chapter 6 conservation of energy. mfmcgrawch06 - energy - revised: 2/20/102 conservation of energy...
TRANSCRIPT
Chapter 6
Conservation of Energy
MFMcGraw Ch06 - Energy - Revised: 2/20/10 2
Conservation of Energy
• Work by a Constant Force
• Kinetic Energy
• Potential Energy
• Work by a Variable Force
• Springs and Hooke’s Law
• Conservation of Energy
• Power
MFMcGraw Ch06 - Energy - Revised: 2/20/10 3
The Law of Conservation of Energy
The total energy of the Universe is unchanged by any physical process.
The three kinds of energy are:
kinetic energy, potential energy, and rest energy.
Energy may be converted from one form to another or transferred between bodies.
MFMcGraw Ch06 - Energy - Revised: 2/20/10 4
MFMcGraw Ch06 - Energy - Revised: 2/20/10 5
Work by a Constant Force
Work is an energy transfer by the application of a force.
For work to be done there must be a nonzero displacement.
The unit of work and energy is the joule (J).
1 J = 1 Nm = 1 kg m2/s2.
MFMcGraw Ch06 - Energy - Revised: 2/20/10 6
Only the force in the direction of the displacement that does work.
An FBD for the box at left:
x
y
Fw
N
The work done by the force F is:
xFrFW xxF cos
rx
Frx
Work - Example
MFMcGraw Ch06 - Energy - Revised: 2/20/10 7
The work done by the normal force N is: 0NW
The normal force is perpendicular to the displacement.
The work done by gravity (w) is: 0gW
The force of gravity is perpendicular to the displacement.
Work - Example
MFMcGraw Ch06 - Energy - Revised: 2/20/10 8
The net work done on the box is:
xF
xF
WWWW gNF
cos
00cos
net
Work - Example
MFMcGraw Ch06 - Energy - Revised: 2/20/10 9
In general, the work done by a force F is defined as
cosrFW
where F is the magnitude of the force, r is the magnitude of the object’s displacement, and is the angle between F and r (drawn tail-to-tail).
Work Done
MFMcGraw Ch06 - Energy - Revised: 2/20/10 10
Example: A ball is tossed straight up. What is the work done by the force of gravity on the ball as it rises?
FBD for rising ball:
x
y
w
r
ymg
ywWg
180cos
Work - Example
MFMcGraw Ch06 - Energy - Revised: 2/20/10 11
A box of mass m is towed up a frictionless incline at constant speed. The applied force F is parallel to the incline.
Question: What is the net work done on the box?
F
w
NFx
y
0cos
0sin
wNF
wFF
y
x
Apply Newton’s 2nd Law:
An FBD for
the box:
Inclined Plane-V Constant
MFMcGraw Ch06 - Energy - Revised: 2/20/10 12
The magnitude of F is: sinmgF
If the box travels along the ramp a distance of x the work by the force F is
sin0cos xmgxFWF
The work by gravity is
sin90cos xmgxwWg
Example continued:
Inclined Plane-V Constant
MFMcGraw Ch06 - Energy - Revised: 2/20/10 13
Example continued:
The work by the normal force is:
090cos xNWN
The net work done on the box is:
0
0sinsin
net
xmgxmg
WWWW NgF
Inclined Plane-V Constant
MFMcGraw Ch06 - Energy - Revised: 2/20/10 14
Example: What is the net work done on the box in the previous example if the box is not pulled at constant speed?
sin
sin
wmaF
mawFFx
Proceeding as before:
xFxma
xmgxmgma
WWWW NgF
net
net
0sinsin
Inclined Plane-Acceleration
New Term
MFMcGraw Ch06 - Energy - Revised: 2/20/10 15
Kinetic Energy
2
2
1mvK is an object’s translational
kinetic energy.
This is the energy an object has because of its state of motion.
It can be shown that, in general
Net Work = Change in K
.net KW
MFMcGraw Ch06 - Energy - Revised: 2/20/10 16
Example: The extinction of the dinosaurs and the majority of species on Earth in the Cretaceous Period (65 Myr ago) is thought to have been caused by an asteroid striking the Earth near the Yucatan Peninsula. The
resulting ejecta caused widespread global climate change.
If the mass of the asteroid was 1016 kg (diameter in the range of 4-9 miles) and had a speed of 30.0 km/sec, what was the asteroid’s kinetic energy?
J 105.4
m/s 1030kg 102
1
2
1
24
23162
mvK
This is equivalent to ~109 Megatons of TNT.
