string and suspended, as shown in

figure below. Object Q is held at rest in such a way that the

string is just about to be pulled downwards by P. Q is then

released. Determine the speed of Q at the instant just before

it strikes the floor. [ 3.61 ms-1]

Question 6


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The figure below shows a stationary marble of mass 20 g sliding down

from a point A along a smooth track. After reaching the bottom of the

track, the marble then moves up another smooth track to C and then

along a rough horizontal plane until it stops at D. The height of A is 8.0 m

whereas the height of C is 0.3 m from the base. Assuming the

acceleration due to gravity is 10 ms-2, and the average frictional force

along CD is 0.2 N, calculate

a) ThegravitationalpotentialenergyofthemarbleatA

b) Thespeedofthemarbleat B

c) ThekineticenergyofthemarbleatC

d) ThedistanceCD

Question 8

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Question 9

Q i 10


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Question 10


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Power


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1. Power is defined as the rate of doing

work, that is

ta entime

doneworPPower !,

t

W!

S.I unit: Js-1 = watt, W


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3. Sincethecarhasaconstantspeedt

dv !

FvtdF

tFdP !!! )(

Note that power is directly proportional to both the force and the speed.

Q1


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Q1

To pass a slow-moving truck, you want your

fancy, 1.30 x 103kg car to accelerate from

13.4 ms-1 to 17.9 ms-1 in 3.0 s. What is the

minimum power required for this pass?

Q2


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Q2

You raise a bucket of water from the bottom

of a deep well. If your power output is 108

W, and the mass of the bucket and the

water in it is 5.00 kg, with what speed canyou raise the bucket? Ignore the weight of

the rope.

Q4


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Q4

The power of a motorcycle varies with its velocity as

shown in the following figure. If the total mass of the

motorcycle and the rider is 500 kg, what is the

acceleration of the motorcycle? [8 ms-1]

Q5


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Q5

The figure below shows a wheel being rotated by amotor so that the two weights 100 N and 60 N which

are attached to the two ends of a belt over the wheel

appears to be stationary. If the circumference of the

wheel is 0.5 m and the wheel is rotating at 50

revolutions per second, what is the output power?

[1000 W]

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4 Work and Energy Osw