Energy, Work and Power

Energy

Energy: the currency of the universe. Just like money, it comes in many forms!

Everything that is accomplished has to be “paid for” with some form of energy.

Energy can’t be created or destroyed, but it can be transformed from one kind into another and it can be transferred from one object to another.

• Doing WORK is one way to transfer energy from one object to another.

Work = Force x displacement

W = F∙d• Unit for work is Newton x meter. One

Newton-meter is also called a Joule, J.

Work- the transfer of energy

Work = Force x displacement• Work is not done unless there is a displacement. • If you hold an object a long time, you may get tired,

but NO work was done on the object.• If you push against a solid wall for hours, there is

still NO work done on the wall.

• For work to be done, the displacement of the object must be along the same direction as the applied force. They must be parallel.

• If the force and the displacement are perpendicular to each other, NO work is done by the force.

• For example, in lifting a book, the force exerted by your hands is upward and the displacement is upward- work is done.

• Similarly, in lowering a book, the force exerted by your hands is still upward, and the displacement is downward.

• The force and the displacement are STILL parallel, so work is still done.

• But since they are in opposite directions, now it is NEGATIVE work.

F

F

d

d

• On the other hand, while carrying a book down the hallway, the force from your hands is vertical, and the displacement of the book is horizontal.

• Therefore, NO work is done by your hands.

• Since the book is obviously moving, what force IS doing work???

The static friction force between your hands and the book is acting parallel to the displacement and IS doing work!

F

d

ExampleHow much work is done to push a 5 kg

cat with a force of 25 N to the top of a ramp that is 7 meters long and 3 meters tall?

W = Force x displacement

Which measurement is parallel to the force- the length of the ramp or the height of the ramp?

W = 25 N x 7 m

W = 175 J 7 m

3 mF = 25 N

ExampleHow much work is done to carry a 5 kg

cat to the top of a ramp that is 7 meters long and 3 meters tall?

W = Force x displacement

What force is required to carry the cat?

Force = weight of the cat

Which is parallel to the weight vector- the length of the ramp or the height?

d = height NOT length

W = mg x h

W = 5 x 10 x 3

W = 150 J

7 m

3 m

• And,….while carrying yourself when climbing stairs or walking up an incline, only the height is used to calculate the work you do to get yourself to the top!

• The force required is your weight!

Horizontal component of d

Ver

tical

com

pone

nt o

f d

Yo

ur

Fo

rce

How much work do you do on a 30 kg cat to carry it from one side of the room to the other if the room is 10 meters long?

ZERO, because your Force is vertical, but the displacement is horizontal.

Pre-AP only…

ExampleA boy pushes a

lawnmower 20 meters across the yard. If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees, how much work did he do?

q

F

Displacement = 20 m

F cos q

W = (F cos q )dW = (200 cos 50˚) 20 mW = 2571 J

A 5.0 kg box is pulled 6 m across a rough horizontal floor (m = 0.4) with a force of 80 N at an angle of 35 degrees above the horizontal. What is the work done by EACH force exerted on it? What is the NET work done?

●Does the gravitational force do any work? NO! It is perpendicular to the displacement.● Does the Normal force do any work? No! It is perpendicular to the displacement.● Does the applied Force do any work? Yes, but ONLY its horizontal component!

WF = FAcosq x d = 80cos 35˚ x 6 m = 393.19 J● Does friction do any work? Yes, but first, what is the normal force? It’s NOT mg!

Normal = mg – FAsinq

Wf = -f x d = -mN∙d = -m(mg – FAsin )q ∙d = -7.47 J● What is the NET work done?393.19 J – 7.47 J = 385.72 J

mg

Normal

FA

qf

Watch for those “key words”

NOTE: If while pushing an object, it is moving at a constant velocity,

the NET force must be zero.

So….. Your applied force must be exactly equal to any resistant forces like friction.

• Energy and Work have no direction associated with them and are therefore scalar quantities, not vectors.

