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WORK AND POWER
Work (W) is done when a force causes displacement of an object.
1
Sample 1: Lifting a box takes work. The higher a box is or the heavier a box is the
re work it takes to lift it.
Which situation causes more work to be done?
1. Lifting a 1 kg box up! flight of stairs or lifting a 1 kg box up 2 flights of stairs?
2: Lifting a 2 kg box up 1 flight of stairs or lifting a 1 kg box up 1 flight of stairs?
3. Lifting a 1 kg box up 2 flights of stairs or lifting a 2 kg box up 1 flight of stairs?
4. Running a 1 kg box up I flight of stairs or walking a 1 kg box up I flight of stairs?
;.,
• Work is Sc.ÿ
a
ÿ &v"
lquantity. /ÿk.ÿ ÿ- poj,ÿyÿ-
to a displacement.
Work isdone only when components of a force are Iÿ rÿ/le /
• Fl = force acting on object (N)
• dl = displacement of objectI (m)
• 0 = angle between direction of force and direction of displacement (o)
• Units for work=
.ÿ =1 ÿ
l(
)
Note; If "F" and "d" are in the same direction, what is O? ÿ-ÿ_ÿo
In this case W = ÿ-J e.o.ÿ ÿ = Coÿ O ÿ .!
Samples:
.
Bud, a very large man of mass 130 kg, stands on a pogo stick. How much work is done as
Bud compresses the spring of the pogo stick 0.50 m? (637 J)!
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=
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A 100. kg object is moved a distance of 20. m with a
force of 40. N at an angle of 30.° above the
horizontal. How much work is done? (693 J)
.
,
100. kg
On his way off to college, Russ drags his suitcase 15.0 m from the door of his house to his
car at a constant speed with a horizontal force of 95 N. How much work does Russ have
to do to overcome the force of friction? If the floor has just been waxed, does he have
to do more work or less work to move the suitcase? Explain.
Power is the rate at which work gets done, OR amount of work that is done per unit of time.
P = W
At
.
•
W=work, J
At = time interval, s
Starting with the equation above, derive another equation for
ower (P) in terms of velocity:
(
I'=Fdÿ thÿsP=FV I "
t
At
At
i p = W ] Rearrange IP---FdcÿsO Rÿrrarÿ
Note that
o power is scalar (no direction)
o units for power are watts (W), 1 W = 1 J/s
o for electricity, power expressed in kilowatts (kW); 1 kW = 1000 W
o for engines, power expressed in horsepower (HP); 1 HP = 746 W
Samples:
1. A 65 kg student travels from the 1st floor to the 4th floor of a school (total height of 15
m).
a. What total work did she do climbing the stairs? (9555 J)
b.
How long would this trip last if the student produced 480 W of power? (20. s)
0ÿ
o£
P
.
.-
...
2. How much power does a crane develop, doing 60,000 J of work in 5.00 rain? (200 W)
fÿ
W--ÿ
ÿ+)
uÿ?
•
" -
11 ÿ-'ÿ "
/ /O-Vÿ/ l=ÿ*J
ÿ/z'ÿ- ÿ
---ÿ 2_oo
"
ÿ"
3. How long does it take a 2.5 kW electric motor to do 75,000 J of work? (30. s)
p..
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-;-
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4. How much workcan a 500 W electric mixer do in 2.Stain? (75,000 J)
LO o¢Iÿ
Energy is defined as the ability to do work, which can be done on or by an object if there
is a displacement in the direction of a component of the net force. Power is the rate of
work done or the rate of energy transfer. Since energy is conserved, energy can present
within a given system in various forms such as internal kinetic (work), mechanical,
potential, and thermal (heat).
Vocabulary _.
work-energy theorem, power, internal energy, mechanical energy, kinetic molecular
theory, potential energy, kinetic energy, power, work, elastic potential energy,
gravitational potential energy law of conservation of energy, Joule, Watt, internal
energy, entropy, thermodynamics, specific heat, conduction, convection, and radiation,
Iÿeat
Essential Questions:
• What are the variables that impact mechanical energy and power?
• How does alforce applied over a distance relate to the kinetic energy of an object?
• How do you apply the work-theorem to calculate transfers of energy?
• 1What relationships exist between variables in the work-energy theorem?
• What is the relationship between roller coasters and the conservation of energy?
