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Physics 121B - Mechanics
Lecture 12 (T&M: 7.1-3)
Mass-Energy & Momentum
April 28, 2010
John G. Cramer
Professor Emeritus of Physics
B451 PAB
jcramer@uw.edu
Announcements
Homework Assignment #5 should be submitted
on the Tycho system by 11:59 PM on Friday, April 30.
On Friday, May 7, we will have Exam 2 covering
T&M Chapters 5-8. Sections will be Lecture
multiple-choice (55 pts), Laboratory (25 pts), and
Tutorial (20 pts). There will be assigned seating.
Send E-mail if you have a new seating preference.
As of 9:10 AM today, 153/175 Physics 121B
students have registered their clickers. If you have
not already done so, register your clicker at:
http://courses.washington.edu/p121bs10/clicker.htm
April 28, 2010 Physics 121B - Lecture 12 2/28
Lecture Schedule (Part 2)
Physics 121B - Prof. John G. Cramer - 11:30-12:20 PM MWF - A118 PAB
Textbook: Physics for Scientists and Engineers, 6th Edition, Paul A. Tipler and Gene Mosca, W. H. Freeman & Co. (2008)
Week Date L# Lecture Topic Text Reading Pages Slides Hwk Tutorial Lab
Forces Free Fall & Projectiles
16-Apr-10 E1 EXAM 1 - Chapters 1, 2, 3, and 4 HW#3
19-Apr-10 8 Friction, Circular Motion Ch. 5.1,2,3 5 29
4 21-Apr-10 9 Center of Mass; Work Ch. 5.5, 6.1-2 19 32 Newton's 2nd & 3rd 1-D Dynamics
23-Apr-10 10 Energy;Potential Energy Ch. 6.3-5, 7.1 10 19 HW#4
26-Apr-10 11 Conservation of Energy Ch. 7.2 to 7.3 12 32
5 28-Apr-10 12 Mass-Energy; Momentum Ch. 7.4 to 8.1 26 28 Work and KE Newton's Laws, Tension
30-Apr-10 13 Kinetic En; Impulse & Colliison Chs. 8.2 to 8.5 16 24 HW#5
3-May-10 14 Explosions & Rockets Ch. 8.4 to 8.5 19 21
We are here!
6 5-May-10 15 Review 2 22 Cons. of Energy Work & Energy
7-May-10 E2 EXAM 2 - Chapters 5, 6, 7, and 8 HW#6
Cons. of Momentum Momentum & Collisions
10-May-10 16 Rotational Kinematics & Energy Ch. 9.1,2,3 6
April 28, 2010 Physics 121B - Lecture 12 3/28
Potential
Energy and Force
dU ( x)
F ( x)
dx
April 28, 2010 Physics 121B - Lecture 12 4/28
Clicker Question 1
A particle moves along the x-axis
with the potential energy shown.
What is the x component of the
force on the particle when it is at
x=4 m ?
(a) 2 N (b) 1 N (c) 4 N (d) 2 N (e) 1 N
April 28, 2010 Physics 121B - Lecture 12 5/28
Stable and Unstable Equilibrium
Condition for Stable Equilibrium: In stable
equilibrium, a small displacement in any direction
results in a restoring force that accelerates the
particle back to its equilibrium position. d2U/dx2 > 0.
Condition for Unstable Equilibrium: In unstable
equilibrium, a small displacement in any direction
results in a force that accelerates the particle
away from its equilibrium position. d2U/dx2 < 0.
Condition for Neutral Equilibrium: In neutral
equilibrium, a small displacement in any direction
results in a zero force and the particle remains in
equilibrium. d2U/dx2 = 0.
April 28, 2010 Physics 121B - Lecture 12 6/28
Example: Force and
the Potential-Energy Function
In the region –a < x < a the force on a
particle is represented by the potential
energy function: U(x) = -b[1/(a+x) +1/(a-x)],
where a and b are positive constants.
(a) Find the force Fx in the region –a < x < a.
(b) At what value of x is the force 0?