YEAH!!

• Power is the rate at which work is done- how fast you

do work. Power = work / time

P = W / t• You may be able to do a lot

of work, but if it takes you a long time, you are not very powerful.

• The faster you can do work, the more powerful you are.

• The unit for power is Joule / seconds which is also called a Watt, W

(just like the rating for light bulbs)

In the US, we usually measure power developed in motors in “horsepower”

1 hp = 746 W

Example

A power lifter picks up a 80 kg barbell above his head a distance of 2 meters in 0.5 seconds. How powerful was he?

P = W / t

W = Fd

W = mg x h

W = 80 kg x 10 m/s2 x 2 m = 1600 J

P = 1600 J / 0.5 s

P = 3200 W

Another way of looking at Power:

Power = Force x velocity

powerwork

time

power = (force x displacement)

time

power force x displacement

time

power force x velocity

Kinds of Energy

Kinetic Energy

the energy of motion

K = ½ mv2

Kinetic Energy

the energy of motion

K = ½ mv2

Potential EnergyStored energy

It is called potential energy because it has the potential to do work.

Different kinds of Potential (stored) Energy

• Example 1: Spring potential energy, SPE, in the stretched string of a bow or spring or rubber band. SPE = ½ kx2

• Example 2: Chemical potential energy in fuels- gasoline, propane, batteries, food!

• Example 3: Gravitational potential energy, GPE- stored in an object due to its position from a chosen reference point.

Gravitational potential energyGPE = weight x

heightGPE = mgh

Since you can measure height from more than one reference

point, it is important to specify the

location from which you are

measuring.

• The GPE may be negative. For example, if your reference point is the top of a cliff and the object is at its base, its “height” would be negative, so mgh would also be negative.

• The GPE only depends on the weight and the height, not on the path that it took to get to that height.

Many different forms of Energy…

Thermal Energy

Solar Energy

Atomic Energy

Sound Energy

Electromagnetic Energy

Nuclear Energy

Electrical Energy

E = mc2

Work and EnergyOften, some force must do work

to give an object potential or kinetic energy.

“Work” is the transfer of energy!! You push a wagon and it starts

moving kinetic energy. You stretch a spring and you transform your work energy spring potential energy.

Or, you lift an object to a certain height- you transfer your work energy into the object in the form of gravitational potential energy.

The Work-Kinetic Energy Theorem

NET Work done by all forces = D Kinetic Energy

Wnet = ½ mv2f – ½ mv2

o

NET Work = D Kinetic Energy

How much more distance is required to stop if a car is going twice as fast (all other things remaining the same)?

The work done by the forces stopping the car = the change in the kinetic energy

Fd = D½ mv2

With TWICE the speed, the car hasFOUR times the kinetic energy.

Therefore it takes FOUR times the stopping distance.(What FORCE is doing the work??)

A car going 25 km/h will skid to a stop over a distance of 7 meters.

If the same car was moving at 50 km/h, how many meters would be required for it to come to a stop?

The velocity DOUBLED, therefore the stopping distance is FOUR times the original distance:

7 meters x 4 = 28 meters

Example, Wnet = Fnetd = DKA 500 kg car moving at 15 m/s skids 20 m to a

stop.

How much kinetic energy did the car lose?

DK = ½ mvf2 – ½ mvo

2 (but vf = 0!)

DK = -½ (500 kg)(15 m/s) 2

DK = -56250 J

What force was applied to stop the car?

F·d = DK

F = DK / d

F = -56250 J / 20 m

F = -2812.5 N

Example Wnet = Fdnet = DKA 500 kg car moving at 15 m/s slows to 10 m/s.

How much kinetic energy did the car lose?

DK = ½ mvf2 – ½ mvo

2

DK = ½ (500 kg)(10 m/s)2 - ½ (500 kg)(15 m/s)2

DK = -31250 J

What force was applied to slow the car if the distance moved was 12 m?

