Transcript Lecture10-09

Chapter 8
Potential Energy and
Conservation of Energy
Copyright © 2010 Pearson Education, Inc.
Reading and Review
Copyright © 2010 Pearson Education, Inc.
Force and Work
a) one force
A box is being pulled up a rough
b) two forces
incline by a rope connected to a
c) three forces
pulley. How many forces are
doing work on the box?
Copyright © 2010 Pearson Education, Inc.
d) four forces
e) no forces are doing work
Force and Work
a) one force
A box is being pulled up a rough
b) two forces
incline by a rope connected to a
c) three forces
pulley. How many forces are
doing work on the box?
d) four forces
e) no forces are doing work
Any force not perpendicular
to the motion will do work:
N does no work
N
T
T does positive work
f
f does negative work
mg does negative work
Copyright © 2010 Pearson Education, Inc.
mg
Free Fall I
Two stones, one twice the
mass of the other, are dropped
from a cliff. Just before hitting
the ground, what is the kinetic
energy of the heavy stone
compared to the light one?
Copyright © 2010 Pearson Education, Inc.
a) quarter as much
b) half as much
c) the same
d) twice as much
e) four times as much
Free Fall I
Two stones, one twice the
mass of the other, are dropped
from a cliff. Just before hitting
the ground, what is the kinetic
energy of the heavy stone
compared to the light one?
a) quarter as much
b) half as much
c) the same
d) twice as much
e) four times as much
Consider the work done by gravity to make the stone
fall distance d:
DKE = Wnet = F d cosq
DKE = mg d
Thus, the stone with the greater mass has the greater
KE, which is twice as big for the heavy stone.
Follow-up: How do the initial values of gravitational PE compare?
Copyright © 2010 Pearson Education, Inc.
Free Fall II
a) quarter as much
In the previous question, just
before hitting the ground, what is
the final speed of the heavy stone
compared to the light one?
Copyright © 2010 Pearson Education, Inc.
b) half as much
c) the same
d) twice as much
e) four times as much
Free Fall II
a) quarter as much
In the previous question, just
before hitting the ground, what is
the final speed of the heavy stone
compared to the light one?
b) half as much
c) the same
d) twice as much
e) four times as much
All freely falling objects fall at the same rate, which is g.
Because the acceleration is the same for both, and the
distance is the same, then the final speeds will be the same for
both stones.
Copyright © 2010 Pearson Education, Inc.
Topics of Chapter 8
• Conservative and Nonconservative
Forces
• Potential Energy and the Work Done by
Conservative Forces
• Conservation of Mechanical Energy
• Work Done by Nonconservative Forces
• Potential Energy Curves and
Equipotentials
Copyright © 2010 Pearson Education, Inc.
8-1 Conservative and Nonconservative
Forces
Conservative force: the work it does is stored in
the form of energy that can be released at a later
time
Example of a conservative force: gravity
Example of a nonconservative force: friction
Also: the work done by a conservative force
moving an object around a closed path is zero;
this is not true for a nonconservative force
Copyright © 2010 Pearson Education, Inc.
8-1 Conservative and Nonconservative
Forces
Work done by gravity on a closed path is zero:
Copyright © 2010 Pearson Education, Inc.
8-1 Conservative and Nonconservative
Forces
Work done by friction on a closed path is not
zero:
Copyright © 2010 Pearson Education, Inc.
8-1 Conservative and Nonconservative
Forces
The work done by a conservative force is zero
on any closed path:
Copyright © 2010 Pearson Education, Inc.
8-2 The Work Done by Conservative Forces
If we pick up a ball and put it on the shelf, we
have done work on the ball. We can get that
energy back if the ball falls back off the shelf; in
the meantime, we say the energy is stored as
potential energy.
(8-1)
Copyright © 2010 Pearson Education, Inc.
8-2 The Work Done by Conservative Forces
Gravitational potential energy:
Copyright © 2010 Pearson Education, Inc.
Sign of the Energy II
Is it possible for the
a) yes
gravitational potential
b) no
energy of an object to
be negative?
Copyright © 2010 Pearson Education, Inc.
Sign of the Energy II
Is it possible for the
a) yes
gravitational potential
b) no
energy of an object to
be negative?
Gravitational PE is mgh, where height h is measured relative to
some arbitrary reference level where PE = 0. For example, a
book on a table has positive PE if the zero reference level is
chosen to be the floor. However, if the ceiling is the zero level,
then the book has negative PE on the table. Only differences (or
changes) in PE have any physical meaning.
