Transcript Lecture 13.Work

Work
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
Is it possible to do
work on an object
that remains at rest?
A) yes
B) no
C) only in southern
hemisphere
D) only in northern
hemisphere
Reading Quiz
Is it possible to do
work on an object
that remains at rest?
A) yes
B) no
C) only in southern
hemisphere
D) only in northern
hemisphere
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.
Last Time
Orbital maneuvers
Ocean tides
Geophysical applications
Free floating in space
Satellites and weightlessness
Principle of Equivalence
Black holes
Today
Discuss 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: joule
1J=1Nm
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:
1 J = 1 N ×m
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.
We use component of
force in direction of
displacement.
What happens when force is not in
same direction as displacement?
W  ( F cos )d  Fd cos
W = Fd cos. Work is a scalar. We can determine work
two ways: Work is the
1) 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.
( F cos  ) d
or
( d cos  ) F
y
x
Fx cos
W  F  d = Fd cos scalar or dot product
Work can be negative. See right.
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:
A×B = AB cos q
Therefore, we can write:
W = F ×d = Fd cos q
Usefulness of scalar products
for general vectors:
A  Ax i  Ay j  Az k
B  Bx i  By j  Bz k
A B   Ax i  Ay j  Az k 
 Bxi  By j  Bzk 
 Ax Bx  Ay By  Az Bz
Notes about work
•Work is a scalar, not a vector.
•Work can be positive, negative, or zero.
•The angle  is always the angle between F
and d . 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.
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 F  d is negative and that F
and d are such that  > 900.
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)
B)
C)
D)
W
Zero
-W
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)
B)
C)
D)
is zero.
is equal to the work done by friction.
increases the kinetic energy of the block.
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
F
force on the block is
f
A)
B)
C)
D)
zero.
equal to the work done by friction.
increases the kinetic energy of the block.
decreases the kinetic energy of the block.
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, F   is not constant!

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.
F1 cos q1
In the limit that the pieces become infinitesimally
narrow, the work is the area under the curve:
or:
b
W  F
a
d
Fspring = - kx
The spring force
varies with
position. It is a
good example of
the kind of force
we are
considering.
k is the spring
constant
Work Needed to Stretch a Spring a Distance x
W   Fdx   kxdx  k  xdx
1 2
W  kx
2
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 Quiz
A box is being
pulled across a
rough floor at a
constant speed.
What can you say
about the work
done by friction?
A) friction does no work
at all
B) friction does negative
work
C) friction does positive
work
Conceptual Quiz
A box is being
pulled across a
rough floor at a
constant speed.
What can you say
about the work
done by friction?
A) friction does no work
at all
B) friction does negative
work
C) friction does positive
work
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.
N Displacement
Pull
f
mg
Conceptual Quiz
A ball tied to a
string is being
whirled around in a
circle. What can
you say about the
work done by
tension?
A) tension does no work at all
B) tension does negative work
C) tension does positive work
Conceptual Quiz
A ball tied to a
string is being
whirled around in a
circle. What can
you say about the
work done by
tension?
A) tension does no work at all
B) tension does negative work
C) tension does positive work
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.
T
v
Follow-up: Is there a force in the direction of the velocity?
Conceptual Quiz
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?
A) one force
B) two forces
C) three forces
D) four forces
E) no forces are doing
work
Conceptual Quiz
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?
A) one force
B) two forces
C) three forces
D) four forces
E) no forces are doing
work
Any force not perpendicular
to the motion will do work:
N does no work
N
T
T does positive work
f
f does negative work
mg does negative work
mg