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Work
Lecturer: Professor Stephen T. Thornton
Is it possible to do
work on an object
that remains at rest?
A) yesB) noC) only in southern
hemisphereD) only in northern
hemisphere
Reading Quiz
Is it possible to do
work on an object
that remains at rest?
A) yesB) noC) only in southern
hemisphereD) only in northern
hemisphere
Work requires that a force acts force acts over a distanceover a distance. If an object does not move at all, there is no no displacementdisplacement, and therefore no no work donework done.
Reading Quiz
Last TimeOrbital maneuvers
Ocean tides
Geophysical applications
Free floating in space
Satellites and weightlessnessPrinciple of EquivalenceBlack holes
TodayDiscuss the concept of work
Let the force and displacement be in the same direction (for now).
W = Fd called Work unit: N m, newton meter
Work is so important that we give it its own unit: joule1 J = 1 N m
How much work is a joule?
• Lifting an apple about 1 meter is a joule.
• Total annual U.S. energy use is ~1020 J.
• Lifting our text book 1 m.
F = (2.6 kg)(9.81 m/s2) = 26 N
W = (26 N)(1 m) = 26 J
In the SI system, the units of work are joules:
As long as this person does not lift or lower the bag of groceries, he is doing no work on it. The force he exerts has no component in the direction of motion.
1 J = 1 N m×
What happens when force is not in same direction as displacement?
( cos ) cosW F d Fd
We use component of force in direction of displacement.
W = Fd cos. Work is a scalar. We can determine work two ways: Work is the1) component of force in the direction of displacement times the magnitude of displacement. 2) component of displacement in the direction of the force times the magnitude of the force.
cosx F
( cos )
or
( cos )
F d
d F
x
y
Work can be negative. See right.
productF d = cos scalar or dotW Fd
Gravity does work on the apple as it falls. The apple accelerates. Gravity does negative work on the apple if we throw it up.
Definition of the scalar, or dot, product:
Therefore, we can write:
cosW F d Fd q= × =
A B = cosAB q×
Usefulness of scalar products for general vectors:
x y z
x y z
x y z x y z
x x y y z z
A A A A
B B B B
A B A A A B B B
A B A B A B
i j k
i j k
i j k i j k
Notes about work
•Work is a scalar, not a vector.
•Work can be positive, negative, or zero.
•The angle is always the angle between
and . Be careful about this.•There is often more than one force acting
on an object. The total net work is the sum
of the work done by all the forces.
F
d
Conceptual Quiz: Negative work means A) the applied force is variable.B) the applied force is perpendicular to the displacement.C) component of the applied force is opposite to the displacement.D) nothing, as there is no such thing as negative work.
Answer: C
Negative work simply means the product of is negative and that and are such that > 900.
F d
F
d
F
d
Conceptual Quiz:A block slides a certain distance down a straight incline. The work done by gravity is W. Next the block slides up a longer, curved incline of the same height. What is the work done by gravity for the second slope?
A) WB) ZeroC) - WD) Gravity cannot do work: some other force does the work.E) The work cannot be determined unless the distance traveled is given.
Answer: C -W
In the second case the displacement is somewhat opposite to the force of gravity, so the work is negative. Because the block goes back to the same height, the magnitude is the same.
Conceptual Quiz:A force F pushes a block along a horizontal surface against the force of friction f. If the block undergoes a displacement d at constant velocity, the work done by the net force on the block (hint on next slide)
A) is zero.B) is equal to the work done by friction.C) increases the kinetic energy of the block.D) decreases the kinetic energy of the block.
Conceptual Quiz:A force F pushes a block along a horizontal surface against the force of friction f. If the block undergoes a displacement d at constant velocity, the work done by the net force on the block is A) zero.B) equal to the work done by friction.C) increases the kinetic energy of the block.D) decreases the kinetic energy of the block.
F
f
Answer: A
The key here is that the force F does positive work and the friction f does negative work. Because the velocity is constant, there is no net force and the forces F and f are equal and opposite. The total work done by the net force is zero.
Graphical Representation of the Work Done by a Constant Force
Work Done by a Non-constant Force
Work Done by a Varying Force
Particle acted on by a varying force. Clearly, is not constant!F
For a force that varies, the work can be approximated by dividing the distance up into small pieces, finding the work done during each, and adding them up.
1 1cosF q
In the limit that the pieces become infinitesimally narrow, the work is the area under the curve:
or:
b
aW dF
The spring force varies with position. It is a good example of the kind of force we are considering.
k is the spring constant
springF kx=-
Work Needed to Stretch a Spring a Distance x
21
2
W Fdx kxdx k xdx
W kx
The Work Done by a Spring Can Be Positive or Negative
Work done by spring is positive in this case.
Compressed spring is pushing out.
The Work Done by a Spring Can Be Positive or Negative
Work done by spring is negative in this case.
Block is moving against compressed spring compressing it more.
Work Done By Spring. If it requires 5.0 J of work to stretch a particular spring by 2.0 cm from its equilibrium length, how much more work will be required to stretch it an additional 4.0 cm?
Conceptual QuizA) friction does no work at all B) friction does negative
workC) friction does positive
work
A box is being A box is being pulled across a pulled across a rough floor at a rough floor at a constant speed. constant speed. What can you say What can you say about the work about the work done by friction?done by friction?
f
N
mg
Displacement
Pull
Friction acts in the oppositeopposite direction to the displacement, so the work is negativenegative. Or using the definition of work (WW = = F d cos F d cos ), because = = 180180ºº, then W < W < 00.
Conceptual QuizA) friction does no work at all B) friction does negative
workC) friction does positive
work
A box is being A box is being pulled across a pulled across a rough floor at a rough floor at a constant speed. constant speed. What can you say What can you say about the work about the work done by friction?done by friction?
Conceptual QuizA) tension does no work at all B) tension does negative workC) tension does positive work
A ball tied to a string A ball tied to a string is being whirled is being whirled around in a circle. around in a circle. What can you say What can you say about the work done about the work done by tension?by tension?
Conceptual QuizA) tension does no work at all B) tension does negative workC) tension does positive work
A ball tied to a string A ball tied to a string is being whirled is being whirled around in a circle. around in a circle. What can you say What can you say about the work done about the work done by tension?by tension?
v v
TT
No work is done because the force acts in a perpendicularperpendicular direction to the displacement.
Or using the definition of work (WW = = F d cos F d cos ), because = = 180180ºº, then W < W < 00.
Follow-up:Follow-up: Is there a force in the direction of the velocity? Is there a force in the direction of the velocity?
Conceptual QuizA) one forceA) one forceB) two forcesB) two forcesC) three forcesC) three forcesD) four forcesD) four forcesE) no forces are doing E) no forces are doing
workwork
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?
Conceptual Quiz
N
f
T
mg
displacementAny force not perpendicular
to the motion will do work:
N does no workno work
T does positivepositive work
f does negative work
mg does negative work
A) one forceA) one forceB) two forcesB) two forcesC) three forcesC) three forcesD) four forcesD) four forcesE) no forces are doing E) no forces are doing
workwork
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?
