![Page 1: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/1.jpg)
Work&Energy
![Page 2: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/2.jpg)
4.1 Work Done by a Constant Force
![Page 3: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/3.jpg)
4.1Work Done by a Constant Force
When force and displacement are in the same direction
The equation is
Work Applet
FxW
![Page 4: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/4.jpg)
4.1 Work Done by a Constant Force
If Force is at an angle
The more general Equation is
More commonly writtenxFW cos
cosFxW
![Page 5: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/5.jpg)
4.1 Work Done by a Constant Force
Work is done only if displacement is in the direction of force
Two situation where no work is done
![Page 6: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/6.jpg)
4.1 Work Done by a Constant Force
Negative work is done, if the force is opposite to the direction of the displacement
Force of man – positive work
Force of gravity – negative work
FM
W
y
![Page 7: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/7.jpg)
4.1 Work Done by a Constant Force
A car of mass m=2000 kg coasts down a hill inclined at an angle of 30o below the horizontal. The car is acted on by three forces; (i) the normal force; (ii) a force due to air resistance (F=1000 N); (iii) the force of weight. Find work done by each force, and the total work done on the car as it travels a distance of 25 m down the slope.
![Page 8: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/8.jpg)
4.1 Work Done by a Constant Force
A car of mass m=2000 kg coasts down a hill inclined at an angle of 30o below the horizontal. The car is acted on by three forces; (i) the normal force; (ii) a force due to air resistance (F=1000 N); (iii) the force of weight. Find work done by each force, and the total work done on the car as it travels a distance of 25 m down the slope.
Work done by the Normal NFair
W
090cos
cos
N
N
N
WNxWFxW
![Page 9: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/9.jpg)
4.1 Work Done by a Constant Force
A car of mass m=2000 kg coasts down a hill inclined at an angle of 30o below the horizontal. The car is acted on by three forces; (i) the normal force; (ii) a force due to air resistance (F=1000 N); (iii) the force of weight. Find work done by each force, and the total work done on the car as it travels a distance of 25 m down the slope.
Work done by the Air Resistance NFair
W
JWW
FxW
F
F
F
25000180cos)25(1000
cos
0NW
![Page 10: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/10.jpg)
4.1 Work Done by a Constant Force
A car of mass m=2000 kg coasts down a hill inclined at an angle of 30o below the horizontal. The car is acted on by three forces; (i) the normal force; (ii) a force due to air resistance (F=100 N); (iii) the force of weight. Find work done by each force, and the total work done on the car as it travels a distance of 25 m down the slope.
Work done by the Weight (parallel component)NFair
W
JWW
xmgW
F
W
W
245000)25)(30sin()8.9)(2000(
sin
JWW FN 25000250000
![Page 11: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/11.jpg)
4.1 Work Done by a Constant Force
A car of mass m=2000 kg coasts down a hill inclined at an angle of 30o below the horizontal. The car is acted on by three forces; (i) the normal force; (ii) a force due to air resistance (F=100 N); (iii) the force of weight. Find work done by each force, and the total work done on the car as it travels a distance of 25 m down the slope.
Total Work NFair
W
JWWW WFN 220000245000250000
![Page 12: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/12.jpg)
4.2 Work Kinetic Energy Theorem
![Page 13: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/13.jpg)
4.2 Work Kinetic Energy Theorem
Suppose an object falls from the side of building.
The acceleration is calculated as
We can also solve acceleration as related to velocity
mFa
maF
xvva
axvv
2
220
2
20
2
![Page 14: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/14.jpg)
4.2 Work Kinetic Energy Theorem
Combining the two equations
With a little manipulationmFa
maF
xvva
axvv
2
220
2
20
2
xvv
mF
2
20
2
202
1221
20
221 )(
mvmvFx
vvmFx
![Page 15: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/15.jpg)
4.2 Work Kinetic Energy Theorem
Combine with our definition of work
This is the work kinetic energy theorem
The quantity is called kinetic energy
202
1221 mvmvWtot
221 mv
221 mvK
![Page 16: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/16.jpg)
4.2 Work Kinetic Energy Theorem
A child is pulling a sled with a force of 11N at 29o above the horizontal. The sled has a mass of 6.4 kg. Find the work done by the boy and the final speed of the sled after it moves 2 m. The sled starts out moving at 0.5 m/s.
N
B
W
![Page 17: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/17.jpg)
4.2 Work Kinetic Energy Theorem
A child is pulling a sled with a force of 11N at 29o above the horizontal. The sled has a mass of 6.4 kg. Find the work done by the boy and the final speed of the sled after it moves 2 m. The sled starts out moving at 0.5 m/s.
Work done by boy
Velocity
N
B
W
JWFxW
2.19)29cos()2)(11(cos
smv
v
mvmvW
5.2)5)(.4.6()4.6(2.19 2
212
21
202
1221
![Page 18: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/18.jpg)
4.3 Work Done by a Variable Force
![Page 19: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/19.jpg)
4.3 Work Done by a Variable Force
If we make a graph of constant force and displacement
The area under the line is work
0
2
4
6
8
10
12
14
16
1 2 3 4 5 6
![Page 20: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/20.jpg)
4.3 Work Done by a Variable Force
If the force varies at a constant rate
The area is still work
![Page 21: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/21.jpg)
4.3 Work Done by a Variable Force
For a more complex variation
The area is still work. Calculated using calculus.
