This Week

Work, Energy, PowerEnergy makes our everyday world work

Where does energy go?Are we using it up?

How can one store energy?Where does energy come from. The heat of the earthEscape velocity

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Work, Energy and PowerWe all use the words Work, Energy and Power and indeed our usage is generally correct. Once again, however, we need to write down simple definitions and be able to do calculations.

Energy comes in a wide variety of forms. For example if you go on a trip in your car energy is being supplied by the gasoline. Initially some of the energy is used to give the car speed but when you stop gasoline has been used but the car now has no energy. The energy went into the air you passed through, dissipated heat in the tires, brakes and engine and so on.

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Energy ConservationIf we take a closed system, that is one that nothing can enter or leave, then there is a physical law that energy is conserved.

We will define various forms of energy and if we examine the system as a function of time energy may change into different forms but the total is constant. Energy does not have direction just a magnitude and units.

Conservation of Energy follows directly from the statement that physical laws do not change as a function of time.

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Forms of mechanical energyOne obvious form of energy is the energy of a moving object this is kinetic energy = 1/2mv2

A second form of energy is what is called Potential Energy. This energy is the energy stored in a compressed spring or stretched elastic or in an object that is held at rest above the earths surface.

When the spring or elastic or the object is released one gets kinetic energy appearing from the stored energy. In the case of a pendulum there is a continual storage and release of energy as the pendulum swings.

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Work and energyIf an object initially at rest is acted on by a net force F it will accelerate and after time t will have moved a distance d

We define Work W = Fd units are joules Both F and d can be + of – so W can be positive or negative

Now take our usual equations v = v0 + at d = v0t +1/2at2 and F = maFd = ma(1/2at2) = ma(1/2av2/a2) = 1/2mv2 kinetic energy

F is the net force in the direction of motion

F

d

+

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Negative WorkIf F is in the opposite direction to the motion then Fd is negative.

Remember F and d have magnitude and direction and can be positive or negative.

If the work is negative energy is being removed from the object

Friction always opposes motion and the work Ff does is negative

F

d

Ff

F

W = Fd - Ffd

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Net force and Work

If there is more than oneforce acting we have to find the work done by eachforce and the work done bythe net force

Net force F – Ff work = (F – Ff)d = 1/2mv2

The work the force F does is Fd and if we write the equation as Fd = Ffd + 1/2mv2

we can see that some work goes into heat and some into kinetic energy and we can account for all the work and energy

F

dFf

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1M-04 Pile Driver

The potential energy of the pendulum is turned into kinetic energy. Then if

the collision is perfectly elastic all the kinetic energy is transferred to the block and then the energy is turned

into heat through friction.Mgh = Ffd

Ff is the average frictional force between the block and the wood.

What happens to the Potential Energy of the

Mass M ?

The kinetic energy of a pendulum is transferred to a block which then slides to rest

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1M-05 Pile Driver

POTENTIAL ENERGY CHANGES TO KINETIC ENERGY, KINETIC ENERGY CHANGES TO WORK.

Work-Energy Relationshipmg(h+y) = fy

f is the average friction force between the nail and the wood.

What happens to the Potential Energy of the

Mass M ?

A Pile Driver does work on a nail

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Potential energyIf we raise an object a height h so that it starts and finishes at rest then the average force = mg and the work done = mgh.

This energy is stored as potential energy since if the mass is allowed to fall back to it’s original point then

v2 = v02 + 2gh

and mgh = 1/2mv2

So the original work in lifting is stored and then returned as kinetic energy

Similarly for a spring stored energy = 1/2kx2

Where x is the distance stretched

F = mghg

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Potential energyUnlike kinetic energy for Potential energy we have to define where zero is.

A block is at a height h above the floor and d above the desk.

Potential energy is mgh with respect to the floor but mgd with respect to the desk. If we dropped block it would have more kinetic energy hitting the floor than hitting the desk

hd

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OscillationsMany simple systems oscillate with a continual transfer from KE to PE and PE to KE with the sum of the two remaining constant.

