exam 3 information - physics and astronomy at...

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

Chapter 13

Gravitation

11/25/2014 2 Physics 218

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

Figure 13.18

The First Law

Figure 13.19

Kepler’s Second Law

m

Lvrrv

dt

dr

dt

dA

221

21

2

21

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 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


4

7

2411 4

4

24

32

32

X

mXr

XGmTr

E

Chapter 14

Periodic Motion

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


Energy diagrams for SHM


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


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.


I

tCtBtAt

kdt

dI

)cos()cos()sin()(

,2

2


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.


Binomial Expansion

32

!3

)2)(1(

!2

)1(11

1ufor Expansion Binomial

unnn

unn

nuu)( n


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.)


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


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.)


Physical Pendulum

I

mgd

tCtBtAt

dt

dImgd

mgddt

dIImgddmgz

)cos()cos()sin()(

)(

)( ,)(sin)(

2

2

2

2


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


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.


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.)


Tacoma Narrows Bridge collapse

https://www.youtube.com/watch?v=xox9BVSu7Ok

An Example of driven harmonic motion with resonance…….

https://www.youtube.com/watch?v=xox9BVSu7Ok

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