Download Definition of Work Resultant Work (Net Work) Resultant Work Work

11/30/2014
Resultant Work
So far: Motion analysis with forces.
NOW: An alternative analysis using the
concepts of Work & Energy.
This box is moved 4m with the Net Force
Resultant work is also equal to the work of the
RESULTANT force.
4m
80 N
-10 N
Easier? Yes
Conservation of Energy: NOT a new
law! Just Newton’s Laws in a different
language.
Work = (F - f) x
Work = (80 - 10 N)(4 m)
Work = 280 N m
Box gains 280N m of work
Definition of Work
Work by a Constant Force
Work is a scalar quantity equal to the
product of the displacement x and the
component of the force Fx in the direction
of the displacement.
Work = Force component d
displacement
Work = Fx d
Work = (F cos ) x
WORK AND ENERGY ARE SCALAR
Force must be in direction of displacement
If one dimension (Cos 0 = 1) W = F d
Work is negative if force is opposite of direction
Resultant Work (Net Work)
Resultant work is the algebraic sum of
the individual works of each force.
x
Ff
Fa
Fa = 80 N, Ff = -10 N and x = 4 m
I do 80N while Friction does -10 over 4m
Net Work = (80 N)(4 m) + (-10 N)(4 m)
Work of a Force at an Angle
F = 70 N
Work = Fx x
60o
x = 12 m
Work = (F cos ) x
Work = (70 N) Cos 600 (12 m) = 420 J
Work = 420 N m
Only the x-component of
the force does work!
Net Work on box= 280 Nm
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UNIT OF WORK / ENERGY
If a bag is pulled angle of 37 deg, d = 50 m, and an of F = 30 N,
what is the work done on the bag?
Find the work done if m = 50 kg, FP = 1000 N, Ffr = 500
N, θ = 37º
Find the work done if m = 50 kg, Ffr = 0 N, θ = 45º, h=20m
Work done only depends on force against gravity, could have lifted it and done same work
W = Fd = Fdcosθ
• Can exert a force & do no work!
Could have d = 0
W=0
Or
Could have F  d
 θ = 90º, cosθ = 0
W=0
Eight books, each 4.3 cm thick with mass 1.7 kg, lie flat on a table.
How much work is required to stack them one on top of another?
.
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A (frictionless) lawn mower is pushed a horizontal
distance of 20 m by a force of 200 N directed at an angle
of 300 with the ground. What is the work of this force?
x = 20 m
F
300
F = 200 N
Work is positive since Fx and x are in the same direction.
You drag a vacuum cleaner 3.5 m by pulling on the
hose – it makes an angle of 30.0 with a frictionless
surface. The tension is 55.0 N. What is the work?
T 55 N
ff
n
rope makes an angle of 350 with the floor and
the work done on the block? By Friction
T = 40 N; x = 8 m, uk = 0.2;  = 350; m = 4 kg
fk

mg
x
A 40-N force pulls a 4-kg Tire a horizontal distance of 8 m. The
+x
3.5 m
x

k
n
= 0.2. What is
T 40 N

mg
+x
8m
WORK AND INCLINE
PROBLEMS
• OK GO THIS TOO SHALL PASS
http://www.youtube.com/watch?v=qybUFnY
7Y8w
• HONDA VIDEO
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A 330-kg piano slides 3.6 m down a 28º incline and is kept from
accelerating by a man who is pushing (Fp) back on it parallel to the incline
The effective coefficient of kinetic friction is 0.40. Calculate:
(a)the force exerted by the man, (b) the work done by the man on the piano,
(c) the work done by the friction force, (d) the work done by (Fp AKA G),
and (e) the net work done on the piano.
(a)
WORK AGAINST A FORCE KE
+ STORED ENERGY
• Chair example lifting const vel against
gravity (all work goes into gravity. Then
accelerating it against gravity extra work
goes into motion
• Now push against friction const rate
- then push w acceleration (great force
(b)
(c)
The angle between friction and the direction of motion is 180o
WORK CLIMING
STAIRS (+ WORK –
WORK)
(d)
(e)
What is the minimum work (zero a) needed to for a woman to push a 950-kg car
810 m up along a 9.0º incline? (a) Ignore friction. (b) Assume the effective
coefficient of friction retarding the car is 0.25.
(a)
You pull a 55.5 kg wooden box at 250N with a rope that makes a 28.0 angle
with the horizontal. The coefficient of kinetic friction between the box and the
deck is 0.1. You pull the crate a distance of 2.25 m. How much work was
250 N
done?
fk
mg
(b)
x
n
T

+x
2.25 m
.