• How are the macroscopic properties of a thermodynamic system such as temperature,
specific heat, and pressure relate to the molecular level of matter?
• What are some real world examples of conduction, convection and radiation?
/
Elastic Potential Energy (EPE) - the energy stored in elastic materials as a result of stretching or
compressing.
• .... K = spring constant N/m
•
X = amount of compression m
Work-Energy Theorem - states that work ÿ energy (interchangeable), so the amount of work done
equals the overall change in energy level
11
Work: ÿPE :-rÿPEf - GPE,
lÿ i ÿ ÿlÿl ÿ lÿ
......
ÿlÿ lÿ
Work = AKE = ½ m(vÿ2- vÿ2)
Note: When applying theorem, only the
initialand fi°naFconditions of the
system are important (A), not what
Work = ÿEPE = ½ k(xfz - xÿÿ)
happens in between.
Samples:
What potential energy does the rock possess
t. A t5-lkg rock is on the edge of a 120 m cliff.
relative to the base of thee cliff?
•
/
'
.
,
,
k.ÿ Iÿ
1
o
What is the
dnetic energy of a rock of mass 12 kgl sliding across the ice at 2.0 m/s?
l';
d.-
__f--- of ur_
"Zÿ
3ÿ
How much work must be done to accelerate an 800 kg car from 15 m/s to 30 m/s?
wt= ÿ0o/%
oÿ ÿo'ÿ/ÿzÿ
,. 7 ,,uo s'.) ..-/'
I,Oÿ?
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A hoist is used to raise a 40 kg block of ice from a platform 8 m above the ground, to a
.
platform 20 m above the ground. For the block of ice:
A. What is the initial potential energy? (3136 J)
- ,ÿt 3+:+
oÿ ! ÿlÿE; ÿ wÿk- I tJÿ ÿ q.ÿÿ
cÿ Pfÿ.ÿ What i the final potential energy? (7840 J)
7ÿ ; 7F?'o3
fÿ.° ÿVhat the change in potential energy? (4704 J)
& fÿ,:-_ 7¢ÿ'0
D. How much work does the hoist do as the block is raised to the higher platform?
E. A horizontal spring with k = 100 N/m is compressed by 10 cm by a 100 g mass.
a. How much elastic potential energy does the compressed spring store?
x+_,/,,,
€ÿ,a-..7
.
ÿ-
)z_ . ÿJ °ÿ [,ÿ€
4ÿ ,/o o/ÿb. How much stic potential energy would the compressed spring store if it were
compressed the same distance by a 300 g mass?
"S.! ...ÿe/ÿ- ÿ '--"-"-" s,,ÿ-:+,,, .+ ÿ :ÿ, '+' f"+ "-'+ "#"""- ' ÿ +,''/ i.
V'.sÿ- ÿvo ÿvu,.d..
Practice Problems
1. A student pushed a box 27.0 m across a smooth, horizontal floor using a constant force of 113 N. If the force was
applied for 9.00 s, how much power was developed, to the nearest watt?
: ,,+# +7
p. ÿ ÿ4
,,+ÿ "-?
*%?'A skateboarder travels back and forth on a U-shaped track during a time trial at a competition. The graph shows the
skateboarder's speed as a function of time during the trial. Which labeled point on the graph identifies the time
during the trial that the skateboarder most likely has equal amounts of kinetic energy and potential energy?
m me+
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.
A warehouse employee is pushing a 30.0 kg desk across a floor at a constant speed of 0.50 m/s. How much work
must the employee do on the desk to change the speed to 1.00 m/s?
2-
8
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10. The diagram below shows a closed system of two tanks that each contains water. When the valve between the two
tanks of water is opened, the temperature of the water in each tank changes. What is the equilibrium temperature to
the nearest whole degree Celsius?
11. In which of the following systems is entropy increasing?
A. An egg is broken and scrambled.
B. A cluttered room is cleaned and organized.
C. A thin stick I placed in a glass of sugar-saturated water, and sugar crystals form on the stick.
D. Water freezing in the freezer.
12. If a student places a metal spoon into a burning candle flame, what form of heat transfer allows heat energy to travel
up the spoon?
A. Radiation
B. Convection
C. Conduction
D. Radiation, convection, and conduction
13. A 0.3 kg metal bar with a specific heat of 460 J / kg'°C is heated from 30°C to 120°C. How much heat is required to
do this?
