Download Chapter4.Presentation.ICAM.Work,Power_and_Energy

NAZARIN B. NORDIN
[email protected]
What you will learn:
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Define work, power and energy
Potential energy
Kinetic energy
Work-energy principle
Conservation of mechanical energy
The Work-Energy Principle
1 2
mv  Fs
2
The equation we derived is one form of the workenergy theorem (principle). 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 cos
2
If the work is positive, the kinetic energy increases.
Negative work decreases the kinetic energy.
The Work-Energy Principle
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
Potential Energy
An object can have potential energy by virtue of its surroundings.
Familiar examples of potential energy:
• A wound-up spring
• A stretched elastic band
• An object at some height above the ground
Potential Energy
In raising a mass m to a height h, the work done by
the external force is
(6-5a)
We therefore define the gravitational potential
energy:
(6-6)
Potential Energy
This potential energy can become kinetic energy if the object is dropped.
Potential energy is a property of a system as a whole, not just of the object
(because it depends on external forces).
If
, where do we measure y from?
It turns out not to matter, as long as we are consistent about where we choose y
= 0. Only changes in potential energy can be measured.
Potential Energy
Potential energy can also be stored in a
spring when it is compressed; the figure
below shows potential energy yielding kinetic
energy.
Potential Energy
The force required to compress or stretch a
spring is:
where k is called the spring constant, and
needs to be measured for each spring.
(6-8)
Work Done by a Varying Force
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. As the pieces
become very narrow, the work done is the area under the
force vs. distance curve.
Potential Energy
The force increases as the spring is stretched or compressed further. We find that the
potential energy of the compressed or stretched spring, measured from its equilibrium
position, can be written:
(6-9)
Kinetic Energy, and the Work-Energy Principle
Energy was traditionally defined as the ability
to do work.
We now know that not all forces are able to do
work; however, we are dealing in these
chapters with mechanical energy, which does
follow this definition.
Kinetic Energy, and the Work-Energy Principle
If we write the acceleration in terms of the velocity and the distance, we find
that the work done here is
We define the kinetic energy:
(6-2)
(6-3)
Kinetic Energy, and the Work-Energy Principle
This means that the work done is equal to the change in the kinetic energy:
(6-4)
• If the net work is positive, the kinetic energy increases.
• If the net work is negative, the kinetic energy decreases.
Kinetic Energy, and the Work-Energy Principle
Because work and kinetic energy can be equated, they must have the same
units: kinetic energy is measured in joules.
The Work-Energy Principle
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 Principle
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 Principle
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
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.
Mechanical Energy and Its Conservation
The principle of conservation of mechanical
energy:
If only conservative forces are doing work, the total
mechanical energy of a system neither increases nor
decreases in any process. It stays constant—it is
conserved.
Problem Solving Using Conservation of Mechanical
Energy
In the image on the left, the total
mechanical energy at any point is:
Problem Solving Using Conservation of Mechanical Energy
Example 3: Falling rock.
If the original height of the rock is y1 = h = 3.0
m, calculate the rock’s speed when it has
fallen to 1.0 m above the ground.
Problem Solving Using Conservation of Mechanical Energy
Example 4: Roller-coaster car speed using
energy conservation. Assuming the height of the
hill is 40 m, and the roller-coaster car starts from
rest at the top, calculate (a) the speed of the
roller-coaster car at the bottom of the hill, and
(b) at what height it will have half this speed.
Take y = 0 at the bottom of the hill.
Problem Solving Using Conservation of Mechanical Energy
Conceptual Example 5: Speeds on two water slides.
Two water slides at a pool are
shaped differently, but start at
the same height h. Two riders,
Paul and Joanna, start from
rest at the same time on
different slides. (a) Which rider,
Paul or Kathleen, is traveling
faster at the bottom? (b) Which
rider makes it to the bottom
first? Ignore friction and
assume both slides have the
same path length.
Problem Solving Using Conservation of Mechanical
Energy
Which to use for solving problems?
Newton’s laws: best when forces are
constant
Work and energy: good when forces are
constant; also may succeed when forces are
not constant
Problem Solving Using Conservation of
Mechanical Energy
Example 6: Pole vault.
Estimate the kinetic energy
and the speed required for a
70-kg pole vaulter to just pass
over a bar 5.0 m high. Assume
the vaulter’s center of mass is
initially 0.90 m off the ground
and reaches its maximum
height at the level of the bar
itself.
Problem Solving Using Conservation of
Mechanical Energy
For an elastic force, conservation of energy tells us:
Example 8-7: Toy dart gun.
A dart of mass 0.100 kg is pressed
against the spring of a toy dart gun.
The spring (with spring stiffness
constant k = 250 N/m and ignorable
mass) is compressed 6.0 cm and
released. If the dart detaches from
the spring when the spring reaches
its natural length (x = 0), what
speed does the dart acquire?
Problem Solving Using Conservation of
Mechanical Energy
Example 8-8: Two kinds of potential energy.
A ball of mass m = 2.60 kg,
starting from rest, falls a vertical
distance h = 55.0 cm before
striking a vertical coiled spring,
which it compresses an amount Y
= 15.0 cm. Determine the spring
stiffness constant of the spring.
Assume the spring has negligible
mass, and ignore air resistance.
Measure all distances from the
point where the ball first touches
the uncompressed spring (y = 0 at
this point).
Problem Solving Using Conservation of Mechanical Energy
This simple pendulum consists of a small bob of mass m
suspended by a massless cord of length l. The bob is
released (without a push) at t = 0, where the cord makes
an angle θ = θ0 to the vertical.
Example 9: A swinging pendulum.
(a) Describe the motion of the bob
in terms of kinetic energy and
potential energy. Then determine
the speed of the bob (b) as a
function of position θ as it swings
back and forth, and (c) at the
lowest point of the swing. (d) Find
the tension in the cord,
FT. Ignore friction and air
resistance.