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You pull a 100 kg wooden box with a rope that makes a 40.0 angle with the horizontal at
a constant speed. The coefficient of kinetic friction between the box and the deck is 0.2.
You pull the crate a distance of 5 m. How much work was done on the box?
Work-Energy Principle
Combine all three
Net work done on an object by a net force is
equal to the change in its Kinetic Energy.
Kinetic Energy; Work-Energy Principle
• Energy  The ability to do work
• Kinetic Energy  The energy of motion
“Kinetic”  Greek word for motion
An object in motion has the ability to do work
Work-Energy Principle
• Net work on an object = Change in KE.
Wnet = KE
– Opposing forces (work) store energy
/reduces KE
– Wnet is a scalar.
– Wnet can be positive or negative
– (because KE can be both + & -)
– Units are Joules for both work & KE.
Kinetic Energy
Kinetic Energy:
Work-Energy Principle
• IF Work >0
• IF Work <0
• IF Work =0
KE INCREASES
KE DECREASES
KE DOES NOT CHANGE
CLIMING STAIRS EXAMPLE:
The Work-Energy Principle
If displacement is upwards the work by gravity is –
the work by you is +
If displacement is downwards the work by gravity is +
the work by you is -
Net work done on an object by a net force is
equal to the change in its Kinetic Energy.
Do not confuse getting tired with doing work
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A net 6500 N force is applied to a resting 1500 kg
car, moving it forward. What is its kinetic energy
and velocity after 150 m?
A circus manager pounds a 1.0 kg tent stake with a
wooden mallet that weighs 12.5 Kg at 8 m/s . What
is the initial velocity of the stake?
since car begins at rest
= 975 000 J
• Moving hammer can do work on nail!
Crider pushes a frictionless tire of mass 5.0 kg across a floor at a rate of
2.0 m/s2 for 7.0 s. Find the net work done on the tire. Use the KE
formula to determine the energy in the tire.
For hammer:
Wh = KEh = -Fd
= 0 – (½)mh(vh)2
For nail:
Wn = KEn = Fd
= (½)mn(vn)2 - 0
A quarterback throws a football (from rest) at 10 m/s. The football
weighs 1.5 kilograms. A receiver catches it. How much
work/energy was used to accomplish both tasks?
How much work must be done to stop a 1250-kg car traveling at 105 km/hr?
The work done on the car is equal to the change in its kinetic energy, and so
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Mechanical Universe
• Lesson 14: Potential Energy
• The nature of stability. Potential energy provides a clue, and a
powerful model, for understanding why the world has worked the
same way since the beginning of time.
• Text Assignment: Chapter 14 (Advanced text -- Chapter 10)
• Instructional Objectives
• Be able to calculate the potential-energy function associated with a
given conservative force.
• Be able to find the force F(x) from the potential-energy function U(x).
• Be able to locate equilibrium points and discuss their stability from a
graph of the potential-energy function U(x).
• Be able to use the gravitational potential-energy and conservation of
energy to solve the problems of escape velocity.
Potential Energy (PE or U)
Potential Energy: stored energy
http://www.youtube.com/watch?v=ujXnhCfrjX8
Net work done by falling object
Reference Levels
http://www.youtube.com/watch?v=alo_XWCqNUQ
Potential Energy
Potential Energy
• Potential Energy (PE)  Energy associated with
position or configuration of a mass.
• Gravitational Potential Energy:
h = distance above Earth
m has potential to do work
mgh when it falls
Y positions are chosen for ease of problem
solving: Set the bottom displacement = 0
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Gravitational PE
Potential energy in the case depends only on vertical displacement
Consider a problem in which the height of a
mass above the Earth changes from y1 to y2:
Change in Gravitational PE is:
(PE)grav = mg(y2 - y1)
Work done on the mass: W = (PE)grav
y = distance above Earth
Where we choose y = 0 is arbitrary, since
we take the difference in 2 y’s in (PE)grav
A 12,500 kg boulder is 157 m above a cabin.
What is its PE above the cabin? What will be the KE
As the boulder crashes onto the house.
What will be the velocity?
y=0

Mass is not important, change in velocity is only impacted by h
y=0

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IF THE Rollercoaster weighs 1000kg find the PE change
A 7.0-kg monkey swings from one branch to another 1.2
m higher. What is the change in potential energy?
By how much does the gravitational potential energy of
a 64-kg pole vaulter change if her center of mass rises
about 4.0 m during the jump?