![Page 22: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/22.jpg)
4.3 Work Done by a Variable Force
Springs are important cases of variable forces
The force is kxF
![Page 23: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/23.jpg)
4.3 Work Done by a Variable Force
If we graph this
The equation for W is
Forc
e
Displacement
W=Area
kx
221
221
21
21
)(
kxU
kxW
xkxWFxW
s
![Page 24: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/24.jpg)
4.3 Work Done by a Variable Force
So for a spring
Spring potential is greatest at maximum displacement
Force is maximum at maximum displacement
Velocity is greatest at equilibrium (all energy is now kinetic)
![Page 25: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/25.jpg)
m
x
Fs
4.3 Work Done by a Variable Force
F = max
Us = max
K = 0
v= 0
![Page 26: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/26.jpg)
m
x=0
equilibrium position
x=0
Fs=0
4.3 Work Done by a Variable Force
F = 0
Us = 0
K = max
v= max
![Page 27: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/27.jpg)
m
x=0
equilibrium position
x
Fs
4.3 Work Done by a Variable Force
F = kx
Us = ½kx2
K = ½mv2
v= depends on K
![Page 28: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/28.jpg)
4.4 Power
![Page 29: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/29.jpg)
4.4 Power
Power
How fast work is done
FvtFx
tWP
![Page 30: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/30.jpg)
4.4 Power
Measured in watts (j/s)
1 horsepower = 746 W
![Page 31: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/31.jpg)
4.4 Power
You are out driving in your Porsche and you want to pass a slow moving truck. The car has a mass of 1300 kg and nees to accelerate from 13.4 m/s to 17.9 m/s in 3 s. What is the minimum power needed?
![Page 32: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/32.jpg)
4.4 Power
You are out driving in your Porsche and you want to pass a slow moving truck. The car has a mass of 1300 kg and nees to accelerate from 13.4 m/s to 17.9 m/s in 3 s. What is the minimum power needed?
WP
JWW
mvmvWtWP
305003
9155091550
)4.13)(1300()9.17)(1300( 2212
21
202
1221
hpP 41
![Page 33: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/33.jpg)
4.5 Conservative and Nonconservative Forces
![Page 34: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/34.jpg)
4.5 Conservative and Nonconservative Forces
Conservative force – energy is stored and can be released
Nonconservative – energy can not be recovered as kinetic energy (loss due to friction)
![Page 35: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/35.jpg)
4.5 Conservative and Nonconservative Forces
If a slope has no friction
All the energy is stored
This situation is conservative
![Page 36: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/36.jpg)
4.5 Conservative and Nonconservative Forces
If a slope has friction
Some energy is lost as heat
It is nonconservative
![Page 37: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/37.jpg)
4.5 Conservative and Nonconservative Forces
Conservative forces are independent of the pathway
Friction (nonconservative) more work as the distance increases
![Page 38: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/38.jpg)
4.6 Potential Energy
![Page 39: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/39.jpg)
4.6 Potential Energy
Work is transfer of energy
If an object is lifted, the work done is
This energy can be converted to kinetic energy if the object is then allowed to fall back to its original position
Stored Energy is called Potential Energy so
mgyFxW
mgyU g
![Page 40: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/40.jpg)
4.6 Potential Energy
Using the same logic
The work done by a spring is
When the spring shoots out, the work is converted to kinetic energy
221 kxFxW
221 kxU s
![Page 41: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/41.jpg)
4.7 Conservation of Mechanical Energy
![Page 42: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/42.jpg)
4.6 Potential Energy
Law of conservation of energy – energy can not be created or destroyed, it only changes form
If we only include conservative forces (mechanical energy)
ExpandsEE 0
KUUKUU sgsg 000
![Page 43: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/43.jpg)
4.8 Work Done by Nonconservative Forces
![Page 44: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/44.jpg)
4.7 Work Done by Nonconservative Forces
Nonconservative forces take energy away from the total mechanical energy
At the end, total mechanical energy decreases because of the work done by friction
![Page 45: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/45.jpg)
4.7 Work Done by Nonconservative Forces
We can think of this as energy that is used up (although it just goes into a nonmechanical form)
Expand this and our working equation becomes
fWEE 0
fsgsg WKUUKUU 000
![Page 46: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/46.jpg)
4.7 Work Done by Nonconservative Forces
Remember that the work due to friction is still defined as
mgxWNxWfxWFxW
![Page 47: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/47.jpg)
4.7 Work Done by Nonconservative Forces
One other helpful concept for the energy of a pendulum
How do you calculate y for
potential energy?
If an angle is given and the
string is defined a L
The height of the triangle is
y
L
Ly
cosL
![Page 48: Work&EnergyWork&Energy. 4.1 Work Done by a Constant Force](https://reader035.vdocuments.mx/reader035/viewer/2022062412/5a4d1aec7f8b9ab05997b670/html5/thumbnails/48.jpg)
4.7 Work Done by Nonconservative Forces
Now the change in height is
Which we can rewrite as
y
L
Ly
cosL
cosLyL
cosLLy