Work and Energy
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Work and kinetic energy
total work-kinetic energy theorem
Potential energy
Mechanic energy
Conservation of energy
power
1.
Work
A. Definition
In science, the word work has a different meaning than you may be familiar
with.
The scientific definition of work is: using a force to displace an object (when
both the force and the motion of the object are in the same direction.)
Work can only be done to a
system by an external force;
a force from something that
is not a part of the system.
So if our system is a plane on
an aircraft carrier and we come
along and push the plane, we
can increase the energy of the
plane…
We are essentially doing work
on the plane.
B. Calculating Work
The amount of work done on an object can be expressed by the
given by the equation:
𝑊 = 𝐹 ∙ ∆𝑥 = 𝐹 ∙ ∆𝑥 ∙ 𝑐𝑜𝑠𝜃
Here θ is the angle between force 𝐹 and displacement ∆𝑥 .
Meaning, work is the product of the force applied which moves the
object a parallel displacement ( we will use “d” for the displacement)
W = Fdparallel
There are some important points to
understand about this equation.
Work
If the object that is experiencing the force does not move
(if d = 0) then no work is done.
parallel
Negative Work
If the object moves in the direction opposite the direction of the force (for instance if force and
displacement are in opposite directions)
then the work is negative: W < 0.
This can happen when a force is applied to slow and object down.
Direction of motion
F
mass
Displacement
Positive Work
If the object moves in the same direction as the direction of the force
(for instance if force and displacement are in the same direction)
then the work is positive: W > 0.
This happens when you push an object to the right to move it rightward.
F
mass
Displacement
If there are several forces acting on an object,
The total work done by all these forces can be
calculated as:
𝑊𝑡𝑜𝑡𝑎𝑙 =𝑊𝐹1 +𝑊𝐹2 + 𝑊𝐹3 +…….
𝑊𝑡𝑜𝑡𝑎𝑙 =𝑊𝑛𝑒𝑡 𝐹𝑜𝑟𝑐𝑒
Units of Work and Energy
W =Fd𝒄𝒐𝒔𝜽= Fd
parallel
This equation gives us the units of work. Since force is measured in Newtons (N) and
displacement is measured in meters (m) the unit of work is the Newton-meter (N-m).And
since N = kg-m/s ; a N-m also equals a kg-m /s .
2
2
2
However, in honor of James Joule, who made critical contributions in developing the idea of
energy, the unit of energy is also know as a Joule (J).
J
Joule
=
N-m
Newton-meter
=
kg-m /s
2
2
kilogram-meter2/second2
A +24 N force is applied to an object that moves 10 m in the same
direction during the time that the force is applied. How much work is
done to the object?
answer
1
W= Fd
W= (24N)(10m)
W= 240J
A +24 N force is applied to an object that moves 10 m in the opposite
direction during the time that the force is applied. How much work is
done to the object?
answer
2
W= Fd
W= (24N)(-10m)
W= -240J
A +24 N force is applied to an object that is stationary during the time
that the force is applied. How much work is done to the object?
answer
3
W= Fd
W= (24N)(0m)
W= 0 J
How much force must be applied to an object such that it gains 100J
of energy (which means 100 J of work is done) over a distance of 20
m?
answer
4
W= Fd
F = W/d
F= 100/20
F= 5N
Over what distance must a 400 N force be applied to an object such
that it gains 1600J of energy (meaning 1600 J of work is done)?
answer
5
W= Fd
d = W/F
d= 1600/400
d= 4m
A girl pulls a sled at a constant speed 1.2 m/s by applying a force of
350 N. How much work will be done during 100 seconds?
answer
6
d= st
d = (1.2)(100)
d = 120m
W= Fd
W= (350)(120)
W= 42000J
A book is held at a height of 2.0 m for 20 s. How much work is done
on the book?
answer
7
Object remains stationary
therefore no work is done
8
A
mg
B
C
D
E
-mgh
mgh
0
-mg
Hint: Do a free body diagram to
determine a formula for the outside
force (Fapp); then use the formula for
work: W = Fdparallel.
answer
A barbell of mass "m" is lifted vertically upwards, at a constant velocity,
to a distance "h" by an outside force. How much work does that outside
force do on the barbell?
C
2. Energy
A. Definition
It turns out that energy is so fundamental, like space and time, that there is no good
answer to this question. However, just like space and time, that doesn't stop us from
doing very useful calculations with energy.
Energy is defined as the ability to do work.
We may not be able to define energy, but because it is a conserved property of
nature, it's a very useful idea.