F·d = DK

F = DK / d

F = -31250 J / 12 m

F = -2604 N

Fd = D ½ mv2

A 0.02 kg bullet moving at 90 m/s strikes a block of wood. If the bullet comes to a stop at a depth of 2.5 cm inside the wood, how much force did the wood exert on the bullet?

F = 3240 N

Example Wnet = Fnetd = DKA 500 kg car moving on a flat road at 15 m/s skids to

a stop.How much kinetic energy did the car lose?DK = ½ mvf

2 – ½ mvo2

DK = -½ (500 kg)(15 m/s)2

DK = -56250 JHow far did the car skid if the effective coefficient of

friction was = m 0.6?Stopping force = friction = mN = mmgF·d = DK-(mmg)·d = DKd = DK / (-mmg) *be careful to group in the denominator!d = 56250 J / (0.6 · 500 kg · 9.8 m/s2) = 19.13 m

Back to Power…

Since Power = Work / time and

Net work = DK…

Power = DK / time

In fact, Power can be calculated in many ways since Power = Energy / time, and

there are MANY forms of energy!

Conservation of Mechanical Energy1. Draw a sketch and choose a reference point for height.

2. Look at the first position of your object. If it is moving, it has Kinetic energy. If it has some height above or below your reference point, it has Potential energy.

3. Repeat for the second location.

4. If there is no friction or air resistance, set the mechanical energies at each location equal.

E1 = E2

mgh1 + ½mv12 = mgh2 + ½ mv2

2

5. If there is friction or air resistance, use E1 – E2 to find the energy lost.

Graphing Force vs. postion

• If you graph the applied force vs. the position, you can find how much work was done by the force.

Work = Fd = “area under the curve”.Total Work = 2 N x 2 m + 3N x 4m = 16 JArea UNDER the x-axis is NEGATIVE work = - 1N x 2 m

Force, N

Position, m

F

dNet work = 16 J – 2 J = 14 J

Back to the Work-Kinetic Energy Theorem…

According to that theorem,

net work done = the change in the kinetic energy

Wnet = DK

But, if the work can be found by taking the “area under the curve”, then it is also true that

Area under the curve = DK = ½ mvf2 – ½ mvo

2

Therefore, the area can be used to predict the final velocity of an object given its initial velocity

and its mass.

For example…

Suppose from the previous graph

(Area = Wnet = 14 J), the object upon which the forces were exerted had a mass of 3 kg and an initial velocity of 4 m/s. What would be its final velocity?

Area under the curve = ½ mvf2 – ½ mvo

2

14 J = ½(3 kg)vf 2 – ½(3 kg)(4 m/s)2

vf = 5.0 m/s

The Spring Force

If you hang an object from a spring, the gravitational force pulls down on the object and the spring force pulls up.

The Spring Force

The spring force is given by

Fspring = kx

Where x is the amount that the spring

stretched and k is the “spring constant”

which describes how stiff the spring is

The Spring ForceIf the mass is hanging at

rest, then Fspring = mg

Orkx = mg

(this is called “Hooke’s Law)

The easiest way to determine the spring

constant k is to hang a known mass from the

spring and measure how far the spring stretches!

k = mg / x

Graphing the Spring Force

Suppose a certain spring had a spring constant

k = 30 N/m.

Graphing spring force vs. displacement:

On horizontal axis- the displacement of the spring: x

On vertical axis- the spring force = kx = 30x

What would the graph look like?

Fs = kx

In “function” language: f(x) = 30x

x1x2

Spring Force vs. Displacement

x

Fs = 30x

How could you use the graphTo determine the work done by The spring from some x1 to x2?Take the AREA under the curve!

Fs

Analytically…

The work done by the spring is given by

Ws = ½ kxf2 – ½ kxo

2

where x is the distance the spring is stretched or compressed

(Which would yield the same result as taking the area under the curve!)