• WATCH A VIDEO ON KINETIC
POTIENTIAL ENERGY HERE
• DO A DEMO
(a) Find the force required to give a helicopter of mass M an acceleration of 0.10 g
upward. (b) Find the work done by this force as the helicopter moves a distance h
upward.
A 1.60-m tall person lifts a 2.10-kg book from the ground so it is
2.20 m above the ground. What is the potential energy of the book
relative to (a) the ground, and (b) the top of the person’s head?
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A 12,500 kg boulder is 157 m above a cabin.
What is its PE above the cabin? What will be the KE
As the boulder crashes onto the house.
What will be the velocity?
A 2-kg ball at rest is released from a height of 20 m.
What is its velocity when its height has decreased to 5m?
20m
v=0
5m
0
Conservation of Mechanical Energy
OR
E = KE + PE = Constant
If the force of Gravity only force in system:
v
A 3000kg roller coaster starts from rest at a height of 50m and
descends down a frictionless track to a nadir that is 10m high before
returning upwards. What is the coaster’s velocity at the nadir?
Only height differences matter!
Horizontal distance doesn’t matter!
A flippin’ powerful method of calculation!!
h0 = Initial height, v0 = Initial velocity
h = Final height, v = Final velocity
E1 = E2
 PE1 = mgh
KE1 = 0
 KE + PE = same
as at points 1 & 2
KE1 + PE1 = KE2 + PE2
0 + mgh = (½)mv2 + 0
v2 = 2gh
The sum remains constant
 PE2 = 0
KE2 = (½)mv2
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MAKE PROBLEMS FROM THIS WEB SITE
http://www.physicsclassroom.com/mmedia/energy/ie.cfm
MAKE PROBLEMS FROM THIS WEB SITE
http://www.physicsclassroom.com/mmedia/energy/hw.cfm
On an icy day Mr C. is late for class, starting from rest, he slides
down from Tully’s with a large Americano on the 35.0º incline
whose vertical height is 50 m. How fast is he going when he
reaches Stadium?
Physics students try to grab some Tully’s, but can’t walk
up the icy walk. They grab a sled and get a running start
in the courtyard. They are traveling at 9.0 m/s when they
hit the ice. How high (y axis) do they travel?
How far up the hill the 35deg incline is that?
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http://www.youtube.com/watch?v=_7IRSQ39pDQ&safe=active
Sect. 6-5: Conservative Forces
WHEN IS MECHANICAL ENERGY IS NOT CONSERVED
WHEN IS MECHANICAL ENERGY IS CONSERVED
http://www.youtube.com/watch?v=-SbwmooRTBI&feature=related
KE CONSERVED IN AN ELASTIC COLLISION BECAUSE ALL
OF THE WORK/ENERGY IS PASSED ALONG AND NOT
TRANSFERRED INTO HEAT, SOUND, LIGHT, DEFORM ECT…
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CAN FRICTION DO POSITIVE
WORK?
Nonconservative Force - Force leads to
dissipation of mechanical energy into another
form of energy
Conservative Force - Does Not leads to
dissipation of mechanical energy
Push book on table -- work greater if not in
straight line due to friction
Friction is nonconservative.
Work depends on the path!
Energy Conservation
• In any process, total energy is neither created
nor destroyed.
• Energy can be transformed from one form to another
& from one body to another, but the total amount
remains constant.
 Law of Conservation of Energy
• Again: Not exactly the same as the Principle of
Conservation of Mechanical Energy, which holds
for conservative forces only! This is a general
Law!!
• Forms of energy besides mechanical:
– Heat (conversion of heat to mech. energy & visaversa)
– Chemical, electrical, nuclear, ..
Conservative Forces
You will be expected to perform mechanics calculations without information
about the initial velocity or other data we have used so far, just force
Conservative Force  The work done by that
force depends only on initial & final conditions &
not on path taken between the initial & final
positions of the mass.
PE CAN be defined for conservative forces
Non-Conservative Force  The work done by
that force depends on the path taken between
the initial & final positions of the mass.
PE CANNOT be defined for non-conservative
forces
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Conservation of Mechanical Energy
For conservative forces ONLY! In any process
KE + PE = 0
Conservation of Mechanical Energy
E = KE + PE
Conservation of mechanical energy
In any process,E = 0 = KE + PE
E = KE + PE = Constant
In any process, the sum of the KE and the PE is
unchanged (energy changes form from PE to KE or KE
to PE, but the sum remains constant).