2. Energy
B. Types
2. Energy
B. Types
Energy can come in different forms.
1. Non Mechanical Energy
• non mechanical energy is energy that is not related to mechanics; in
other words, no work was done
• There are five examples you should be familiar with:
a.
b.
c.
d.
e.
Thermal (Heat) Energy
Chemical Energy
Electrical Energy
Nuclear Energy
Electromagnetic Radiation (R M I V U X G)
2. Energy
B. Types
2. Mechanical Energy
• When work is done to an object, it acquires energy..
• The energy it acquires is known as mechanical energy.
• There are two forms of mechanical energy:
a. Potential Energy – energy of position
b. Kinetic Energy – energy of motion
3.
Kinetic Energy
The energy an object has by
virtue of its motion is called its
kinetic energy. The symbol we
will be using for kinetic energy is
KE.
Like all forms of energy, it is
measured in Joules (J).
The amount of KE an object has is given by:
KE = ½ mv2
Note: Kinetic energy KE is scalar quantity and will never be negative.
As an object falls, its KE always _____.
A
decreases
B
increases
C
stays the same.
answer
13
B
What is the kinetic energy of a 12 kg object with a velocity of 10 m/s?
KE = 0.5 mv
E = (0.5)(12)(10 )
2
K
answer
15
KE = 600 J
2
What is the mass of an object which has 2400 J of KE when traveling
at 6.0 m/s?
KE = 0.5 mv
= KE / (0.5)(v )
m = 2400 / (0.5)(6 )
2
m
2
2
answer
16
m = 133.33kg
A 3 kg object has 45 J of kinetic energy. What is its velocity?
KE = 0.5 mv
= KE/0.5m
v = 45 / 0.5 (3)
2
v2
2
answer
17
v = 5.48 m/s
If the speed of a car is doubled, the KE of the car is:
A
quadrupled
B
quartered
C
halved
D
doubled
answer
18
A
Which graph best represents the relationship between the KE and
the velocity of an object accelerating in a straight line?
C
A
KE
KE
v
v
B
D
KE
KE
v
swer
19
v
D
20
The data table below lists mass and speed for 4 objects. Which 2
have the same KE?
A
A and D
B and D
C
A and C
D
B and C
answer
B
D
4. Potential Energy
A barbell of mass "m" is lifted vertically upwards a distance "h" by an outside
force. How much work does that outside force do on the barbell?
F
app
mg
W = Fdparallel
Since a = 0, Fapp = mg
W = (mg) dparallel
Since F and d are in the same
W = (mg) h
direction ...and dparallel = h
W = mgh
4.1 Gravitational Potential Energy
The name for this form of energy—when work is done against
gravity--is
Gravitational Potential Energy (GPE).
GPE = mgh
Gravitational Potential Energy, PE=mgh
• Energy of position
• Stored energy
• Symbol: U or PE
• Unit: Joule
• Compared to a Reference
Point
(base level)
example
A flower pot of mass 5kg is
located 15m above the
ground.
A)What is its potential energy
with respect to the ground?
B) What work was done to
raise it to its position?
A: 750J
What is the change of GPE for a 5.0 kg object which is raised from the
floor to a final height of 2.0m above the floor?
answer
9
GPE=mgh
GPE= (5kg)(9.8)(2m)
GPE=98 J
10
As an object falls, its GPE always _____.
increases
B
decreases
C
stays the same
answer
A
B
What is the change of GPE for a 8.0 kg object which is lowered from an
initial height of 2.0 m above the floor to a final height of 1.5m above the
floor?
answer
11
GPE=mgh
GPE= (8)(9.8)(-0.5)
GPE= -39.2 J
What is the change in height of a 2.0 kg object which gained 16 J of
GPE?
answer
12
GPE=mgh
h = GPE/mg
h = 16/(2)(9.8)
h = 0.82m
4.2. Elastic Potential Energy
Energy can be stored in a spring, this energy is called
Elastic Potential Energy.
Robert Hooke first observed the relationship between the
force necessary to compress a spring and how much the
spring was compressed.
Hooke's Law
Fspring = -kx
k represents the spring constant and is measured in N/m.
x represents how much the spring is compressed and is
measured as you would expect, in meters.
The - sign tells us that this is a restorative force.
(if you let the spring go once it is compressed, it
will go back to its original position)
Elastic Potential Energy
The energy imparted to the spring by this work must be stored
in the Elastic Potential Energy (EPE) of the spring:
EPE = ½ k x
2
Like all forms of energy, it is measured in Joules (J).