In practice energy is lost through friction and the motion slows down.

http://www.physics.purdue.edu/class/applets/phe/pendulum.htm

anim0006.mov anim0007.mov anim0009.mov

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1M-01 Bowling Ball Pendulum

NO POSITIVE WORK IS DONE ON THE BALLTHUS, THERE IS NO GAIN IN TOTAL ENERGYTHE BALL WILL NOT GO HIGHER THAN THE INITIAL POSITION

Is it safe to stand here after

I release the bowling ball ?

hmgh

1/2mv2

mgh

mgh = 1/2 mv2

A bowling ball attached to a wire is released like a pendulum

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1M-03 Triple Chute

EACH OF THE STEEL BALLS LANDS AT THE SAME POSITION

Each path is clearly different. Which ball will

travel the farthest ?

The Change in Gravitational Potential Energy does not depend on the Path Traveled

Three Steel Balls travel down different Paths

EACH BALL HAS SAME KINETIC ENERGY AT BOTTOM OF RAMP, REGARDLESS OF THE PATH TAKEN AND HAS THE SAME VELOCITY

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Conservative forcesGravity is an example of a conservative force where total energy is conserved and there is just an interchange between kinetic and potential energy.

In real life frictional forces would cause energy to be lost as heat

For a conservative force if no energy is added or taken out then

E = PE + KE

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1M-08 Galileo Track

AS THE BALL OSCILLATES BACK AND FORTH, THE HEIGHT IS REDUCED BY A LITTLE. WHAT MIGHT ACCOUNT FOR THIS?FRICTION IS SMALL, BUT NOT ZERO.

Will the ball travel to a lower or higher height when going

up the steeper, shorter ramp ?

Conservation of Energy:

mgh = 1/2mv2 = mgh

So, The Ball should return to the same height

Ball travels down one ramp and up a much steeper ramp

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1M-10 Loop-the-Loop

Conservation of Energy:mgh = mg(2R) + 1/2mv2

From what height should the ball be

dropped to just clear the Loop-the-

Loop ?

Ball travels through a Loop-the-Loop

At the top of the loopN + mg = mv2/rThe minimum speed is when N = 0

Therefore h = 5/2R (Friction means in practice H must be larger)

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PowerIt is not only important how much work is done but also the rate atwhich work is done

So the quantityPower = P = W/t (unit is a watt)

is very important.

Generally energy supplies, motorsetc are rated by power and one can determine how much work can be done by multiplying by time.

W = Pt (joules)

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Watts and JoulesJoule is the Unit of Energy and Energy is the fundamental resource that is required for all activity and for life itself. All our energy comes from the sun although there is geothermal energy which was produced by the formation of the earth and tidal motion produced by the motion of the moon.

Practical The unit for electrical usage is the kilowatt –hour. A kilowatt – hour is the energy used by a 1000 watt device for 3600 seconds 1kWHr = 1000*3600 = 3.6 million joules

Watt is the Unit of Power and Power measures the rate at which work is done or energy is used. All appliances, motors etc are rated in Watts so that one can match to the required application.

Example. In order to lift an elevator with a mass of 1000kg to 100 meters requires 1000*9.8*100 joules but we need to do it in say 20 seconds so the power we need is 1000*9.8*100/20 = 49000 Watts so we need to install a motor rated at > 49000 watts

Physics 214 Summer 2017

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Mechanical AdvantageVery often we are limited by the maximum force we can apply and the power we can supply. This is also true of electric motors.

One can design simple arrangements so that for example one can lift a large weight by using a lever or a pulley system that reduces the force.

The total work done is the same as lifting the weight directly but for example using a force which is half the weight but pulling it for twice the distance

http://www.physics.purdue.edu/class/applets/phe/pulleysystem.htm

Our World

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Equinox

http://www.classzone.com/books/earth_science/terc/content/visualizations/es0408/es0408page01.cfm?chapter_no=04

http://www.mesoscale.iastate.edu/agron206/animations/01_EarthSun.html

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The Tropic of Capricorn, or Southern tropic, marks the most southerlylatitude at which the sun can appear directly overhead at noon. This event occurs at the December solstice, when the southern hemisphere is tilted towards the sun to its maximum extent. The Tropic of Cancer, also referred to as the Northern tropic, is the circle of latitude on the Earth that marks the most northerly position at which the Sun may appear directly overhead at its zenith. This event occurs once per year, at the time of the June solstice, when the Northern Hemisphere is tilted toward the Sun to its maximum extent.