(c) At that location, is the equilibrium stable or
unstable?
dU d 1 1 1 1
Fx b b 2
dx dx (a x) (a x) (a x ) (a x )
2
At x 0, Fx 0.
d 2U d2 1 1 1 1
2 b 2b 3
dx 2 dx (a x) (a x) (a x ) (a x )
3
d 2U
At x 0, 4b / a 0, so the equilibrium is unstable.
dx 2
April 28, 2010 Physics 121B - Lecture 12 7/23
Suspension and Equilibrium
By
choosing
several
pivot points,
one can
locate the
CM as the
cross point
of the
several
An object hangs so that the CM is verticals
below the pivot point. from the
pivot points.
April 28, 2010 Physics 121B - Lecture 12 8/28
The Conservation of Energy
The total energy of the universe is constant. Energy can be
converted from one form to another or transmitted from one region
to another, but energy can never be created or destroyed.
Ein Eout Esys
Esys Emech Etherm Echem Eother
The Work-Energy Theorem
Wext Esys Emech Etherm Echem Eother
April 28, 2010 Physics 121B - Lecture 12 9/28
Clicker Question 2
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 ball’s 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
April 28, 2010 Physics 121B - Lecture 12 10/28
Example: Falling Clay
A ball of modeling clay with mass m is released from rest from a height h
and falls to the perfectly rigid floor (thud). Discuss the application of the law
of conservation of energy to
(a) the system consisting of the clay ball alone, and
Wext mgh; Emech 0; Wext Emech Etherm
Etherm mgh
(b) the system consisting of Earth, the floor, and the clay ball.
Wext 0; Wext Emech Etherm 0
Emechi mgh; Emech f 0
Emech 0 mgh mgh
Etherm Emech mgh
April 28, 2010 Physics 121B - Lecture 12 11/28
Problems Involving
Kinetic Friction
Wext 0 Emech Etherm
Emech Kblock Kboard 1
2
mv 2 1 mvi2
f 2 1
2
MV f2 0
fk max ; fk x max x; 2ax x v2 vi2
f
f k x 1 m v2 vi2 1 mv2 1 mvi2
2 f 2 f 2 f k X MAx X 1 M V f2 Vi 2 1 MV f2 0
2 2
fk x X 1 mv2 1 mvi2 1 MVf2
2 f 2 2
f k srel Emech ; f k srel Etherm
Wext Emech Etherm Emech f k srel
April 28, 2010 Physics 121B - Lecture 12 12/28
Example: Pushing a Box
A 4.0-kg box is initially at rest on a horizontal
tabletop. You push the box a distance of 3.0 m
along the tabletop with a horizontal force of 25
N. The coefficient of kinetic friction between the
box and tabletop is 0.35. Find
(a) the external work done on the block–table
system,
(b) the energy dissipated by friction,
(c) the final kinetic energy of the box, and
(d) the speed of the box.
Wext Wby you on block Wby gravity on block Wby gravity on table Wby floor on table
Fpush x 0 0 0 25 N 3.0 m 75 J
Etherm f k x k Fn x k mg x 0.35 4.0 kg 9.81 m/s2 3.0 m 41 J
K f Wext Etherm 75 J 41 J 34 J v f 2K f / m 2 34 J / 4.0 kg 4.1 m/s
April 28, 2010 Physics 121B - Lecture 12 13/28
Example: A Playground Slide
A child of mass 40 kg goes down an 8.0-m-
long slide inclined at 30° with the horizontal.
The coefficient of kinetic friction between the
child and the slide is 0.35. If the child starts
from rest at the top of the slide, how fast is
he traveling when he reaches the bottom?
