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Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html
Lecture 11
Chapter 7
Work and Energy(Work Done by a Constant Force)
10.16.2013Physics I
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Chapter 7
Work done by a constant force Scalar (Dot) product of two vectors Kinetic Energy Work-Kinetic Energy theorem
Outline
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work Done by a Constant Force Work done on an object by a force If a constant force F moves object by a displacement d, work
done by force is given by W = F||d, where F|| is the component of the force along d
• Work has units of N-m, or Joules (J), and is a scalar !!
d
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Example I
A force of 5 N, applied in the direction shown,moves a box across a horizontal distance of 10 m.How much work does the force do on the box?
NF 5
60
d 10m
= 25J
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Scalar Product of Two Vectors
Three ways to multiply vectors
– Multiplication by a scalar
– Scalar (or dot) product
– Vector (or cross) product
vectorAc
scalarBA
vectorBA
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Scalar (Dot) Product of Two Vectors
The scalar product of two vectors is written as:
Therefore, we can say that work is a scalar product of force and displacement
Work done by a force along a displacement:
Is it possible to do work on an
object that remains at rest?A) yes
B) no
Work requires that a force acts over a distance. If
an object does not move at all, there is no
displacement, and therefore no work done.
ConcepTest 1. To Work or Not to Work
The amount of work you actually do may have little relationship to the amount of effort you apply. For example, if you push on a car stuck in a snow drift, you may exert a lot of force (and effort) but if the car does not budge, you have not done any work! In order for work to be done on an object, the object must move some distance as a result of the force you apply.
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work
F
d
F
d
F
90d
W 0 W 0 W 0
f
N
mg
Displacement
Pull
Friction acts in the opposite
direction to the displacement, so
the work is negative. Or using the
definition of work (W = F d cos ),
because = 180º, then W < 0.
ConcepTest 2 Friction and Work I
A) friction does no work at all
B) friction does negative work
C) friction does positive work
A box is being pulled across a rough floor at a constant speed. What can you say about the work done by friction?
In a baseball game, the catcher
stops a 90-mph pitch. What can
you say about the work done by
the catcher on the ball?
A) catcher has done positive work
B) catcher has done negative work
C) catcher has done zero work
The force exerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F d cos ), because = 180º, then W < 0. Note that because the work done on the ball is negative, its speed decreases.
ConcepTest 3 Play Ball!
F
d
0W
Follow-up: What about the work done by the ball on the catcher?
ConcepTest 4 Tension and Work
A) tension does no work at all
B) tension does negative work
C) tension does positive work
A ball tied to a string is being whirled around in a circle. What can you say about the work done by tension?
v T
No work is done because the force
acts in a perpendicular direction to
the displacement. Or using the
definition of work (W = F d cos ),
because = 180º, then W < 0.
Follow-up: Does the Earth do work on the Moon?
ConcepTest 5 Force and Work
N
f
T
mg
Any force not perpendicularto the motion will do work:
N does no work
T does positive work
f does negative work
mg does negative work
1) one force
2) two forces
3) three forces
4) four forces
5) no forces are doing work
A box is being pulled up a rough incline by a rope connected to a pulley. How many forces are doing work on the box?
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work on a backpack. Example 7-2
(a) Determine the work a hiker must do on a 15.0-kg backpack to carry it up a hill of height h = 10.0 m, as shown. Determine also
(b) the work done by gravity on the backpack, and
(c) the net work done on the backpack. For simplicity, assume the motion is smooth and at constant velocity (i.e., acceleration is zero).
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work on a backpack.
FH d
mg
θ
180‐θ
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work on a backpack.
FH d
mg
It is not an accident that the net work is zero.
Wherever, the net force acting on an object is zero, the net work will be zero as well
321 WWWWnet
dFFFdFdFdF
321321 dFnet
0,0 netnet WthenFIf
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Scalar Product (component form)
kAjAiAA zyxˆˆˆ
kBjBiBB zyxˆˆˆ
In component form:
zzyyxx BABABABA
A
B
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Scalar Product Math kAjAiAA zyxˆˆˆ
kBjBiBB zyxˆˆˆ
A
B (Axi Ay j Azk)(Bxi By j Bzk) AxBx (i i ) AxBy (i j) AxBz (i k)
AyBx ( j i ) AyBy ( j j) AyBz ( j k)
AzBx (k i ) AzBy (k j) AzBz (k k)
(i i ) ( j j) (k k) 1
)ˆˆ( ji
Proof:
(i j) (i k) ( j k) 0
90cosˆˆ ji 00
The scalar product of two unit vectors)ˆˆ( ii 0cosˆˆ ii 1
11
Similar andUsing these we can write:
0 0
0 0
0 0
1
1
1
zzyyxx BABABABA
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Properties of Scalar Product
• Commutative property ABBA
CABACBA
• Distributive property
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Example
A constant force F acts on an object as it moves from position x1 to x2. What is the work done by this Force?
kjix ˆ3ˆ4ˆ21 kjix ˆ6ˆ3ˆ52
kjiF ˆ2ˆ5ˆ4
xd
JdFW 29)9)(2()7)(5()3)(4(
12 xx kji ˆ9ˆ7ˆ3
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Translational Kinetic Energy
Kinetic energy (K) is the energy of motion Translational kinetic energy is the energy of motion in a
line or trajectory Rotational kinetic energy (Ch. 10) is the energy of
circular/rotational motion
Kinetic energy is the energy possessed by an object because
of its motion
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Kinetic Energy I
A constant horizontal force F acts on an object on a frictionless surface. The box accelerates, i.e. speeds up.
We want to define a property of the object (called kinetic energy, or energy of motion) which changes as work is done on it by the force.
F Fd
vi vf
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Kinetic Energy II
F Fd
vi
F mavf
2 vi2 2ad
dmFvv if
222
K f Ki W
12
mvf2 1
2mvi
2 Fd
12
mvf2 1
2mvi
2 Fd
mvf2 mvi
2 2Fd K 12
mv2
K f Ki K Wnet
Kinetic energy
Work-Kinetic Energy Principle
The work done is equal to the change in the kinetic energy.
vf
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work-Energy Principle
K f Ki K Wnet
Work-Kinetic Energy Principle
If the net work is positive, the kinetic energy increases.
If the net work is negative, the kinetic energy decreases.
Work and kinetic energy have the same units: joules.
Energy can be considered as the ability to do work.
A child on a skateboard is moving at a speed of 2 m/s. After a force acts on the child, her speed is 3 m/s. What can you say about the work done by the external force on the child?
1) positive work was done 2) negative work was done 3) zero work was done
The kinetic energy of the child increased because her speed increased. This increase in KE was the result of positive work being done.
ConcepTest 6 Work and KE
Follow-up: What does it mean for negative work to be done on the child?
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
ExampleNet Work required to accelerate a 1000kg car from
KW
kJJvvmKW if 800000,800]400[2
1000][2
222
(a) from 20 m/s to 40 m/s?
(b) from 40 m/s to zero?
vi vf
][2
22if vvm
kJJ 600000,600]2040[2
1000 22
K f Ki K Wnet
Work-Kinetic Energy Principle
Department of Physics and Applied Physics95.141, Fall 2013, Lecture 11
Work & Kinetic Energy
The work done by the earth on an apple as the apple falls towards the earth with increasing speed v is positive.So kinetic energy increases.
The work done by the earth on the moon as the moon makes one full revolution around the earth at constant speed v is zero.So there is no change in kinetic energy.