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Where do we get energy? Power comes from the sun 1.35 kilowatts/m2 on the atmosphere and a maximum of about 1 kilowatt/m2 on earth. In one hour 1 kilowatt = 3600 x 103 joules. A toaster is usually 1 to 2 kilowatts. Burning fossil fuels and making new molecules

carbon plus oxygen gives CO2 plus energy Nuclear power plants

breaking very heavy nuclei into lighter nucleiIn 2003, the United States generated 3,848 billion kilowatthours (Kwh) of electricity, coal-fired plants accounted for 53% , nuclear 21%, natural gas 15%, hydroelectricity 7%, oil 3%, geothermal and "other" 1%.Area of the USA is about 1013m2 but efficiency for converting solar power is about 10% and then there is night, clouds etc.

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The heat of the earthFirst we have to define what heat is.Heat is the internal energy stored in an object by the motionof it’s constituent particles (e.g. atoms)

How do we get heat in our everyday life?We can transfer mechanical energy of an object into heat. For example if drop a brick the kinetic energy just beforeimpact is turned into heat.

An object can also be heated by bombarding it with particles of which photons from the sun is a common example.That is why snow and ice can melt even if the temperature is below freezing

About 60% of the heat in the earth comes from the original formation due to loss of potential energy and impact of the material that makes up the earth.About 40% comes from energy emitted in radioactive decays

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Escape velocity

Suppose we want to propel an object to a height of 1 kilometer.

If we assume that g is 9.8m/s2 and no friction then 1/2mvi

2 = mgh so vi

2 = 19600 and v = 140m/s this is about 315mph.

To fire an object so that it never returns requires a speed of 11200m/s or 25000mph. The highest projectile ever fired was from a 16 inch gun with a barrel length of 176feet and it reached an altitude of 112miles or 180km. HARP Project, Barbados

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Summary of Chapter 6W = Fd joules and can be + or –

Power = W/t watts

KE = 1/2mv2 joules

PE = mgh or 1/2 kx2 joules

Conservative E = KE + PE

Gravity, oscillations such as a pendulum or mass on a spring and KE and PE just keep interchanging

F

dF

d

http://www.physics.purdue.edu/class/applets/phe/springpendulum.htm

Worked Questions and Problems

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Questions Chapter 6Q1 Equal forces are used to move blocks A and B across the floor. Block A has twice the mass of block B, but block B moves twice the distance moved by block A. Which block, if either, has the greater amount of work done on it? Explain.

Q3 A string is used to pull a wooden block across the floor without accelerating the block. The string makes an angle to the horizontal.

A. Does the force applied via the string do work on the block?

B. Is the total force involved in doing work or just a portion of the force?

Work is Force times distance so the most work is done on B

F

d

A. Yes B. just the horizontal component

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Q4 In the situation pictured in question 3, if there is a frictional force opposing the motion of the block, does this frictional force do work on the block? Explain.

Yes it does negative work since force is opposite the motion

Q8 A woman uses a pulley, arrangement to lift a heavy crate. She applies a force that is one-fourth the weight of the crate, but moves the rope a distance four times the height that the crate is lifted. Is the work done by the woman greater than, equal to, or less than the work done by the rope on the crate? Explain.

The product Fd is the same for both and the work is equal

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Q12 A child pulls a block across the floor with force applied by a horizontally held string. A smaller frictional force also acts upon the block, yielding a net force on the block that is smaller than the force applied by the string. Does the work done by the force applied by the string equal the change in kinetic energy in this situation?

No energy because is lost to friction. Fd – Ffd = 1/2mv2

Q18 Suppose that work is done on a large crate to tilt the crate so that it is balanced on one edge, as shown in the diagram, rather than sitting squarely on the floor as it was at first. Has the potential energy of the crate increased in this process?

Yes. Work has been put in and the center of mass is now higher

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Q22 A pendulum is pulled back from its equilibrium (center) position and then released.