Wext Emech Etherm (U K ) f k srel
K K f 0 1 mv2 ; Wext 0; U mgh
2
Fn mg cos 0; fk k Fn k mg cos ; h s sin
0 mgh 1 mv2 fk s mgs sin 1 mv2 k mg cos s
2 f 2 f
v f 2 gs sin k cos 2 9.81 (4.0) sin 30 0.35cos30 5.6 m/s
April 28, 2010 Physics 121B - Lecture 12 14/28
Friction and Chemical Energy
Fk srel Etherm
Wext Emech Etherm Emech Fk srel Echem (mgh Etherm )
April 28, 2010 Physics 121B - Lecture 12 15/28
Mass & Energy
E mc2
E
M 2
c
April 28, 2010 Physics 121B - Lecture 12 16/28
Example: Nuclear Binding Energy
A hydrogen atom consisting of a proton and an electron has a binding
energy of 13.6 eV. By what percentage is the mass of a proton plus the
mass of an electron greater than that of the hydrogen atom?
April 28, 2010 Physics 121B - Lecture 12 17/28
Example: Nuclear Fusion
In a typical nuclear fusion reaction, a triton (t) and a deuteron (d) fuse
together to form an alpha particle (a) plus a neutron (n). The reaction is
written d + t → a + n.
How much energy is released per deuteron produced for this fusion
reaction?
April 28, 2010 Physics 121B - Lecture 12 18/28
Nonrelativistic (Newtonian)
Mechanics and Relativity
As the speed of a particle approaches a significant fraction of the speed
of light, Newton’s second law breaks down, and we must modify Newtonian
mechanics according to Einstein’s theory of relativity. The criterion for the
validity of Newtonian mechanics can also be stated in terms of the energy
of a particle.
In nonrelativistic (Newtonian) mechanics, the kinetic energy of a particle
moving with speed v is:
where E0 = mc2 is the rest energy of the particle. Solving for v/c gives:
Nonrelativistic mechanics is valid if the speed v of the particle is much
less than c, the speed of light, or alternatively, if the kinetic energy K of a
particle is much less than its rest energy E0.
April 28, 2010 Physics 121B - Lecture 12 19/28
Quantization of Energy
In microscopic mechanical systems (e.g.,
molecules), because of quantum mechanics, the
system energy cannot change continuously, but
rather can change only in discrete steps to well
defined “state” values. The “ground state” E0 is the
lowest of these energy values.
En (n )hf , n 0,1, 2,3,
1
E0 1 hf
2
2
h Planck's constant 6.636 10-34 J s =4.136 10-15 eV s
Erad E f Ei Ephoton hf
April 28, 2010 Physics 121B - Lecture 12 20/28
Momentum
Momentum:
p mv
Conservation
of Linear Momentum
Momentum: p mv
dp d (mv ) dv dp
m ma Fnet
dt dt dt dt
P mi vi pi Mvcm
sys
i i
dPsys dvcm
F
i
ext Fnet ext
dt
M
dt
Macm
If Fext 0, then P mi vi Mvcm constant Conservation
sys
of Momentum
April 28, 2010 Physics 121B - Lecture 12 22/28
Example: A Space Repair
During repair of the Hubble Space Telescope, an
astronaut replaces a damaged solar panel during a
spacewalk. Pushing the detached panel away into space,
she is propelled in the opposite direction. The astronaut’s
mass is 60 kg, and the panel’s mass is 80 kg. Both
astronaut and panel are initially at rest relative to the
telescope, until the astronaut gives the panel a shove,
giving it a velocity of 0.30 m/s relative to the telescope.
Assuming her tether is slack, what is her velocity
relative to the telescope?
dPsys
F ext 0
dt
, so Psys constant
mP vPf mAv Af mP vPi mAv Ai 0
mP vPf mAv Af
mP (80 kg) ˆ ˆ
v Af vPf (0.30 m/s)i (0.40 m/s)i
mA (60 kg)
April 28, 2010 Physics 121B - Lecture 12 23/28
Kinetic Energy of a System
K K i 1 mi vi2 1 mi (vi vi )
2 2
vi vCM ui
i i i
K 1 mi (vCM ui ) (vCM ui ) 1 mi (v CM u i2 2vCM ui )
2 2
2
i i
K 1 mi v CM 1 mu i2 vCM mi ui
2
i
2
2
i i
mu
i
i i 0
K 1 mi vCM 1 mi ui2 1 MvCM K rel
2
2
2 2
2
i i
In an isolated system, only the relative kinetic energy can change
due to internal forces. I. e., no “space drives”.