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14. In nature, heat tends to flow from regions with higher temperatures to ones with lower temperatures. This tendency
illustrates which of the following laws of thermodynamics?
A. If two systems are both in thermal equilibrium with a third then they are in thermal equilibrium with each other.
B. When two systems are allowed to interact, they will eventually reach a mutual thermodynamic equilibrium.
C. The increase in internal energy of a body is equal to the heat supplied to the body minus the work done by the
body.
Do The entropy of a system approaches a constant value as the temperature of the system approaches zero.
1,
The runner ran up tlae stairs to deliver his message that the British were coming. If he had a mass of
250 kg and the height of the stairs were 12 meters, how much work did he do?
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f2tt-ÿ I/-ÿ,.
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Jill, of Jack and Jill up the hill fame used a bucket attached to a pulley to fetch a pail of water. If the
.
.
well was 5 meters deep and the weight of the pail of water was 7 pounds (1 pound = 4.45 N), how
much work did Jill [o?
G/U
Equation
Sub
. 7ÿ€.,f<
d-. ÿ.,
Solve
C"ÿ
w, c ÿ W
ÿ ,/.4s-) × ÿ-
1ÿ-<. 7ÿd
vO.ÿ ?
Jack pushed his 500 kg car with a horizontal force of 20,000 N over a level distance of 30 meters. If
.
the friction coefficient was 0.6, then what was the total amount of work done?
-Solve
G/U (VVÿ)
0 -ÿ o
Equation
Sub
Equation
Sub
ÿ'ÿ.25ÿooÿ
,,,,k-.?
(Wÿr,ÿ,ion)
Solve
.ÿ /ÿ'0°
NO___WW you can determine Wnet
(ÿ,o ÿI.#-Yÿ
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4. Worker Wilber pushed a box with a force of 40 N directed at an angle of 30° downward from the
horizontal over a level distance of 60 meters. If there is no friction, then what was the total amount
of work done? (Hint WA angle is not 0° this time)
%ÿOe
Equation
Sub
?olve
g.¢o tJ
5. Powerful Pete pulled a sled with a force of 200 N at an angle of 430 to the ground over level ground
for 500 m with no friction. If it took 13.46 minutes, how much power did Powerful Pete exert?
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6. If a 600 W motor was used to lift the 1 ton (1 ton=20001b)car 2 m up to save the puppy, then how
long did it take?
-
(
- ?
ÿ I1ÿ ÿ "/. q'ÿ" 0Oÿ)
o
ÿ-
4--- "-"V'
oo°ÿ-'- "-------
, 2
ÿ'75
11
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The 74kg skier was standing at the top of a 100 m hill. He reached a velocity of 6 m/s. What was his
total (mechanical) energy? (Hint, remember both roller coaster labs)
Solve
Sub
Equation
G/U
I ÿ 7:ÿCÿ +
-, 7ÿ'ÿ
? fx q.8 X/oo
7q, J
k "- Io,,..
1'3 ÿ'1'
8. What was his GPE at the bottom of the hill?
-+ 7 S,ÿ-oo
O ('k = o ,,,) J
9. What was his KE at the top of the hill? 0 { t/= o ,',/9)_J
10. What was his KE half the way down the hill?
J
11. If a 21 kg mass has a gravitational potential energy of 70 J, how high about the ground is the object?
Eqÿuation
Glu
ÿub
t;OIve
Wl-- ÿ! k-ÿ
Aÿÿ _..ÿ._-O_ÿj
- ?
.z/x
12. Kim pushed a box horizontally for 6 m with a force of 50 N at an angle of 25° above the horizontal.
It the frictional force was 10 N, how much work was done?
-{ub
G/U
Solve
Equation
W.- ÿ: o/ coÿ O
fox6 x ÿszÿ
13. A 12 kg ball has a kinetic energy of 900 J. What is its velocity?
Solve
g ¢ - "Ioo3
,¢'z-
!:L.2"M/ÿ
1 5 Equati°ÿ ÿÿÿ/Itÿ i,ÿ
(
12
Sample #1_: Gravitation System (Rock falls from rest.)
a. Label the top and bottom with PE and KE as
120 rn-ÿ------ÿ
maximum or zero values.
b. Find the speed as the rock hits the base of the cliff.
c. Can you work part b) using kinematics?
d. Find the speed at a height of 20 m instead.
