Transcript Dynamics - Where can my students do assignments that require

Work and Energy
Chapter 6
Expectations
After Chapter 6, students will:




understand and apply the definition of work.
solve problems involving kinetic and potential
energy.
use the work-energy theorem to analyze physical
situations.
distinguish between conservative and
nonconservative forces.
Expectations
After Chapter 6, students will:



perform calculations involving work, time, and
power.
understand and apply the principle of conservation of
energy.
be able to graphically represent the work done by a
non-constant force.
The Work Done by a Force
The woman in the picture exerts a force F on her
suitcase, while it is displaced through a distance s.
The force makes an angle q with the displacement
vector.
The Work Done by a Force
The work done by the woman is: W  Fs cosq
Work is a scalar quantity. Dimensions: force·length
SI units: N·m = joule (J)
History/Biography Break
James Prescott Joule
December 24, 1818 –
October 11, 1889
English physicist, son of a
wealthy brewer, born near
Manchester. He was the
first scientist to propose a
kinetic theory of heat.
The Work Done by a Force
Notice that the component of the force vector parallel to
the displacement vector is F cos q. We could say that
the work is done entirely by the force parallel to the
displacement.
W  Fs cosq
The Work Done by a Force
Recalling the definition of the scalar product of two
vectors, we could also write a vector equation:
 
W  F  s (  Fs cos q )
The Work Done by a Force
Work can be either positive or negative.
In both (b) and (c), the man is doing
work.
(b): q = 0°; W  Fs
(c): q = 180°;
W  Fs
A Force Accelerates an Object
Let’s look at what happens when a net force F acts on
an object whose mass is m, starting from rest over a
distance s.
F
m
s
A Force Accelerates an Object
The object accelerates according to Newton’s second
law:
F
a
m
F
m
s
A Force Accelerates an Object
Applying the fourth kinematic equation:
v  v0  2as  2as
2
F
m
2
(v0  0)
F
2 Fs
2
but a  , so : v 
m
m
1 2
mv  Fs
2
s
Kinetic Energy
A closer look at that result:
1 2
mv  Fs
2
1 2
We call the quantity mv kinetic energy.
2
In the equation we derived, it is equal to the work (Fs)
done by the accelerating force.
Kinetic energy, like work, has the dimensions of
force·length and SI units of joules.
The Work-Energy Theorem
1 2
mv  Fs
2
The equation we derived is one form of the workenergy theorem. It states that the work done by a net
force on an object is equal to the change in the
object’s kinetic energy. More generally,


1
2
2
KE  m v  v0   F s cosq
2
If the work is positive, the kinetic energy increases.
Negative work decreases the kinetic energy.
The Work-Energy Theorem
A hand raises a book from height h0 to height hf, at
constant velocity.
hf
Work done by the hand force, F:
WF  F h f  h0   mg h f  h0 
hf – h0
F ( = mg )
Work done by the gravitational force:
h0
WG  mg h f  h0 
mg
The Work-Energy Theorem
Total (net) force exerted
on the book: zero.
hf
Total (net) work done
on the book: zero.
hf – h0
F ( = mg )
Change in book’s kinetic energy:
zero.
h0
mg
The Work-Energy Theorem
Now, we let the book fall freely
from rest at height hf to height h0.
hf
mg
Net force on the book: mg.
Work done by the gravitational
force: W  mg h f  h0


hf – h0
h0
The Work-Energy Theorem
Calculate the book’s final kinetic
energy kinematically:
v  v0  2ax  2 g h f  h0  (v0  0)
2
2
1 2
KE f  mv  mg h f  h0 
2
The book gained a kinetic energy equal
to the work done by the gravitational force
(per the work-energy theorem).
hf
mg
hf – h0
h0
Gravitational Potential Energy


The quantity mg h f  h0
is both work done on the
book and kinetic energy gained by
it. We call this the gravitational
potential energy of the book.
hf
hf – h0
F ( = mg )
h0
mg
Work and the Gravitational Force
The total work done by the
gravitational force does not
depend on the path the book takes.
hf
hf – h0
The work done by the gravitational
force is path-independent. It
depends only on the relative
heights of the starting and
ending points.
F ( = mg )
h0
mg
Work and the Gravitational Force
Over a closed path (starting and ending points the
same), the total work done by the gravitational force
is zero.
Forces and Work
Compare with the frictional force. The longer the path,
the more work the frictional force does. This is true
even if the starting and ending points are the same.
Think about dragging a sled around a race course.
The work done by
the frictional force
is path-dependent.
Conservative Forces
The gravitational force is an example of a conservative
force:
 The work it does is path-independent.
 A form of potential energy is associated with it
(gravitational potential energy).
Other examples of conservative forces:
 The spring force
 The electrical force
Nonconservative Forces
The frictional force is an example of a nonconservative
force:
 The work it does is path-dependent.
 No form of potential energy is associated with it.
Other examples of nonconservative forces:
 normal forces
 tension forces
 viscous forces
Total Mechanical Energy
A man lifts weights upward at a constant velocity.
He does positive work on the weights.
The gravitational force does equal negative work.
The net work done on
the weights is zero.
But …
Conservation of Mechanical Energy
The gravitational potential energy of the weights
PE  mg h f  h0 
increases:
The work done by the nonconservative normal force of
the man’s hands on
the bar changed the
total mechanical
energy of the weights:
E  KE  PE
Conservation of Mechanical Energy
Work done on an object by nonconservative forces
changes its total mechanical energy.
If no (net) work is done by nonconservative forces, the
total mechanical
energy remains
constant (is conserved):
E  KE  PE  WNC
Conservation of Mechanical Energy
E  KE  PE  WNC
This equation is another form of the work-energy
theorem.
Note that it does not require both kinetic and potential
energy to remain constant – only their sum. Work
done by a conservative force often increases one
while decreasing the other. Example: a freely-falling
object.
Conservation of Every Kind of Energy
“Energy is neither created nor destroyed.”
Work done by conservative forces conserves total
mechanical energy. Energy may be interchanged
between kinetic and potential forms.
Work done by nonconservative forces still conserves
total energy. It often converts mechanical energy into
other forms – notably, heat, light, or noise.
Power
Power is defined as the
time rate of doing work.
Since power may not
be constant in time, we
define average power:
SI units: J/s = watt (W)
W
P
t
James Watt
1736 – 1819
Scottish engineer
Invented the first efficient
steam engine, having a
separate condenser for the
“used” steam.
Graphical Analysis of Work
Plot force vs. position (for a constant force):
F cos q, N
area = (F cos q)·s = work
s
position, m
Graphical Analysis of Work
Plot force vs. position (for a variable force):
F cos q, N
area = work
position, m