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Chapter 8
Conservation of Energy
HW5:Chapter 8: Pb.8, Pb.20,
Pb.23, Pb.33, Pb.68, Pb.71
Due Wednesday, Oct.19
7-4 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 as:
7-4 Kinetic Energy and the Work-Energy
Principle
This means that the work done is equal to the change in
the kinetic energy:
This is the Work-Energy Principle:
• If the net work is positive, the kinetic energy increases.
• If the net work is negative, the kinetic energy decreases.
7-4 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.
Energy can be considered as the ability to do work:
7-4 Kinetic Energy and the Work-Energy
Principle
Example 7-7: Kinetic energy and work
done on a baseball.
A 145-g baseball is thrown so that it
acquires a speed of 25 m/s. (a) What
is its kinetic energy? (b) What was
the net work done on the ball to make
it reach this speed, if it started from
rest?
7-4 Kinetic Energy and the Work-Energy
Principle
Example 7-8: Work on a car, to increase its
kinetic energy.
How much net work is required to accelerate
a 1000-kg car from 20 m/s to 30 m/s?
The net work is the increase in kinetic
energy
7-4 Kinetic Energy and the Work-Energy
Principle
Example 7-10: A compressed spring.
A horizontal spring has spring constant k = 360 N/m. (a) How
much work is required to compress it from its uncompressed
length (x = 0) to x = 11.0 cm? (b) If a 1.85-kg block is
placed against the spring and the spring is released, what will
be the speed of the block when it separates from the spring at
x = 0? Ignore friction. (c) Repeat part (b) but assume that the
block is moving on a table and that some kind of constant drag
force FD = 7.0 N is acting to slow it down, such as friction (or
perhaps your finger).
Problem 56
56. (II) An 85-g arrow is fired from a bow whose string exerts
an average force of 105 N on the arrow over a distance of 75 cm.
What is the speed of the arrow as it leaves the bow?
8-1 Conservative and Nonconservative Forces
2
1
Example 8-1: How much work is
needed to move a particle from
position 1 to 2?
8-1 Conservative and Nonconservative Forces
A force is conservative if:
the work done by the force on an object moving from one point
to another depends only on the initial and final positions of the
object, and is independent of the particular path taken.
Example: gravity.
8-1 Conservative and Nonconservative
Forces
Another definition of a conservative force:
a force is conservative if the net work done by the force
on an object moving around any closed path is zero.
(a)
(b)
8-1 Conservative and Nonconservative
Forces
If friction is present, the work done depends not only
on the starting and ending points, but also on the path
taken. Friction is called a non-conservative force.
8-1 Conservative and Nonconservative Forces
8-2 Potential Energy
Example 8-2
What potential energy is needed
to move a block upward with an
external force Fext?
8-2 Potential Energy
In raising a mass m to a height h, the work
done by the external force is
We therefore define the gravitational
potential energy at a height y above some
reference point:
8-2 Potential Energy
Example 8-3: Potential energy changes for a roller coaster.
A 1000-kg roller-coaster car moves from point 1 to point 2 and then to
point 3. (a) What is the gravitational potential energy at points 2 and 3
relative to point 1? That is, take y = 0 at point 1. (b) What is the
change in potential energy when the car goes from point 2 to point 3?
(c) Repeat parts (a) and (b), but take the reference point (y = 0) to be
at point 3.
8-2 Potential Energy
An object can have potential energy by virtue of its
surroundings. Potential energy can only be defined for
conservative forces
Familiar examples of potential energy:
• A wound-up spring
• A stretched elastic band
• An object at some height above the ground
8-2 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 Ugrav = mgy, 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.
8-2 Potential Energy
General definition of gravitational potential
energy:
For any conservative force:
Problem 7
Problem 9
8-2 Potential Energy
A spring has potential energy, called
elastic potential energy, when it is
compressed. 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.
8-2 Potential Energy
Then the potential energy for a spring is:
8-2 Potential Energy
In one dimension,
We can invert this equation to find U (x)
if we know F (x):
In three dimensions:
8-3 Mechanical Energy and Its
Conservation
If the forces are conservative, the sum of the
changes in the kinetic energy and in the potential
energy is zero—the kinetic and potential energy
changes are equal but opposite in sign.
This allows us to define the total mechanical energy:
And its conservation:
.
8-3 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 16
(II) A 72-kg trampoline artist jumps vertically upward from the
top of a platform with a speed of 4.5m/s. (a) How fast is he going
as he lands on the trampoline, 2.0 m below (b) If the trampoline
behaves like a spring of spring constant 5.8 × 104 N/m, how far
does he depress it?