Determine the elastic potential energy stored in a spring whose spring
constant is 250 N/m and which is compressed 8 cm.
EPE = 0.5 kx
PE = 0.5 (250)(0.08 )
2
E
answer
21
EPE = 0.08 J
2
What is the spring constant of a spring that is compressed 5 cm and
has 0.65 J of elastic potential energy stored in it?
EPE = 0.5 kx
= EPE/0.5x
= 0.65 / 0.5 (0.05 )
2
k
k
answer
22
k = 520 N/m
2
2
How much does a spring with a spring constant of 500 N/m need to be
compressed in order to store 1.75 J of elastic potential energy?
EPE = 0.5 kx
= EPE / 0.5k
x = 1.75 / 0.5(500)
2
x2
2
answer
23
x = 0.08m
A 3 kg mass compresses a spring 2.5 cm. What is the spring constant?
answer
24
F = ma F = kx
F = 3(9.8)
k=F/x
F = 29.4N
k = 29.4 / 0.025
k = 1176 N/m
k = 1176 N/m
The same 3 kg mass compresses the same spring 2.5 cm. How much
elastic potential energy is stored in the spring?
EPE = 0.5kx
PE = (0.5)(1176)(0.025 )
2
E
answer
25
EPE = 0.368 J
2
4.4 Mechanical Energy
5. Power
𝑾
𝑷=
𝒕
Average power = Work/Time
Power is the rate at which work is done:
units: 1 J/s =1 watt (W)
Since work is measured in Joules
(J) and time is measured in
seconds (s) the unit of power is
Joules per second (J/s).
However, in honor of James Watt,
who made critical contributions in
developing efficient steam engines,
the unit of power is also know as a
Watt (W).
B. Calculating
Since W = Fd
parallel
Regrouping this becomes
Since v = d/t
So power can be defined as the product of the force
applied and the velocity of the object parallel to that force.
A steam engine does 50 J of work in 12 s. What is the power supplied by
the engine?
answer
26
27
27
A 3.0 kg block is initially at rest on a frictionless, horizontal surface. The
block is moved 8.0m in 2.0s by the application of a 12 N horizontal force,
as shown in the diagram below. What is the power developed when
moving the block?
F = 12 N
24 W
B
32 W
C
48 W
D
96 W
Frictionless
surface
8.0 m
answer
A
3.0 kg
C
How long must a 350 W engine run in order to produce 720 kJ of work?
answer
28
A 12 kW motor runs a vehicle at a speed of 8 m/s. What is the force
supplied by the engine?
answer
29
4. Work-Energy Theorem
W=KE
W=KEf-KEi
The net work done on an object is equal to the
change in its kinetic energy.
Problem
If Robin Hood applies a 15N
force to pull his bow back by
25cm, with what speed will
his 0.150kg arrow leave the
bow?
A: 7.1m/s
Problem 2
What is the work
done by gravity
as an 8 kg object
falls from rest at
15m to 5m?
What is the
object’s speed at
5m?
A: 800J, 14m/s
6. Transformation and Conservation Principles
The most powerful concepts in science are called "conservation principles". These
principles allow us to solve problems without worrying too much about the details of a
process.
Conservation Principles
A good example is a bag of candy.
If you know that there are 50 pieces of candy at the beginning of the day and
you know that none of the pieces have been taken out or added...you know that
there must be 50 pieces at the end of the day.
Conservation Principles
Perhaps during the day you changed the way you arrange them by moving them
around...but you still will have 50 pieces. In that case we would say that the number of
pieces of candy is conserved.
That is, we should always get the same amount, regardless of how they are arranged.
The same amount exists before and after you moved them around.
Conservation Principles
We also have to be clear about the system that we're talking about. If we're
talking about a specific type of candy...we can't suddenly start talking about a
different one and expect to get the same answers.
We must define the “system” whenever
we use a conservation principle.
Conservation of Energy
Energy is a conserved property of
nature. It is not created or destroyed.
Therefore in a closed system we will
always have the same amount of
energy.
The only way the energy of a system
can change is if it is open to the
outside...this means that energy has
been added or taken away.
7. Conservation of Energy
• Energy can neither be created nor destroyed. It
can only change from form to form.
• Mechanical Energy, PE and KE
• Conservation of Mechanical Energy
PE1 + KE1 = PE2 + KE2
Return to
Table of
Contents
https://www.khanacademy.org/science/physics/work-and-energy/workand-energy-tutorial/v/conservation-of-energy
7.1: Conservative and Nonconservative
Forces
The gravitational force has an interesting property that when an object is moved
from one place to another, the work done by the gravitational force does not
depend on the choice of path.