If several forces act (conservative & non-conservative):
The total work done is: Wnet = WC + WNC
WC = work done by conservative forces
WNC = work done by non-conservative forces
The work energy principle still holds:
Wnet = KE
For conservative forces (by definition of PE):
WC = -PE
For conservative forces:
KE = -PE + WNC
OR:
WNC = KE + PE
Law of Conservation of Energy
E(Top)A = E(Bottom)B = E(Top)C = E(Bottom)B = E(Top)D
Not the work energy principle since mechanical
Energy is not necessarily conserved
C
A
D
The total energy is neither decreased nor
increased in any process.
Energy can be transformed from one form to
another & from one body to another, but the
total amount IN SYSTEM remains constant.
B
At D ac must be greater than g
At B the velocity is maximum and the KEB = PEA
If all forces are conservative PEA = PED & KEA = KED
E1 = E2
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A 655 N diver leaps into the water from a height of 10.0 m. What is his
speed 2.00 m above the water?
A bicycle with initial velocity 10 m/s coasts to a net height of 4 m.
What is the velocity at the top, neglecting friction?
A roller coaster boasts a maximum height of 100.0 m.
What is the speed when it reaches its lowest point?
A projectile is fired at an upward angle of 45.0º from the top of a 265-m
cliff with a speed of 185 m/s. What will be its speed when it strikes the
ground below? (Use conservation of energy.)
The image part with relationship ID rId6 was not found in the file.
In the high jump, Fran’s kinetic energy is transformed into gravitational
potential energy without the aid of a pole. With what minimum speed must
Fran leave the ground in order to lift her center of mass 2.10 m and cross the
bar with a speed of 0.70 m/s?
25.0m
Sled and rider together have a mass of 87.0 kg. They are atop a hill
elevated at 42.5. They are at a height of 25.0 m (y axis). Find v at
bottom of hill. Assume no friction.
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Mechanical Universe
A 15.0 kg crate slides down a ramp. The ramp is 1.50 m long and is at a
30.0 angle. The crate starts from rest. It experiences a constant frictional
force of 7.50 N as it slides downwards. What is the speed of the crate
when it reaches the bottom?
• Lesson 13: Conservation of Energy
• The myth of the energy crisis. According to one of the
major laws of physics, energy is neither created nor
destroyed.
• Text Assignment: Chapter 13 (Advanced text - Chapter
10)
• Instructional Objectives
• Know the definitions of work, kinetic energy, and
potential energy.
• Understand the relationship between work and energy.
• Be able to work problems using conservation of energy.
CONSERVATION OF ENERGY WITH NC
WNC = Work done by non-conservative forces
KE = Change in KE
PE = Change in PE (conservative forces)
So, if friction is present, we have (WNC  Wfr)
Mechanical Universe
• Lesson 15: Conservation of Momentum
• If The Mechanical Universe is a perpetual clock, what keeps it
ticking away till the end of time? Taking a cue from Descartes,
momentum -- the product of mass and velocity -- is always
conserved. Newton's laws embody the concept of conservation and
momentum. This law provides a powerful principle for analyzing
collisions, even at the local pool hall.
• Text Assignment: Chapter 19 (Advanced text -- Chapter 11)
• Instructional Objectives
• Recognize conservation of momentum as a consequence of
Newton's Second Law.
• Know when the momentum of a system is conserved.
• Recognize the connection between kinetic energy and momentum.
• Be able to solve problems involving elastic and inelastic collisions.
• Know the relationship between impulse and time average of force.
A 100kg skateboarder comes to a stop (just barely) at the top of a second
hill that is 100m from his starting point. What is his force of friction?
POWER
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Sect. 6-10: Power
• Power  Rate at which work is done or
rate at which energy is transformed:
• Average Power:
P = (Work)/(Time) = (Energy)/(time)
P = W/t
• SI units: Joule/Second = Watt (W)
1 W = 1J/s
A 1250 kg Elevator carries max load of 955 kg. A
constant frictional force of 3850 N exists. What minimum
Power for motor is needed to lift the thing at a
constant speed of 3.50 m/s?
the force is T, so:
– British units: Horsepower (hp). 1hp = 746 W
“Kilowatt-Hours” (from your power bill). Energy!
1 KWh = (103 Watt)  (3600 s) = 3.6  106 W s = 3.6  106 J
LAB TASK
Average Power:

• DETERMINE The s of between your two
surfaces using forces
Its often convenient to write power in
terms of force & speed.
A pump is to lift 18.0 kg of water per minute through a height of
3.60 m. What output rating (watts) should the pump have?
• DETERMINE k between your two
surfaces using energy
SPRINGS
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There are other types of PE besides gravitational.
