Exam 3 information Exam 3 will be held Monday, 11/24 at 7:30 pm in TBA.
The exam is design to take one hour and you will be given 75 minutes to complete your work. The Exam will cover material in chapters 8-11.
Some practice exam problems are available on the class web site.
You will not need a scantron. The exam book and formula sheet will be provided to you.
You will need your TAMU ID and remember your section number and lecturers name.
You will be allowed to use a calculator (simple), but if you have a programmable calculator, you must have cleared its memory BEFORE coming to the exam.
Cell phones should be turned off and stored for the exam period. Any use of cell phones during the exam is prohibited and if used will be considered an Honor Code violation.
11/25/2014 Physics 218 1
Learning Goals
How to calculate the gravitational forces that two bodies exert on each other.
How to relate the weight of an object to the general expression for gravitational force.
How to use and interpret the generalized expression for gravitational potential energy.
How to relate the speed, orbital period, and mechanical energy of a satellite in circular orbit.
The laws that describe the motions of the planets and how to work with these laws (Kepler’s Laws)
11/25/2014 Physics 218 3
Kepler’s Laws
Each Planet moves in an elliptical orbit with the sun at one focus of the ellipse.
A line from the sun to a given planet sweeps out equal areas in equal times.
The periods of the planets are proportional to the 3/2 power of the major axis lengths of their orbits.
11/25/2014 Physics 218 4
Kepler’s Third Law
11/25/2014 Physics 218 7
SGm
aT
23
2
a. length, axismajor -semi with ther, radius, the
replace weorbits lellipiticafor orbit,circular a
forn calculatioearlier our from Following
Calculation the period for our Moon’s Orbit
11/25/2014 Physics 218 8
dsX
X
X
Gm
r
v
rT
E
4.271036.2
)10 X 5.97(1067.6
)1084.3(222
m10 X 3.84 orbit moon of radius
kg10 X 5.97 earth theof mass
kg10 X 7.35 moon theof mass
6
2411
8
8
24
22
23
23
Radius of the Earth’s orbit
11/25/2014 Physics 218 9
mXr
XGmTr
S
11
3011 7
7
24
30
1049.1
)2
)10 X 1.99(1067.610 X 3.16(
2
s10 X 3.16 orbit our of period
kg10 X 5.97 earth theof mass
kg10 X 1.99 sun theof mass
32
32
Geosynchronous Satellites
Period of the orbit =
same as the orbital rotation of the earth
Satellite will the appear to stay fixed in the sky above a point on the earth.
11/25/2014 Physics 218 10
11/25/2014 Physics 218 11
earth theofcenter thefrom miles 1059.2
or 1021.4
)2
)10 X 5.97(1067.610 X 8.64(
2
s10 X 8.64 orbit theof period
kg10 X 5.97 earth theof mass
4
7
2411 4
4
24
32
32
X
mXr
XGmTr
E
Goals for Chapter 14
• To describe oscillations in terms of amplitude, period, frequency and angular frequency
• To do calculations with simple harmonic motion
• To analyze simple harmonic motion using energy
• To apply the ideas of simple harmonic motion to different physical situations
• To analyze the motion of a simple pendulum
• To examine the characteristics of a physical pendulum
• To explore how oscillations die out
• To learn how a driving force can cause resonance
Introduction
• Why do dogs walk faster than humans? Does it have
anything to do with the characteristics of their legs?
• Many kinds of motion (such as a pendulum, musical
vibrations, and pistons in car engines) repeat themselves.
We call such behavior periodic motion or oscillation.
What causes periodic motion?
• If a body attached to a spring is displaced from its equilibrium position, the spring exerts a restoring force on it, which tends to restore the object to the equilibrium position. This force causes oscillation of the system, or periodic motion.
• Figure 14.2 at the right illustrates the restoring force Fx.
Characteristics of periodic motion
• The amplitude, A, is the maximum magnitude of displacement from equilibrium.
• The period, T, is the time for one cycle.