Forces like these are called conservative forces.
Definition Of A Conservative Force
A force is conservative when the work it does on a moving object is independent of
the path between the object's initial and final positions.
Definition Of A nonconservative
Force
A force is nonconservative when the work it does on a moving object is dependent of
the path between the object's initial and final positions.
Examples
Conservative Forces
Gravitational force (Ch. 4)
Elastic spring force (Ch. 10)
Electric force (Ch. 18, 19)
Nonconservative Forces
Static and kinetic frictional forces
Air resistance
Tension
Normal force
Propulsion force of a rocket
7.2 The Conservation of Mechanical
Energy
THE PRINCIPLE OF
CONSERVATION OF MECHANICAL
ENERGY
The total mechanical energy (E = KE + PE) of an object
remains constant as the object moves, provided that
the net work done by external nonconservative forces
is zero.
Conservation of Mechanical Energy
If friction and wind resistance are ignored, a bobsled run illustrates
how kinetic and potential energy can be interconverted, while the
total mechanical energy remains constant.
A Daredevil Motorcyclist
A motorcyclist is trying to leap across the canyon shown in Figure 6.18
by driving horizontally off the cliff at a speed of 38.0 m/s. Ignoring air
resistance, find the speed with which the cycle strikes the ground on
the other side.
Roller Coaster (Ideal)
The tallest and fastest roller coaster in the world is now
the Steel Dragon in Mie, Japan (Figure 6.20). The ride
includes a vertical drop of 93.5 m. The coaster has a
speed of 3.0 m/s at the top of the drop. Neglect friction
and find the speed of the riders at the bottom.
6.6 Nonconservative Forces and
the Work–Energy Theorem
In the roller coaster example, we ignored nonconservative
forces, such as friction. In reality, however, such forces are
present when the roller coaster descends. The actual
speed of the riders at the bottom is 41.0 m/s. Assuming
again that the coaster has a speed of 3.0 m/s at the top,
find the work done by nonconservative forces on a 55.0-kg
rider during the descent.
Problem 5
A 3kg watermelon sits on a table
1.2m above the ground.
a)What is its PE at the top
compared to the ground?
b) What is its KE at the top?
c) With what speed will it hit the
ground?
A: 36J, 0J, 4.9m/s
Problem 6
A penny is at the top of the
Empire State Building 381m
above the ground. It is then
released. With what speed will
it hit the ground?
A: 87.3 m/s
Problem 7-Roller Coaster
A roller coaster car started from point A
at a height of 100m. What is its speed
at point B?
A: 44.7m/s
Problem 8 - Pendulum
a) What is the potential energy
of the bob at point A
compared to the ground?
b) What is the speed of the
bob at the bottom?
A: 2.3J, 1.1m/s
A
Conservation of Energy
A roller coaster is at the top of a track that is 80 m high.
How fast will it be going at the bottom of the hill?
Eo + W = Ef
Eo = Ef
GPE = KE
mgh = 0.5mv2 …… solve for v2
V2 = 2gh
2
v = 2 (9.8) 80
v =39.6 m/s
answer
A spring gun with a spring constant of 250 N/m is
compressed 5 cm. How fast will a 0.025 kg dart go when it
leaves the gun?
answer
A student uses a spring (with a spring constant of 180 N/m)
to launch a marble vertically into the air. The mass of the
marble is 0.004 kg and the spring is compressed 0.03 m.
How high will the marble go?
h = 2.066m
A student uses a spring gun (with a spring constant of 120
N/m) to launch a marble vertically into the air. The mass of
the marble is 0.002 kg and the spring is compressed 0.04 m.
a)How high will the marble go?
b)How fast will it be going when it leaves the gun?
A)
answer
B)
v=9.8m/s
h = 4.897m
answer
A roller coaster has a velocity of 25 m/s at the bottom of the
first hill. How high was the hill?
h=31.9m
answer
A 5 kg rock is dropped a distance of 1 m onto a spring. It
compresses the spring 2 cm. What is the spring constant?
k=245000N/m
answer
A student uses the lab apparatus shown above. A 5 kg block
compresses a spring 6 cm. The spring constant is 300 N/m.
What will the block's velocity be when released?
answer
How much work is done in stopping a 5 kg bowling ball
rolling with velocity of 10 m/s?
answer
How much work is done compressing a spring with a 450
N/m spring constant 2 cm?