A. What form of energy is added to the system prior to its release?

B. At what points in the motion of the pendulum after release is its kinetic energy the greatest?

C. At what point is the potential energy the greatest?

Q28 Suppose that a mass is hanging vertically at the end of a spring. The mass is pulled downward and released to set it into oscillation. Is the potential energy of the system increased or decreased when the mass is lowered?

A. Potential B. at it’s lowest point C. At the highest points where it stops

The potential energy is increased

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Woman does 160 J of work to move table 4m horizontally. What is the magnitude of horizontal force applied?

F

d

Force & displacement in SAME directionW = Fd, 160J = F(4m) F = 40N

Ch 6 E 2

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5.0 kg box lifted (without acceleration) thru height of 2.0 ma) What is increase in potential energy?b) How much work was required to lift box?

a) PE = mgh PE = PEfinal – PEinitial= mg(ho+2.0m) – mgho = mg(2.0m)= (5.0 kg)(9.8 m/s2)(2.0m) = 98J

b) F = ma = 0 = Flift – mgFlift = mg = (5.0kg)(9.8m/s2) = 49NW = Fd = (49N)(2.0m) = 98J

M

Mho+2.0m

M

mg

gFlift

Ch 6 E 8

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To stretch a spring a distance of 0.70 m, 40 J of work is done.What is the increase in potential energy?b) What is the value of the spring constant k?

a) PE = 40J

x=0 x=0.70 m

equilibrium

Ch 6 E 10

b) PE = ½ kx2

k = 2PE/x2 = 80/(0.7)2 - = 296.8n/m

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The frequency of oscillation of a pendulum is 8 cycles/s.What is its period?

x

T

tf = 1/TT = 1/f = 1/(8 cycles/s)T = 0.125 seconds

Ch 6 E 18

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100 kg crate accelerated by net force = 50 N applied for 4 s.a) Use Newton’s 2nd Law to find acceleration?b) If it starts from rest, how far does it travel in 4 s?c) How much work is done if the net force = 50 N?

a) F = ma a = F/m = 50N/100kg = 0/5 m/s2

b) d = v0t + ½at2 = ½(0.5)(4)2 4mc) W = Fd = (50N)(4m) = 200J

d) v = v0 + at = 0 + (0.5 m/s2)(4s) = 2m/se) KE = ½mv2 = ½(100kg)(2m/s)2 = 200 J

work done equals the kinetic energy.

M Fnet

Ch 6 CP 2

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A 0.20 kg mass is oscillating horizontally on a friction-free table on a spring with a constant of k=240 N/m. The spring is originally stretched to 0.12 m from equilibrium and released.a) What is its initial potential energy?b) What is the maximum velocity of the mass? Where does it reach this maximum velocity?c) What are values of PE, KE and velocity of mass when the mass is 0.06 m from equilibrium.d) What is the ratio of velocity in (c) to velocity in (b)

Ch 6 CP 4

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a) PE = 1/2kx2 = ½(240)(0.12)2 = 1.73J

b) No friction so energy is conservedE=PE+KE, maximum KE when PE=0KEmax = 1/2mv2 v = 4.16 m/s. This occurs at the equilibrium position

c) PE = 1/2kx2 = ½(240)(0.06)2 = 0.432JSince total energy = 1.73J then the kinetic energy = 1.73 – 0.432 = 1.3JKE = 1/2mv2 = 1.3 then v = 3.6m/sd) vc/vb = 3.6/4.16 = 0.86

x=0 x=0.12 m

M

Ch 6 CP 4 (con‘t)

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Review Chapters 1 - 6

Units----Length, mass, time SI units m, kg, secondCoordinate systemsAverage speed = distance/time = d/tInstantaneous speed = d/∆tVector quantities---magnitude and directionMagnitude is always positiveVelocity----magnitude is speedAcceleration = change in velocity/time =∆v/∆tForce = ma Newtons