• The frequency, f, is the number of cycles per unit time.
• The angular frequency, , is 2π times the frequency: = 2πf.
• The frequency and period are reciprocals of each other: f = 1/T and T = 1/f.
Simple harmonic motion (SHM)
• When the restoring force is directly proportional to the displacement from equilibrium, the resulting motion is called simple harmonic motion (SHM).
• An ideal spring obeys Hooke’s law, so the restoring force is Fx = –kx, which
results in simple harmonic motion.
Copyright © 2012 Pearson Education Inc.
SHM differential equation: F=ma
)sin()cos()sin()(
,
2
2
2
2
tCtBtAtx
dt
xdmkx
dt
xdmmaFkxF x
Copyright © 2012 Pearson Education Inc.
Energy in SHM
• The total mechanical energy E = K + U is conserved in SHM:
E = 1/2 mvx2 + 1/2 kx2 = 1/2 kA2 = constant
Copyright © 2012 Pearson Education Inc.
Angular SHM
• A coil spring (see Figure 14.19 below) exerts a restoring torque
z = –, where is called the torsion constant of the spring.
• The result is angular simple harmonic motion.
Copyright © 2012 Pearson Education Inc.
Vibrations of molecules
• Figure 14.20 shows two atoms having centers a distance r apart,
with the equilibrium point at r = R0.
• If they are displaced a small distance x from equilibrium, the
restoring force is Fr = –(72U0/R02)x, so k = 72U0/R0
2 and the
motion is SHM.
Copyright © 2012 Pearson Education Inc.
Binomial Expansion
32
!3
)2)(1(
!2
)1(11
1ufor Expansion Binomial
unnn
unn
nuu)( n
Copyright © 2012 Pearson Education Inc.
The simple pendulum
• A simple pendulum
consists of a point mass
(the bob) suspended by a
massless, unstretchable
string.
• If the pendulum swings
with a small amplitude
with the vertical, its
motion is simple
harmonic. (See Figure
14.21 at the right.)
Copyright © 2012 Pearson Education Inc.
Simple Pendulum
L
g
m
Lmg
m
k
tCtBtAtx
dt
xdmx
L
mg
L
xmg
dt
xdmmaFmgF x
)cos()cos()sin()(
,sin
2
2
2
2
Copyright © 2012 Pearson Education Inc.
The physical pendulum
• A physical pendulum is
any real pendulum that
uses an extended body
instead of a point-mass
bob.
• For small amplitudes, its
motion is simple harmonic.
(See Figure 14.23 at the
right.)
Copyright © 2012 Pearson Education Inc.
Physical Pendulum
I
mgd
tCtBtAt
dt
dImgd
mgddt
dIImgddmgz
)cos()cos()sin()(
)(
)( ,)(sin)(
2
2
2
2
Copyright © 2012 Pearson Education Inc.
Harmonic oscillation with damping
2
2
)(
2
2
2
2
4'
)'cos()(
0
,
2
m
b
m
k
tAetx
dt
xdm
dt
dxbkx
dt
dxbkx
dt
xdmmaFbvkxF
t
xxx
mb
Copyright © 2012 Pearson Education Inc.
Damped oscillations
• Real-world systems have
some dissipative forces that
decrease the amplitude.
• The decrease in amplitude is
called damping and the
motion is called damped
oscillation.
• Figure 14.26 at the right
illustrates an oscillator with a
small amount of damping.
• The mechanical energy of a
damped oscillator decreases
continuously.
Copyright © 2012 Pearson Education Inc.
Forced oscillations and resonance
• A forced oscillation occurs if a driving force acts on an oscillator.
• Resonance occurs if the frequency of the driving force is near the
natural frequency of the system. (See Figure 14.28 below.)
Copyright © 2012 Pearson Education Inc.
Tacoma Narrows Bridge collapse
https://www.youtube.com/watch?v=xox9BVSu7Ok
An Example of driven harmonic motion with resonance…….