Physics 121C Mechanics Lecture 16 Energy Diagrams November 8, 2004

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Physics 121C - MechanicsLecture 16Energy
DiagramsNovember 8, 2004

• John G. Cramer
• Professor of Physics
• B451 PAB
• cramer_at_phys.washington.edu

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Announcements

• Homework Assignment 5 is now posted on Tycho
  and is due on Wednesday, November 10.
• Exam 2 will be held on Friday, November 12 in
  this room.
• Exam seating assignments will be posted on Tycho
  several days before the exam. Please look up
  your seat assignment before arriving at the exam.
  You may request a special seat type
  (left-handed, right handed on an aisle, up front,
  table in back, etc.) by sending me E-mail in the
  next few days.

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Lecture Schedule (Part 2)
You are here!
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Hookes Law
The linear proportionality between force and
displacement is found to be valid whether the
spring is stretched or compressed. The force and
displacement are always in opposite directions,
so that it is a restoring force.
Therefore, with k (constant), we write the
force-displacement relation as
This relation for the restoring force of a
spring is sometimes called Hookes Law, named
after Robert Hooke, a contemporary and rival of
Newton. It is not really a law or nature, but
rather a rule of behavior for most springs.
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Elastic Potential Energy
Elastic Potential Energy
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ExampleA Spring-Launched Plastic Ball
A spring-loaded toy gun is used to launch a
10 g plastic ball. The spring, which has a
spring constant of k10 N/m, is compressed by 10
cm as the ball is pushed into the barrel. When
the trigger is pulled, the ball is released and
shoots the ball back out. What is the balls
speed as it leaves the barrel? (Neglect
friction.)
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Improved Energy Model
(conserved quantity)
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ExampleA Spring-Launched Satellite
Prince Harry the Horrible wants to launch a
satellite, but his country cannot afford rockets.
He places a 2.0 kg payload in top of a very
stiff 2.0 m long spring with a spring constant of
k50,000 N/m. Then the Prince has his strongest
men use a winch to crank the spring down to a
length of 80 cm. When released, the spring
shoots the payload straight up. How high
does the payload go?
(not high enough for orbit)
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ExamplePushing Apart
A spring with spring constant k2,000 N/m is
sandwiched between a 1.0 kg block and 2.0 kg
block on a frictionless table. The blocks are
pushed together to compress the spring by 10 cm,
and then released. What are the velocities
of the blocks as they fly apart?
(energy cons.)
(momen. cons.)
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Clicker Question 1
4.0 m/s
A spring-loaded gun shoots a plastic ball
with a speed of 4.0 m/s. If the spring is
compressed twice as far, what is the balls speed?
a) 2.0 m/s b) 4.0 m/s c) 8.0 m/s d)
16.0 m/s e) 32.0 m/s
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Elastic Collisions
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Three Elastic Collisions
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ExampleA Rebounding Pendulum
A 200 g steel ball hangs from a 1.0 m long
string. The ball is pulled sideways so that the
string is at a 450 angle, then released. At the
very bottom of the swing, the ball strikes a 500
g steel block that is resting on a frictionless
steel table. To what angle does the ball
rebound?
The ball drops by h1L(1 cos 450) 0.293
m, so its kinetic energy at the bottom of the
swing is KUgmgh ½m(v1)A2 so (v1)A2gh1½2.40
m/s
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Using Reference Frames
We have a solution for elastic collisions
when one object is initially at rest. But what
if both objects are initially moving?
Solution Transform to the rest frame of the 2nd
object, calculate velocities there, and then
transform back. Example m1200 g, v1i2
m/s, m2100 g, v2i-3 m/s. Transform by 3 m/s
v1i5 m/s, v2i0. Then v1fv1i(2m1)/(m1m2)
(4/3)5 m/s6.67 m/s and v2f
v1i(m1-m2)/(m1m2) (1/3)5 m/s 1.67 m/s.
Transforming back v1f (1.67 m/s)(3 m/s)
-1.33 m/s andv2f (6.67 m/s)(3 m/s) 3.67
m/s.
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Analyzing Elastic Collisions

• Use the Galilean transformation to transform
  the initial velocities of balls 1 and 2 from the
  lab frame S to a reference frame S in which
  ball 2 is at rest
• Use Equations 10.43 to determine the outcome of
  the collision in frame S then
• Transform the final velocities back to the lab
  frame S.

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Energy Diagrams
Energy diagram for a particle in a gravitational
field, so that Ugmgy (PE curve is a straight
line).
Energy diagram for a mass and a horizontal spring
on a frictionless surface, so that Us ½kx2 (PE
curve is a parabola).
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Trajectory of a Ball
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A Mass and Spring
Leq
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Equilibrium Positions

• There are positions in such a potential
  where the particle will remain in a fixed
  position if placed at rest there.
• If there is a restoring force at such a
  position, it is stable equilibrium.
• If the force makes a displacement larger, it is
  unstable equilibrium.

A particle moves in a more general potential
with maxima and minima.
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Interpreting Energy Diagrams

• The distance from the axis to the PE curve is the
  particles potential energy. The distance from
  the PE curve to the TE line is its kinetic
  energy. These are transformed as the position
  changes, causing the particle to speed up or slow
  down, but the sum doesnt change.
• A point where the TE line crosses the PE curve is
  a turning point. The particle reverses
  direction.
• The particle cannot be at a point where the PE
  curve is above the TE line.
• The PE curve is determined by the properties of
  the systemmass, spring constant, and the like.
  You cannot change the PE curve. However, you can
  raise or lower the TE line simply by changing the
  initial conditions to give the particle more or
  less total energy.
• A minimum in the PE curve is a point of stable
  equilibrium. A maximum in the PE curve is a point
  of unstable equilibrium.

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ExampleBalancing a Mass on a Spring
A spring of length Lo and spring constant k
is standing on one end. A block of mass m is
place on it, compressing it. What is the
compressed length of the spring?
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Clicker Question 2
A particle with the potential energy shown
in the above graph is moving to the right. It
has 1 J of kinetic energy at x 1m. Where
is the particles turning point?
a) x 0 m b) x 1 m c) x 2 m d) x
4 m e) x 6 m
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Clicker Question 2
A particle with the potential energy shown
in the above graph is moving to the right. It
has 1 J of kinetic energy at x 1m. Where
is the particles turning point?
a) x 0 m b) x 1 m c) x 2 m d) x
4 m e) x 6 m
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Molecular Bonds
Cl
H
At small separation distances, the potential
is repulsive. At large separation distances,
the potential is attractive. The potential
minimum is at about 0.13 nm. This is the
equilibrium separation of the atoms of the
molecule.
Energy diagram for the diatomic molecule HCl.
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Chapter 10 Summary (1)
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Chapter 10 Summary (2)
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Chapter 10 Summary (3)
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End of Lecture 16

• Before the next lecture, read Knight, Sections
  10.5 through 10.7.
• Homework Assignment 5 is posted on Tycho and is
  due on Wednesday, November 10.
• Exam 2 will be held on Friday, November 12 in
  this room.
• Exam seating assignments will be posted on Tycho
  several days before Exam 2. Please look up your
  seat assignment before arriving at the exam. You
  may request a special seat type (left-handed,
  right handed on an aisle, up front, table in
  back, etc.) by sending me E-mail before Monday.