20m
Om
Sample #2: Pendulum (mass = 0.50 kg)
Label where PE and KE are maximum and zero values.
a.
What is the potential energy at position A?
What is the maximum speed of the pendulum?
C.
d. What is the potential energy at C?
What is the speed at position B?
e,
b.
A
C
Sample #3: Roller Coaster (Coaster starts from rest at A.)
The mass of the small car (including contents) is 400
kg and the height of point is 20 meters above point C.
Points B and D are at the same height and are 10ÿ
A
B
meters above point C.
a°
b.
D
What potential energy does the car have at A?
If the car starts at point A, what velocity will it have
C
at C?
C.
If the car starts at point A, what velocity will it have
at D?
D°
A 2 kg ball is dropped onto a vertical spring with a k = 1000 N/ml
=rom what height did the
ball drop if the spring was compressed by 25 cm?
E°
o
A horizontal spring is used to launch a 2 kg ball. The spring is compressed by .ZD m and the
k = 1000 N/m. What is the maximum speed of the ÿall?
O
iLxÿ
ÿ
iÿ× ÿ-
11 ÿPe
. Iÿ ÿ.
Work, Power, Energy: In, Class Practice
A plane moves down a runway for take-off after starting from rest. The mass of the jet is
0.2 X 103 kg. If the jet's engines do 2.55 X 104j of work to speed it up for take-off, what
.
is thetakeoff speed (final velocity)? 15.97 m/s)
I/ÿÿ
ÿl
1
A pendulum, of mass 4.00 kg, is pulled from its equilibrium position, and then released. If
the pendulum has a velocity of 3.5 m/s as it passesthrough the equilibrium position, find
the height, h. (.625 m)
ÿt - ÿ"ÿ-?'ÿ
.
llz
ÿ
= O, 62fÿ-
k=7
A man pulls his son acrosÿa fÿ with a constant force of 53 N. What work
.
does the man do pulling him 159 m across the floor? (8427 J)
,
A race car driver slams on his •brakes. The mass of the car is 3,500 kg. He is able to slow
his car from 51.8 m/s to 39.7 m/s. What is the magnitude of the work done by the car's
brakes to slow the. car down? (
:)
ÿ/eÿoC?ÿ ÿ" ÿSÿ,/./
%
f "Z.,
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5. A worker lifts a box from the third shelf to the top shelf. The top shelf is 15 m above the
second shelf. The clerk does 589 J of work lifting the box. What is the weight of the
box? (39.27 N)
j ÿ ÿ ÿ'£ ?_ ÿ"ÿ ÿ tÿ,.ÿ.W ;J ÿ,-.//
A construction worker drops a .9 kg hammer from a roof that is 12 m high. What is the
.
velocity of the hammer when it strikes the ground? (15.34 m/s)
÷
o
o
x<J"-
"-"
V; ÿo
,
1
8.
The net force used is 30 N. What
A crate is pushed with a velocity of 3 m/s horizor "ally.
is the power used to move the crate at that velocity? (90 W)
A .6 kg arrow is shot straight up in the air with a velocity of 8 m/s. What is the height of
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9.
A car with a mass ofÿstarts aÿand accelerates to a speed of 60 m/s. What is
the cars iÿ?ÿ
10. What is the car's Final KE? (450,000 J)
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24
11. A woman runs up a hill that is 67 m high. Her mass is 15 kg. What is her change in PE?
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<
(9849 J)
12. What is the mass of a car that has a change in KE of 45,000 J and speeds up from 3 m/s
13. A man pushes a cart with an angle of 45 degrees above the floor with a force of 350N.
The man does 13,000 J of work. How far does the cart go? (52.53 m)
lÿooÿ
13ÿ°°_-...-
o
14. A 2.0 kg ball rolls off a 3.0 m high table traveling at i m/s. If it lands directly on top of
an un-stretched sprÿ ng with a k = 40 N/m, how far does the spring compress?
V- !./ÿ
15. A .70 kg ball is placed on a vertical 200 N/m spring that is compressed 40 cm. When the
spring is released, how hÿh above its starting point:ill the ball go?
_ÿ, 7 ,"-ÿ. ÿ'
.
25