- + xd

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Conversions, prefixes and scientific notation

giga 1,000,000,000 109 billion

mega 1,000,000 106 million

kilo 1,000 103 thousand

centi 1/100 0.01 10-

2hundredth

milli 1/1000 0.001

10-

3thousandth

micro 1/1,000,000 1/106 10-

6millionth

nano 1/1,000,000,000 1/109 10-

9billionth

1 in 2.54cm

1cm 0.394in

1ft 30.5cm

1m 39.4in 3.281ft

1km 0.621mi

1mi 5280ft 1.609km

1lb 0.4536kg g =9.8

1kg 2.205lbs g=9.8

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Speed, velocity and accelerationv = ∆d/∆t a = ∆v/∆t

The magnitude of a is not related to the magnitude of v

the direction of a is not related to the direction of v

12 3 4

v = v0 + at constant accelerationd = v0t + 1/2at2

d = 1/2(v + v0) t v2 = v0

2 + 2add,v0 v,a can be + or –independently

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One dimensional motion and gravityv = v0 + at d = v0t + 1/2at2 d = ½(v + v0)t v2 = v0

2 + 2ad

+

+

g = -9.8m/s2

At the top v = 0 and t = v0/9.8 At the bottom t = 2v0/9.8

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Equationsv = v0 + at d = v0t + 1/2at2 d = ½(v + v0)t v2 = v0

2 + 2adSometimes you have to use two equations.

v0 = 15m/s v = 50m/s What is h?v = v0 + at 50 = 15 + 9.8t t = 3.57 s

` h = v0t + 1/2at2

h = 15 x 3.57 + 1/2x9.8x3.572

= 116mh = ½(15 + 50) x 3.57 = 116m

v0

v

gh

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Projectile Motionaxis 1 v1 = constant and d1 = v1taxis 2 vv = v0v + at and d = v0vt + 1/2at2

9.8m/s2

v1

g

hv

R

Use + down so g is + and h is + h = v0vt + 1/2at2

v0v = 0, t2 = 2h/a R = v1t v = v0v + at

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Complete Projectilev1

v1

v0v

v1

9.8m/s2

highest point the vertical velocity is zero

vv = v0v + at so t = v0v/9.8 h = v0vt + 1/2at2

end t = 2v0v/9.8 and R = v1 x 2v0v/9.8 and the vertical velocity is minus v0v

v0v

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Newton’s Second and First LawSecond Law F = ma unit is a Newton (or pound) First Law F = 0 a = 0 so v = constant Third law For every force there is an equal and opposite

reaction force

FFFf

Ff

mg

N

Weight = mg

F = ma v = v0 + at d = v0t + ½ at2 d = ½(v + v0)t v2 = v02 + 2ad

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Examples

30 – 8 – T = 4aT – 6 = 2a30 – 8 – 6 = 6a mg

N

g

+

N – mg = maa + N > mga – N < mg

T

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ForcesForces are responsible for all physical phenomena

Gravitation and the electromagnetic force are responsible for all the phenomena we normally observe in our everyday life.

Newton’s laws F = ma where F is net force

v = v0 + at d = v0t + ½ at2 d = ½(v + v0)t v2 = v02 + 2ad

Every force produces an equal and opposite reactionWeight = mg where g = 9.8m/s2 locallyApparent weight in an elevator depends on the acceleration

a up weight is higher a down weight is lower

If your weight becomes zero it’s time to worry because you are in free fall!!

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Circular motion, gravitation

F = ma = mv2/r

Ff

Rear

Ferris wheel

Bottom N - mg = mv2/rtop Mg – N = mv2/r

Ff = mv2/rW = mg

Nv

Mg –N = mv2/r

GmM/r2 = mv2/r

T2/r3 = 4π2/GMs

v2 = GM/r T = 2πr/v

T2 = 4π2r2/v2 = 4π2r3/GMs

Gravitation

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Examples of circular motion

W = mg

N

v

mg – N = mv2/r

N

mgN - mg = mv2/r

v

Looking down

N

N = mv2/r

Side

mg

Ff

mg = Ff

Vertical motion

v

Tmg

mg + T = mv2/r top

T - mg = mv2/r bottom

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Work energy and PowerKinetic energy = 1/2mv2

W = Fd and can be + or –F is net force parallel to d.Units are joulesPower = W/t watts

Potential energy = mghSpring = 1/2kx2

Oscillations Transfer of KE PE

Conservative forceTransfer of KE PE

F

d

F = mgh

g

v