Kinetic Energy
MFMcGraw Ch06 - Energy - Revised: 2/20/10 17
Gravitational Potential Energy Part 1- Close to Earth’s Surface
Potential energy is an energy of position.
There are potential energies associated with different forces. Forces that have a potential energy associated
with them are called conservative forces.
In general UW cons
Not all forces are conservative, i.e. Friction.
MFMcGraw Ch06 - Energy - Revised: 2/20/10 18
The change in gravitational potential energy (only near the surface of the Earth) is
ymgU g
where y is the change in the object’s vertical position with respect to some reference point.
You are free to choose to location of this where ever it is convenient.
Gravitational Potential Energy
MFMcGraw Ch06 - Energy - Revised: 2/20/10 19
The table is 1.0 m tall and the mass of the box is 1.0 kg.
Ques: What is the change in gravitational potential energy of the box if it is placed on the table?
J 8.9m 0m 0.1m/s 8.9kg 0.1 2
ifg yymgymgU
First: Choose the reference level at the floor. U = 0 here.
GPE Problem
U = 0
MFMcGraw Ch06 - Energy - Revised: 2/20/10 20
Example continued:
J 8.9m 0.1 m0.0m/s 8.9kg 1 2
ifg yymgymgU
Now take the reference level (U = 0) to be on top of the table so that yi = 1.0 m and yf = 0.0 m.
The results do not depend on the location of U = 0.
GPE Problem
MFMcGraw Ch06 - Energy - Revised: 2/20/10 21
Mechanical energy is UKE
The total mechanical energy of a system is conserved whenever nonconservative forces do no work.
That is Ei = Ef or K = U.
Total Mechanical Energy
Then if K increases U decreases and vice versa
MFMcGraw Ch06 - Energy - Revised: 2/20/10 22
A cart starts from position 4 with v = 15.0 m/s to the left. Find the speed of the cart at positions 1, 2, and 3. Ignore friction.
m/s 5.202
2
1
2
1
34243
233
244
3344
34
yygvv
mvmgymvmgy
KUKU
EE
Mechanical Energy Problem
MFMcGraw Ch06 - Energy - Revised: 2/20/10 23
m/s 0.182
2
1
2
1
24242
222
244
2244
24
yygvv
mvmgymvmgy
KUKU
EE
m/s 8.242
2
1
2
1
14241
211
244
1144
14
yygvv
mvmgymvmgy
KUKU
EE
Or use E3=E2
Or use E3=E1
E2=E1
Mechanical Energy Problem
MFMcGraw Ch06 - Energy - Revised: 2/20/10 24
A roller coaster car is about to roll down a track. Ignore friction and air resistance.
40 m 20
m y=0
m = 988 kg
(a) At what speed does the car reach the top of the loop?
m/s 8.192
2
10 2
fif
ffi
ffii
fi
yygv
mvmgymgy
KUKU
EE
Roller Coaster Problem
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(b) What is the force exerted on the car by the track at the top of the loop?
N w
y
x
N 109.2 42
2
2
mgr
vmN
r
vmwN
r
vmmawNF ry
Example continued:
FBD for the car:Apply Newton’s Second Law:
Roller Coaster Problem
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(c) From what minimum height above the bottom of the track can the car be released so that it does not lose contact with the track at the top of the loop?
Example continued:
2min2
10 mvmgymgy
KUKU
EE
fi
ffii
fi
Using conservation of mechanical energy:
g
vyy fi 2
2minSolve for the starting height
Roller Coaster Problem
MFMcGraw Ch06 - Energy - Revised: 2/20/10 27
Example continued:
v = vmin when N = 0. This means that
grv
r
vmmgw
r
vmwN
min
2min
2
What is vmin?
m 0.255.22
22
2min r
g
grr
g
vyy fi
The initial height must be
Roller Coaster Problem
MFMcGraw Ch06 - Energy - Revised: 2/20/10 28
What do you do when there are nonconservative forces? For example, if friction is present
fricWEEE if
The work done by friction.
Nonconservative Forces
MFMcGraw Ch06 - Energy - Revised: 2/20/10 29
Gravitational Potential Energy Part 2 - Away from Earth’s Surface
The general expression for gravitational potential energy is:
0 where
21
rUr
MGMrU
MFMcGraw Ch06 - Energy - Revised: 2/20/10 30
Example:
What is the gravitational potential energy of a body of mass m on the surface of the Earth?
e
ee R
mGM
r
MGMRrU 21
Gravitational Potential Energy
MFMcGraw Ch06 - Energy - Revised: 2/20/10 31
A planet of mass m has an elliptical orbit around the Sun. The elliptical nature of the orbit means that the distance between the planet and Sun varies as the planet follows its orbital path. Take the planet to move counterclockwise from its initial location.
