Transcript Chapter 8. Potential Energy and Energy Conservation

Chapter 8. Potential Energy and
Energy Conservation
8.1. What is Physics?
8.2. Work and Potential Energy
8.3. Path Independence of Conservative
Forces
8.4. Determining Potential Energy Values
8.5. Conservation of Mechanical Energy
8.6. Reading a Potential Energy Curve
8.7. Work Done on a System by an External
Force
8.8. Conservation of Energy
Introduction
In Chapter 7 we introduced the concepts of
work and kinetic energy. We then derived
a net work-kinetic energy theorem to
describe what happens to the kinetic
energy of a single rigid object when work
is done on it. In this chapter we will
consider a systems composed of several
objects that interact with one another.
What is Physics?
(1) The system consists of an Earth–
barbell system that has its
arrangement changed when a weight
lifter (outside of the system) pulls the
barbell and the Earth apart by pulling
up on the barbell with his arms and
pushing down on the Earth with his
feet
(2) The system consists of two crates
and a floor. This system is
rearranged by a person (again,
outside the system) who pushes the
crates apart by pushing on one crate
with her back and the other with her
feet
There is an obvious difference between these two situations. The work the
weight lifter did has been stored in the new configuration of the Earth-barbell
system, and the work done by the woman separating the crates seem to be
lost rather than stored away.
How do we determine whether the work
done by a particular type of force is
“stored” or “used up.”?
The Path Independence Test for a
Gravitational Force
The net work done on the skier as she travels down the
ramp is given by
It does not depend on the shape of the ramp but only on the
vertical component of the gravitational force and the initial
and final positions of her center of mass.
Path Dependence of Work Done by a
Friction Force
• The work done by friction along that path 1→2 is given
by
• The work done by the friction force along path
1→4→3→2 is given by
Conservative Forces and Path
Independence
• conservative forces are the forces that
do path independent work;
• Non-conservative forces are the forces
that do path dependent work;
The work done by a conservative force along any
closed path is zero.
Test of a System's Ability to Store Work
Done by Internal Forces: the work done by a
conservative internal force can be stored in the
system as potential energy, and the work done
by a non-conservative internal force will be
“used up”
U  Wc
EXAMPLE 1: Cheese on a Track
Figure a shows a 2.0 kg block of slippery cheese that
slides along a frictionless track from point 1 to point 2.
The cheese travels through a total distance of 2.0 m
along the track, and a net vertical distance of 0.80 m.
How much work is done on the cheese by the
gravitational force during the slide?
Determining Potential Energy Values
Consider a particle-like object that is part of a system in
which a conservative force acts. When that force does work
W on the object,
the change in the potential energy associated with the
system is the negative of the work done
Gravitational Potential Energy
GRAVITATIONAL POTENTIAL ENERGY
• The gravitational potential energy U is the
energy that an object of mass m has by
virtue of its position relative to the surface
of the earth. That position is measured by
the height h of the object relative to an
arbitrary zero level: U G  mgh
• SI Unit of Gravitational Potential
Energy: joule (J)
Elastic Potential Energy
or
we choose the reference
configuration to be when the
spring is at its relaxed length and
the block is at .
Sample Problem 2
A 2.0 kg sloth hangs 5.0 m
above the ground (Fig. 8-6).
• a) What is the gravitational
potential energy U of the sloth–
Earth system if we take the
reference point y=0 to be (1) at
the ground, (2) at a balcony floor
that is 3.0 m above the ground,
(3) at the limb, and (4) 1.0 m
above the limb? Take the
gravitational potential energy to
be zero at y=0.
• (b) The sloth drops to the
ground. For each choice of
reference point, what is the
change in the potential energy
of the sloth–Earth system due to
the fall?
What is mechanical energy of a system?
The mechanical energy is the sum of
kinetic energy and potential energies:
Esys  K sys  U sys
mec
For example,
Esys  K sys  U sys
mec
1 2
1 2
 mv  mgh  kx
2
2
Conservation of Mechanical Energy
In a system where (1) no work is done on it by
external forces and (2) only conservative internal
forces act on the system elements, then the
internal forces in the system can cause energy
to be transferred between kinetic energy and
potential energy, but their sum, the mechanical
energy Emec of the system, cannot change.
E
mec
sys
 Ksys  U sys  0
An isolated system: is a system that there is no net work is
done on the system by external forces.