Calculus Connection
Compressed springs store energy
x
Force by person to compress spring:
x  spring constant
F
m
K is a measure of how “stiff” the spring is.
Force exerted by spring is:
Hooke’s Law
A 4-kg mass suspended from a spring produces a
displacement of 20 cm. What is the spring constant?
20 cm
F
m
Derivative of WS is FS
k = 196 N/m
The units of k are really kg/s2
Compressing or Stretching a
Spring Initially at Rest: WORK
Two forces are
always present: the
outside force Fout
ON spring and the
reaction force Fs BY
the spring.
x
What work is required to stretch this spring (k = 196 N/m)
from x = 0 to x = 30 cm?
x
m
Compressing
m
Stretching
Compression: FMAN does positive work
and Fs does negative work
Stretching: Fout does positive work
and Fs does negative work
F
30 cm
Note: The work to stretch an additional 30 cm is
greater due to a greater average force.
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• In a problem in which compression or stretching
distance of spring changes from x1 to x2.
A 62-kg bungee jumper jumps from a bridge. She is tied to a bungee cord
whose unstretched length is 12 m, and falls a total of 31 m. Draw a FBD. Calculate the
spring stiffness constant k of the bungee cord, assuming Hooke’s law applies.
• The change in PE is:
(PE)elastic = (½)k(x2)2 - (½)k(x1)2
• The work done: W = - (PE)elastic
• The PE belongs to the system, not to
individual objects
Note: If the height of the platform and the extension height are given you will have PEg for both sides.
Conservation of Energy:
Energy In A Spring
A 0.450 kg silver arrow, resting in a frictionless dart gun is
pushed 3.00 cm into a light spring, k = 75.0 N/m. It is then
released. What is the velocity of the arrow as it just leaves the
spring?
Conservation of Energy:
this is the PE
stored in spring
.450
Add to Conservation of mechanical energy equation:
A 1200-kg car rolling on a horizontal surface has speed
v= 18 m/s when it strikes a horizontal coiled spring and
is brought to rest in a distance of 2.2 m. What is the spring
stiffness constant of the spring?
How far up the frictionless 30o-incline will the 2-kg block move after
release? The spring constant is 2000 N/m and it is compressed by 8 cm.
Conservation of Energy:
Conservation of Energy:
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How far up the frictionless 30o-incline will the 2-kg block move after
release? The spring constant is 2000 N/m and it is compressed by 8 cm.
Sitting on a table that is 100 cm above the deck is a spring that has a spring constant of k =
50 N/m. The spring is compressed a distance of 50 cm with a 0.756 kg ball. (a) What is the
kinetic energy stored in the spring? (b) When the spring extends, what is the velocity of the
ball? (c) When the ball rolls off the table, how much time does it take till it hits the deck?
(d) How far does the ball travel horizontally from the edge of the table before it hits the
deck? (e) What is the kinetic energy of the ball just before it hits the deck?
(a)
Conservation of Energy:
(b)
(c)
(a)
A vertical spring (ignore its mass), whose spring stiffness constant is 950 N/m is attached to a
table and is compressed down 0.150 m. (a) What upward speed can it give to a 0.30-kg ball when
released? (b) How high above its original position (spring compressed) will the ball fly?
The level of the ball on the uncompressed spring taken as the zero
location for both gravitational PE and elastic PE. So the compressed point
It a negative value for gravitational potential.
Conservation of Energy:
.
A ball mass 5.kg falls a distance of 1.2m before hitting a spring. The spring
compresses 20cm and stops. Find the v it strikes the spring and solve for k.
emember that the compression of the spring is - 0.2m.
TARZAN
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FORMULAS ON AP
Crazy Crider is above the courtyard with a 20.0m rope that makes 40 degrees with
the vertical. He swings out on a rope and lets go of the thing when he is at the
lowest point of the swing. At this point, he is 15.0 m above the ground. How far
Will he travel from the window?
EXAM TIME
Jane, looking for Tarzan, is running at top speed 5.3 m/s and grabs a vine
hanging vertically from a tall tree in the jungle. How high can she swing upward?
Does the length of the vine affect your answer?
The only forces acting on Jane are gravity and the vine
tension. The tension pulls in a centripetal direction,
and so can do no work – the tension force is
perpendicular at all times to her motion. So Jane’s
mechanical energy is conserved.
No, the length of the vine does not enter into the calculation, unless the vine is
less than 0.7 m long. If that were the case, she could not rise 1.4 m high.
Instead she would wrap the vine around the tree branch.
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