QUES: How does the speed of a planet vary as it orbits the Sun once?
The mechanical energy of the planet-sun system is:
constant2
1 sun2 r
GmMmvE
rB A
Planetary Motion
MFMcGraw Ch06 - Energy - Revised: 2/20/10 32
constant2
1 sun2 r
GmMmvE
Starting from point “B” (where the planet moves the slowest), as the planet moves in its orbit r begins to decrease. As it decreases the planet moves faster.
At point “A” the planet reaches its fastest speed. As the planet moves past point A in its orbit, r begins to increase and the planet
moves slower.
rB A
Planetary Motion
At point “B” the planet is the farthest from the Sun. At point “A” the planet is at its closest approach to the sun.
MFMcGraw Ch06 - Energy - Revised: 2/20/10 33
Work by a Variable Force
Work can be calculated by finding the area underneath a plot of the applied force in the direction of the displacement versus the displacement.
MFMcGraw Ch06 - Energy - Revised: 2/20/10 34
x (m)
Fx (N)F3
F2
F1
x3x2x1
The work done by F1 is 0111 xFW
Example: What is the work done by the variable force shown below?
The net work is then W1+W2+W3.
The work done by F2 is 1222 xxFW
The work done by F3 is 2333 xxFW
MFMcGraw Ch06 - Energy - Revised: 2/20/10 35
By hanging masses on a spring we find that stretch applied force. This is Hooke’s law.
For an ideal spring: Fx = kx
Fx is the magnitude of the force exerted by the free end of the spring, x is the measured stretch of the spring, and k is the spring constant (a constant of proportionality; its units are N/m).
A larger value of k implies a stiffer spring.
Spring Force
MFMcGraw Ch06 - Energy - Revised: 2/20/10 36
(a) A force of 5.0 N applied to the end of a spring cause the spring to stretch 3.5 cm from its relaxed length.
Ques: How far does a force of 7.0 N cause the same spring to stretch?
For springs Fx. This allows us to write .2
1
2
1
x
x
F
F
Solving for x2: cm. 9.4cm 5.3N 5.0
N 0.71
1
22
x
F
Fx
Spring Force
MFMcGraw Ch06 - Energy - Revised: 2/20/10 37
Example continued:
N/cm. 43.1cm 3.5
N 0.5
1
1 x
Fk
(b) What is the spring constant of this spring?
N/cm. 43.1cm 4.9
N 0.7
2
2 x
Fk
Or
Spring Force
MFMcGraw Ch06 - Energy - Revised: 2/20/10 38
An ideal spring has k = 20.0 N/m. What is the amount of work done (by
an external agent) to stretch the spring 0.40 m from its relaxed length?
J 6.1m 4.0N/m 0.202
1
2
1
2
1
curveunder Area
22111
kxxkx
W
Spring Force
MFMcGraw Ch06 - Energy - Revised: 2/20/10 39
The work done in stretching or compressing a spring transfers energy to the spring.
Below is the equation of the spring potential energy.
2
2
1kxU s
Elastic Potential Energy
The spring is considered the system
MFMcGraw Ch06 - Energy - Revised: 2/20/10 40
A box of mass 0.25 kg slides along a horizontal, frictionless surface with a speed of 3.0 m/s. The box encounters a spring with k = 200 N/m.
Ques: How far is the spring compressed when the box is brought to rest?
m 11.0
02
1
2
10 22
vk
mx
kxmv
KUKU
EE
ffii
fi
Elastic Potential Energy
MFMcGraw Ch06 - Energy - Revised: 2/20/10 41
Power
cosFvP
Average Power
The unit of power is the watt. 1 watt = 1 J/s = 1 W.
Instantaneous Power
t
EP
av
Power is the rate of energy transfer.
MFMcGraw Ch06 - Energy - Revised: 2/20/10 42
A race car with a mass of 500.0 kg completes a quarter-mile (402 m) race in a time of 4.2 s starting from rest. The car’s final speed is 125 m/s. (Neglect friction and air resistance.)
Ques: What is the engine’s average power output?
watts103.921
5
2
av
t
mv
t
K
t
KU
t
EP
f
Power - Car Example
MFMcGraw Ch06 - Energy - Revised: 2/20/10 43
Summary
• Conservation of Energy
• Calculation of Work Done by a Constant or Variable Force
• Kinetic Energy
• Potential Energy (gravitational, elastic)
• Power