April 28, 2010 Physics 121B - Lecture 12 24/28
Example: Hubble Repair
During repair of the Hubble Space Telescope, an
astronaut replaces a damaged solar panel during a
spacewalk. Pushing the detached panel away into
space, she is propelled in the opposite direction. The
astronaut’s mass is 60 kg and the panel’s mass is 80
kg. Both the astronaut and the panel initially are at
rest relative to the telescope. The astronaut then
gives the panel a shove. After the shove it is moving
at 0.30 m/s relative to the Hubble Telescope.
What is her subsequent velocity relative to the
telescope? (During this operation the astronaut is
tethered to the ship; for our calculations assume
that the tether remains slack.)
April 28, 2010 Physics 121B - Lecture 12 25/28
Example:
A Runaway Railroad Car
A runaway 14,000-kg railroad car is rolling
horizontally at 4.00 m/s toward a switchyard. As
it passes by a grain elevator, 2000 kg of grain
suddenly drops into the car.
How long does it take the car to cover the
500-m distance from the elevator to the
switchyard? Assume that the grain falls straight
down and that slowing due to rolling friction or air
drag is negligible.
April 28, 2010 Physics 121B - Lecture 12 26/28
Example: Radioactive Decay
A thorium-227 nucleus (mass 227 u)
at rest decays into a radium-223 nucleus
(mass 223 u) by emitting an alpha particle
(mass 4.00 u) . The kinetic energy of the a particle is measured to be 6.00
MeV. What is the kinetic energy of the recoiling radium nucleus?
April 28, 2010 Physics 121B - Lecture 12 27/28
End of Lecture 12
For the next lecture, read T&M Chapter 8.2-3.
Homework Assignment #5 should be submitted
on the Tycho system by 11:59 PM on Friday, April 30.
As of 9:10 AM today, 152/176 Physics 121B
students have registered their clickers. If you have
not already done so, register your clicker at:
http://courses.washington.edu/p121bs10/clicker.htm
Isolation
Emech Eth Esys Wext
April 28, 2010 Physics 121B - Lecture 12 29/28
Problem Solving Strategy
Picture: Determe that the net external force Fext (or Fext x) on
the system is negligible for some time interval. (If the net force is
NOT determined to be negligible, do not proceed.)
Solve:
1. Draw a sketch showing the system before and after the time
interval. Include coordinate axes and label the initial and final
velocity vectors.
2. Equate the initial momentum to the final momentum and express
this as a vector equation (or one or more scalar equations involving x,
y, and z components.)
3. Substitute the given information into the equation(s) and solve
for the quantity or quantities of interest.
Check: Make sure you include any minus signs that accompany
velocity components, because momentum can have either sign.
April 28, 2010 Physics 121B - Lecture 12 30/28
Example: Moving a Sled
A sled is coasting on a horizontal snow-covered surface with an initial
speed of 4.0 m/s. If the coefficient of friction between the sled and the
snow is 0.14, how far will the sled travel before coming to rest?
Wext Emech Etherm (U K ) f k srel ; f k k Fn k mg
0 0 K k mgsrel
srel
1
2
mv 2
v2 4.0 m/s 5.8 m
k mg 2k g 2 0.14 9.81 m/s
2
April 28, 2010 Physics 121B - Lecture 12 31/28
Example:
A Skateboard Workout
A 40.0-kg skateboarder on a
3.00-kg board is training with two
5.00-kg weights. Beginning from rest,
she throws the weights horizontally,
one at a time, from her board
The speed of each weight is 7.00 m/s relative to her
after it is thrown. Assume the board rolls without friction.
(a) How fast is she moving in the opposite
direction after throwing the first weight?
(b) After throwing the second weight?
April 28, 2010 Physics 121B - Lecture 12 32/28