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Mechanical Energy

Mechanical Energy = Kinetic Energy + Potential Energy

E = ½ mv2 + mgh

“Conservative” forces - mechanical energy is conserved if these are the only forces acting on an object.

The two main conservative forces are: Gravity, spring forces

“Non-conservative” forces - mechanical energy is NOT conserved if these forces are acting on an object.

Forces like kinetic friction, air resistance (which is really friction!)

Conservation of Mechanical EnergyIf there is no kinetic friction or air resistance, then the

total mechanical energy of an object remains the same.

If the object loses kinetic energy, it gains potential energy.

If it loses potential energy, it gains kinetic energy.

For example: tossing a ball upward

Conservation of Mechanical Energy

The ball starts with kinetic energy…

Which changes to potential energy….

Which changes back to kinetic energy

K = ½ mv2

PE = mgh

K = ½ mv2

Energybottom = Energytop

½ mvb2 = mght

What about the energy when it is not at the top or bottom?

E = ½ mv2 + mgh

Examples

• dropping an object• box sliding down an incline• tossing a ball upwards• a pendulum swinging back and forth• A block attached to a spring oscillating

back and forth

First, let’s look at examples where there is NO friction and NO air resistance…..

Example of Conservation of Mechanical Energy

Rapunzel dropped her hairbrush from the top of the castle where she was held captive. If the window was 80 m high, how fast was the brush moving just before it hit the ground? (g = 10 m/s2)

mgh1 + ½ mv12 = mgh2 + ½ mv2

2

mgh = ½ mv2

gh = ½ v2

2gh = v2

Don’t forget to take the square root!

Enter your answer on #1

Now… do one on your own

#2 An apple falls from a tree that is 1.8 m tall. How fast is it moving just before it hits the ground? (g = 10 m/s2)

mgh1 + ½ mv12 = mgh2 + ½ mv2

2

mgh = ½ mv2

And another one…#3 A woman throws a ball straight up with an initial velocity of 12 m/s. How high above the release point will the ball rise? g = 10 m/s2

mgh1 + ½ mv12 = mgh2 + ½ mv2

2

½ mv2 = mgh

h = ½ v2 / g

And another one…

#4

Mario, the pizza man, tosses the dough upward at 8 m/s. How high above the release point will the dough rise?

g = 10 m/s2

mgh1 + ½ mv12 = mgh2 + ½ mv2

2

Conservation of Mechanical Energy- another lookA skater has a kinetic energy of 57 J at position 1, the bottom of the ramp (and NO potential energy)At his very highest position, 3, he comes to a stop for just a moment so that he has 57 J of potential energy (and NO kinetic energy)Mechanical energy = KE + PE

#5 What is his kinetic energy at position 2, if his potential energy at position 2 is 25.7 J?

E = 57 J

E = 57 J

PE = 25.7 JKE = ??

Conservation of Mechanical Energy… more difficult

A stork, at a height of 80 m flying at 18 m/s, releases his “package”. How fast will the baby be moving just before he hits the ground?

Energyoriginal = Energyfinal

mgh + ½ mvo2 = ½ mvf

2

 

Vf = 43.5 m/s

#6 Now you do one …

The car on a roller coaster starts from rest at the top of a hill that is 60 m high. How fast will the car be moving at a height of 10 m? (use g = 9.8 m/s2)

mgh1 + ½ mv12 = mgh2 + ½ mv2

2

mgh1 = mgh2 + ½ mv22

# 6 Enter your answer with ONE decimal place.

If there is kinetic friction or air resistance, mechanical energy will not be conserved.

Mechanical energy will be lost in the form of heat energy.

The DIFFERENCE between the

original energy and the final energy

is the amount of mechanical energy lost due to heat.