Copyright © 2010 Pearson Education, Inc.
Question 8.2 KE and PE
You and your friend both solve a
problem involving a skier going
down a slope, starting from rest.
The two of you have chosen
different levels for y = 0 in this
problem. Which of the following
quantities will you and your friend
agree on?
A) skier’s PE
Copyright © 2010 Pearson Education, Inc.
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
Question 8.2 KE and PE
You and your friend both solve a
problem involving a skier going
down a slope, starting from rest.
The two of you have chosen
different levels for y = 0 in this
problem. Which of the following
quantities will you and your friend
agree on?
A) skier’s PE
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
The gravitational PE depends upon the reference level, but
the difference D PE does not! The work done by gravity
must be the same in the two solutions, so DPE and DKE
should be the same.
Follow-up: Does anything change physically by the choice of y = 0?
Copyright © 2010 Pearson Education, Inc.
8-2 The Work Done by Conservative Forces
Springs:
Copyright © 2010 Pearson Education, Inc.
(8-4)
8-3 Conservation of Mechanical Energy
Definition of mechanical energy:
(8-6)
Using this definition and considering only
conservative forces, we find:
Or equivalently:
Copyright © 2010 Pearson Education, Inc.
8-3 Conservation of Mechanical Energy
Energy conservation can make kinematics
problems much easier to solve:
Copyright © 2010 Pearson Education, Inc.
Question 8.2 KE and PE
You and your friend both solve a
problem involving a skier going
down a slope, starting from rest.
The two of you have chosen
different levels for y = 0 in this
problem. Which of the following
quantities will you and your friend
agree on?
A) skier’s PE
Copyright © 2010 Pearson Education, Inc.
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
KE and PE
You and your friend both solve a
problem involving a skier going
down a slope, starting from rest.
The two of you have chosen
different levels for y = 0 in this
problem. Which of the following
quantities will you and your friend
agree on?
A) skier’s PE
B) skier’s change in PE
a) only B
b) only C
c) A, B, and C
d) only A and C
e) only B and C
C) skier’s final KE
The gravitational PE depends upon the reference level, but
the difference D PE does not! The work done by gravity
must be the same in the two solutions, so DPE and DKE
should be the same.
Follow-up: Does anything change physically by the choice of y = 0?
Copyright © 2010 Pearson Education, Inc.
Example: Two water slides are shaped differently, but start at the
same height h and are of equal length. Two rides, Paul and Kathy,
start from rest at the same time on different slides.
a) Which is travelling faster at the bottom?
b) Which makes it to the bottom first?
Copyright © 2010 Pearson Education, Inc.
8-4 Work Done by Nonconservative Forces
In the presence of nonconservative forces, the
total mechanical energy is not conserved:
Solving,
(8-9)
Copyright © 2010 Pearson Education, Inc.
8-4 Work Done by Nonconservative Forces
In this example, the
nonconservative force
is water resistance:
Copyright © 2010 Pearson Education, Inc.
8-5 Potential Energy Curves and
Equipotentials
The curve of a hill or a roller coaster is itself
essentially a plot of the gravitational
potential energy:
Copyright © 2010 Pearson Education, Inc.
8-5 Potential Energy Curves and
Equipotentials
The potential energy curve for a spring:
Copyright © 2010 Pearson Education, Inc.
8-5 Potential Energy Curves and
Equipotentials
Contour maps are also a form of potential
energy curve:
Copyright © 2010 Pearson Education, Inc.
Summary of Chapter 8
• Conservative forces conserve mechanical
energy
• Nonconservative forces convert mechanical
energy into other forms
• Conservative force does zero work on any
closed path
• Work done by a conservative force is
independent of path
• Conservative forces: gravity, spring
Copyright © 2010 Pearson Education, Inc.
Summary of Chapter 8
• Work done by nonconservative force on closed
path is not zero, and depends on the path
• Nonconservative forces: friction, air
resistance, tension
• Energy in the form of potential energy can be
converted to kinetic or other forms
• Work done by a conservative force is the
negative of the change in the potential energy
• Gravity: U = mgy
• Spring: U = ½ kx2
Copyright © 2010 Pearson Education, Inc.
Summary of Chapter 8
• Mechanical energy is the sum of the kinetic and
potential energies; it is conserved only in
systems with purely conservative forces
• Nonconservative forces change a system’s
mechanical energy
• Work done by nonconservative forces equals
change in a system’s mechanical energy
• Potential energy curve: U vs. position
Copyright © 2010 Pearson Education, Inc.