Example 3
Check Your Understanding
Some of the following situations are consistent with the
principle of conservation of mechanical energy, and
some are not. Which ones are consistent with the
principle?
(a) An object moves uphill with an increasing speed.
(b) An object moves uphill with a decreasing speed.
(c) An object moves uphill with a constant speed.
(d) An object moves downhill with an increasing speed.
(e) An object moves downhill with a decreasing speed.
(f) An object moves downhill with a constant speed.
Example 4 A Daredevil Motorcyclist
A motorcyclist is trying to leap across the
canyon shown in Figure by driving
horizontally off the cliff at a speed of 38.0
m/s. Ignoring air resistance, find the speed
with which the cycle strikes the ground on
the other side.
EXAMPLE 5: Bungee Jumper
A 61.0 kg bungee-cord jumper
is on a bridge 45.0 m above a
river. The elastic bungee cord
has a relaxed length of
L = 25.0 m. Assume that the
cord obeys Hooke's law, with
a spring constant of 160 N/m.
If the jumper stops before
reaching the water, what is
the height h of her feet above
the water at her lowest point?
EXAMPLE 6
• In Fig., a 2.0 kg package of tamales slides along a floor
with speed v1=4.0 m/s. It then runs into and
compresses a spring, until the package momentarily
stops. Its path to the initially relaxed spring is
frictionless, but as it compresses the spring, a kinetic
frictional force from the floor, of magnitude 15 N, acts
on it. The spring constant is 10 000 N/m. By what
distance d is the spring compressed when the package
stops?
Net Work on a system
Internal Work on a single rigid object
W
int
0
Internal Work on a system
Since Newton's Third Law tells us that
the internal work is given by the integral of y
the internal work on a system is not zero in general
Work-Energy Theorem
W
ext
W
int
NC
 Ksys  U sys  E
mec
sys
Example 7 Fireworks
• A 0.20-kg rocket in a fireworks
display is launched from rest
and follows an erratic flight path
to reach the point P, as Figure
shows. Point P is 29 m above
the starting point. In the process,
425 J of work is done on the
rocket by the nonconservative
force generated by the burning
propellant. Ignoring air
resistance and the mass lost
due to the burning propellant,
find the speed vf of the rocket at
the point P.
Reading a Potential Energy Curve
Finding the Force Analytically
Solving for F(x) and passing to the differential limit yield
Reading a Potential Energy Curve
• Turning Points: a place where K=0
(because U=E ) and the particle changes
direction.
• Neutral equilibrium: the place where the
particle has no kinetic energy and no
force acts on it, and so it must be
stationary.
• unstable equilibrium: a point at which .
If the particle is located exactly there, the
force on it is also zero, and the particle
remains stationary. However, if it is
displaced even slightly in either direction,
a nonzero force pushes it farther in the
same direction, and the particle continues
to move
• stable equilibrium: a point where a
particle cannot move left or right on its
own because to do so would require a
negative kinetic energy
Sample Problem
A 2.00 kg particle moves along an x axis in onedimensional motion while a conservative force along that
axis acts on it. The potential energy U(x) associated with
the force is plotted in Fig. 8-10a. That is, if the particle
were placed at any position between x=0 and x=7m , it
would have the plotted value of U. At x=6.5m , the
particle has velocity v0=(-4.0m/s)i . (a) determine the
particle’s speed at x1=4.5m. (b) Where is the particle’s
turning point located? (c) Evaluate the force acting on the
particle when it is in the region 1.9m<x<4.0m.
General Energy Conservation
int
mec
W ext  WNC
 K sys  U sys  Esys
int
WNC
 E NC
mec
int
mec
W ext  Esys
 WNC
 Esys
 E NC
For a isolated system where Wext is zero, it energy is conserved.
mec
Esys
 E NC  0
THE PRINCIPLE OF CONSERVATION
OF ENERGY: Energy can neither be
created nor destroyed, but can only be
converted from one form to another.
Example
In Fig. 8-58, a block slides along a path that is without
friction until the block reaches the section of length
L=0.75m, which begins at height h=2.0m on a ramp of
angle θ=30o . In that section, the coefficient of kinetic
friction is 0.40. The block passes through point A with a
speed of 8.0 m/s. If the block can reach point B (where
the friction ends), what is its speed there, and if it
cannot, what is its greatest height above A?