Final energy – original energy = energy loss

Let’s try one…#7 A 2 kg cannonball is shot straight up from the ground at 18 m/s. It reaches a highest point of 14 m. How much mechanical energy was lost due to the air resistance? g = 10 m/s2

Final energy – original energy = Energy loss

mgh – ½ mv2 = Heat loss

2 kg(10 m/s2)(14 m) – ½ (2 kg)(18 m/s)2 = ??

And one more…

#8A 1 kg flying squirrel drops from the top of a tree 5 m tall. Just before he hits the ground, his speed is 9 m/s. How much mechanical energy was lost due to the air resistance?

g = 10 m/s2

Final energy – original energy = Energy loss

Sometimes, mechanical energy is actually INCREASED!

For example: A bomb sitting on the floor explodes.

Initially:Kinetic energy = 0 Gravitational Potential energy = 0 Mechanical Energy = 0After the explosion, there’s lots of kinetic

and gravitational potential energy!!Did we break the laws of the universe and

create energy???Of course not! NO ONE, NO ONE, NO ONE

can break the laws!The mechanical energy that now appears

came from the chemical potential energy stored within the bomb itself!

Don’t even think about it…

According to the Law of Conservation of Energy

energy cannot be created or destroyed.

But one form of energy may be transformed into another form as conditions change.

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Simple Machines and Efficiency

Machine: A device that HELPS do work.

A machine cannot produce more WORK ENERGY than the energy you put into it- that would break the Law of Conservation of Energy-

but it can make your work easier to do.

• Some common “simple machines” include levers, pulleys, wheels and axles, and inclined planes

• Ideally, with no friction, the work energy you get out of a machine equals the work energy you put into it.

Ideally:

Work energy in = work energy out

Work = Force x distance

The work you put into a machine is called EFFORT work.

The work you get out of the machine- is called RESISTANCE work, so ideally

Effort Work = Resistance Work

Feffortdeffort = Fresistancedresistance

(if there’s no NON-conservative forces!)

Try one…

# 1. Hercules pushes a 500 kg boulder up a hill a distance of 25 m using a force of 6000 N. How much work did Hercules do?

Effort Work = Resistance WorkFeffortdeffort = Fresistancedresistance

Levers

• The RESISTANCE force is the weight of the load being lifted.

#2 Which arrangement will require the least EFFORT force?

Effort forceEffort force Effort forceA B C

Levers

3. How do you “pay” for a small effort force?A) You push harderB) You push just the sameC) You push a smaller distanceD) You push a greater distance

Effort forceEffort force Effort force

Inclined Planes

#4. Which arrangement will require the least EFFORT force?

How do you “pay” for a smaller effort force?

Effort Force

Effort Distance Height =

Resistance Distance

Weight =Resistance Force

A B C

Two pulleys with a belt

A motor is attached to one of the pulleys so that as it turns, the belt causes the second pulley to turn.

To have the least effort force, the effort distance must be the greatest. In this case the effort distance is the number of turns around – the ROTATIONS!

Which pulley will have to go around more times? This is the pulley that the motor should be attached to for the least effort force.

# 5 Which pulley should the motor be attached to so that it requires the least effort force from the motor?

A B

Efficiency

No machine is perfect. That is reflected in the “efficiency” of the machine. In the real world, the efficiency will always be less that 100%. It is found by

)effort(inwork

)cetanresis(outwork

inEnergy

outEnergyEfficiency

Some practice…

#6 While using a simple machine, you put in 4500 J of work energy. The machine puts out 3690 J of energy. What was the percent efficiency of the machine?

)effort(inwork

)cetanresis(outwork

inEnergy

outEnergyEfficiency

A man pushes a 48 kg box up a 12 m long incline that is 4.2 meters high by applying a force of 240 N. (g = 10 m/s2)

What is the effort (input) work?

Weffort = Feffortdeffort

#7 We = ?What is the resistance (output) work?

Wresistance = Frdr

W = mg x h

#8 Wr = ?

#9 What is the percent efficiency of the incline?

)effort(inwork

)cetanresis(outwork

inEnergy

outEnergyEfficiency