Work

1
Work
and
Energy
2
Lets start with a definition

• Work a scalar quantity (it may be or -)that
  is associated with a force acting on an object as
  it moves through some displacement.
• If the force tends to make the object go in the
  direction it traveled, then the force did work.
  
• If the force opposed the motion, it did - work.
• If the force acts perpendicular to the motion,
  it does zero work.

Work (component of force in direction of
motion)(displacement size)
units of work are
or Joules (J)
3
F115 N
8 N
10 N
F2
5 m
4
The previous definition is valid if the force
does not change over the displacement. More
generally, we can restrict our displacement to
one that is very small (infinitesimal). Then the
force will not change over the small
displacement. Let us consider the case in one
dimension with a force that depends on position
F
x
x2
?x
x1
As ?x gets smaller, the area of the rectangle has
less sticking out above the curve. Thus we
expect
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Fundamental Theorem of Calculus
Thus integration and differentiation are inverses
of each other, sort of.
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Exercise A spring exerts a restoring force on
an object attached to it directly proportional to
the amount of stretch or compression.
unstretched
stretched
x
How much work is done by the spring as the object
is stretched from x1 to x2 ?
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In more than one dimension, we must take the
direction into account
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Power
If we calculate the work done over an interval,
and divide by the time interval, we get the rate
at which work is done, the power supplied (or
consumed) by the force.
In one dimension we have simply
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Why do we bother with the concept of
work? Because when we look at the work done by
the net force, i.e., the total work, a useful
result is obtained.
vo
v
net F
net F
x
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We then define the kinetic energy of an object as
Since our object started with KE , the
work done by the net F, i.e. the total work done,
is equal to the change in the KE
Thus work is important because it is directly
related to how the kinetic energy changes.
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Energy of motion
KE
Total work done by all forces to get object from
rest up to speed v
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vo 4m/s
v 6 m/s
T20 N
T20 N
fk
fk
3 m.
What was the change in the KE for our 4 kg
mass? ?KE 1/2(4)(6)2 - 1/2(4)(4)2 ?KE 72 -
32 40 J
Find the value of fk
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Conservation of Energy
Suppose the system is under the influence of a
single force F
Here the ? refers to the interval, i.e., the
kinetic energy changed from K0 to K as the force
F did work over the interval. Let us suppose that
the work done by F does not depend on the
particular path taken between the ends of the
interval, but depends only on the end points
themselves. Such a force is said to be
conservative. For such a force, we can always
express the work done over an interval as the
difference in the value of a new function at the
ends of the interval
The function U is called the potential energy.
14
Then we may write
Define the total mechanical energy
Total mechanical energy is conserved (stays the
same) if only conservative forces act.
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Conservative Forces Examples
For a force to be conservative, the work it does
must be independent of the path taken between
the endpoints implying that the work depends on
just the endpoints.
Example 1 Any force that acts in 1-D and
depends only on the position x. Denote the
force as f(x).
Clearly this depends only on the endpoints.
Example 2 A mass changes position near the
surface of the earth.
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Depends only on endpoinjts
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It is possible to do work on an object in such a
way as to never give it any appreciable
KE. Example lift a mass slowing to some
height. Example slowly pull out a mass attached
to a spring.
In both examples some agent is applying a force
just equal to the opposing force (gravity or
spring force) and does work in positioning the
object. Intuitively we might expect that the work
done by the agent can be gotten back by
allowing the system to return to where it
started let the mass fall release the
mass/spring
We say there is potential energy stored in the
system. This PE is associated with the position
of the system, not its motion.
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Energy due to position
Formulae
U
Work that some agent must do to slowly put the
system into position starting from some
(arbitrary) reference point.
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Potential Energy Formulae
Lets stick to 1-D
Only differences in energy can be measured, so we
can pick the value of U(a) to cancel
Then we have
Essentially an indefinite integral
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For gravity we can choose the y 0 point
wherever convenient.
For the spring we will always use x 0 as the
reference point, although in principle we could
use other points..
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If a system moves only under the influence of
gravity, elastic, or electric forces, then any
change in KE always comes at the expense of PE in
such a way that the total of the two remains the
same. This is the principle of Conservation of
Energy.
Total mechanical energy
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ExampleA 4 kg ball is dropped from a height of 8
m. How fast is it moving just as it hits the
ground?
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Example A spring with k 50 N/m has a 4 kg
mass attached such that it can move horizontally
on a smooth surface. It is observed that when the
spring is stretched 1.2 m, the mass is moving
outward with a speed of 20 m/s. How much further
out will the mass travel?
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Force from Potential Energy
We can find the various components of the force
by taking the appropriate derivative.
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Exercise A 3-D potential energy is given by
Find Fx at the point (1, 2, 4).
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Graphs of Potential Energy
Given the total energy and a graph of U vs x, one
can qualitatively describe the types of motions
available to the system
E
K
U
Negative slope of curve is force
Turning points
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Exercise Describe possible motions for
different total energies. Find force at x 1 m