# Lab M4: The Torsional Pendulum and Moment of Inertia

Size: px
Start display at page:

Transcription

1 M4.1 Lab M4: The Tosional Pendulum and Moment of netia ntoduction A tosional pendulum, o tosional oscillato, consists of a disk-like mass suspended fom a thin od o wie. When the mass is twisted about the axis of the wie, the wie exets a toque on the mass, tending to otate it back to its oiginal position. f twisted and eleased, the mass will oscillate back and foth, executing simple hamonic motion. This is the angula vesion of the bouncing mass hanging fom a sping. This lab will give you a bette gasp of the meaning of moment of inetia. n Pats 1-3 you will investigate the moment of inetia of disks and ings and the tosional sping constant of a od. n Pat 4, you will use you data to discove how the sping constant depends upon the diamete of the od. Conside a thin od with one end fixed in position and the othe end twisted though an angle about the od s axis. untwisted twisted This end fixed. τ f the angle is sufficiently small that the od is not plastically defomed, the od exets a toque τ popotional to the angle, (1) τ = κ (like F = k x fo a sping) whee κ (Geek lette kappa) is called the tosion constant. The minus sign indicates that the diection of the toque vecto τ is opposite to the angle vecto, so the toque tends to undo the twist. This is just like Hooke s Law fo spings. f a mass with moment of inetia is attached to the od, the toque will give the mass an angula acceleation α accoding to Fall 004

2 M4. () τ = α = d (like F = ma = m d x ) Combining (1) and () yields the equation of motion fo the tosional pendulum, d = κ (like m d x kx = ) (4) d κ = The solution to this diffeential equation is (5) () t = m cos( ωt + φ) (like xt () = xm cos( ωt + φ )) (6) ω = κ (like ω= k m ) whee m and ϕ ae constants which depend on the initial position and angula velocity of the mass. (The equation of motion is a second ode diffeential equation so its solution must have two constants of integation.) m is the maximum angle; oscillates between + m and - m. The constant ω is elated to the fequency f and the peiod T of the simple hamonic motion by (7) ω = πf = π T. So, fom (6) and (7), the peiod T is given by (8) T = π κ. t s always good to stae at a fomula like this until it makes physical sense. f the wie is vey stiff (lage κ) then the mass oscillates apidly and peiod T is shot. f the mass is lage so that is lage, then its oscillates slowly and T is long. The tosion constant can be detemined fom measuements of T if is known Fall 004

3 M4.3 (9) κ = 4π T. Convesely, if κ is known, the moment of inetia can be detemined fom measuements of T, T (10) =κ 4 π. Expeiment n this fist pat of this expeiment, you will compute the moments of inetia of a disk and an annulus fom measuements of the masses and dimensions. n the second pat, you will detemine the tosion constant of a od fom measuements of T, using a solid disk as a tosion mass. Fom the definition of moment of inetia, (11) = mi i it can be shown (you will show it!) that fo a solid disk of mass M disk and adius R with the axis of otation along the symmety axis, i, (1) M disk = 1 disk R. Thus disk can be computed fom measuements of M and R. Notice that the thickness of the disk does not ente into eq n (1). od annulus n the thid pat, you will add to the disk an annula ing with inne adius R 1 and oute adius R. f the mass of the annulus is M ing, its moment of inetia is (13) ing = 1 Ming ( R + ) R. 1 disk [This fomula can be undestood qualitatively as follows: Fom (11), the of a ing of negligible adial thickness is MR. The fomula (13) is the aveage of two ings, one of adius R 1 and one of adius R.] Fall 004

4 M4.4 Finally, in the last pat of the expeiment, you will detemine the tosion constant κ of seveal wies of diffeent adius and expeimentally discove the functional dependence of κ on. Pocedue Thee ae 4 tosion ods used in this expeiment. Thee of them ae made of steel and one is made of bass. To detemine which is the bass od, use a magnet it will attact the steel ods, but not the bass od. All the ods have the same length, but the thee steel ods have diffeent diametes. The thickest steel od and the bass od have the same diamete. Pat 1. Calculation of disk, ing Begin by weighing the solid disk and the annula ing. They ae quite heavy (a few kilogams) so use the 6000-gam-capacity scales only. [Do not attempt to weigh them on the smalle capacity scales! They ae heavy enough to damage those scales.] With a mete stick, measue the adius R of the disk and R 1, R of the annulus. (To get the adius, measue the diamete and divide by.) Then compute disk and ing. Fo this lab, it is pefeable to use cgs units(centimete-gam-second), athe than mks units, because the values of in this lab ae vey small in mks units. As usual, ecod you computed values fo with the coect numbe of significant figues. The disk has a small cylindical potusion at its hub which is needed to attach the tosion od. n one of the pe-lab questions, you will compute the moment of inetia of this little hub. s its contibution to the total moment of inetia of the disk significant o negligible? Pat. Detemination of κ of thinnest od Choose the thinnest od and use it to suspend the solid disk fom the backet on the wall as shown on the diagam below. Note that the ends of the od have indentations whee the set scews on the backet and the disk should seat. Tighten the set scews caefully so you can feel them seat popely. To pevent the disk fom wobbling when it is oscillating, the axis of the disk can be kept fixed with a centeing post, consisting of a pointed od that spins feely on a mount. Place the centeing post exactly below the cente of the disk and aise the post so that its pointed end is seated in the hole in the cente of the disk. Thee ae two ways to measue the peiod of the oscillation. You can use a photogate time in PEROD mode and tape a piece of pape to the disk so that it inteupts the photogate. Use the eset switch to quickly make seveal measuements and estimate the best value of T. An altenative way to measue the peiod is to use a stop watch and time the inteval fo seveal complete oscillations. f the time fo N complete peiods (N 0 o moe) is T total, then the peiod is T = T total. [DO NOT measue one peiod twenty N times! Measue the time fo twenty peiods once.] By measuing the time fo seveal peiods, the uncetainty in T, due to you eaction time, is educed by a facto of N. Fall 004

5 M4.5 δttotal = human eaction time 0.1 sec. δt = δttotal N Fo this pat of the lab, measue the peiod both with the photogate time and with the stopwatch and compae the two esults. Fo subsequent measuements of T, you can choose one o the othe. Now, using you computed disk and you measued T, compute the tosion constant κ of the thin od. wall backet set scew od pointed set scew seated in indentation solid disk centeing post set scew pape slip low-fiction pivot photogate time Pat 3. Measuement of ing With both the disk and the annulus suspended fom the thin od, measue the new peiod T. Fom you measuements of the peiod with and without the annulus and you computed disk, detemine ing, as shown below, and compae with you ealie computed value of ing. Fall 004

6 M4.6 Td = peiod with disk only T d = peiod with disk + ing d d + T κ = 4π = 4 d Td π ( ) Td Td = d + d = 1 + d T d = d Td 1 Pat 4. Dependence of κ on adius of od. Using the pecision calipes available (the metal ones in the gay plastic case), detemine the adius of each of the fou ods. Using the disk alone (no annulus), measue the peiod T of the tosion oscillato with each of the thee thicke ods. Fom you measued T s and computed disk, compute the tosion constant κ fo each of the fou ods. (κ fo the thinnest od was aleady detemined in pat.) Fo a fixed length od and a fixed composition, the tosion constant κ vaies as some powe of the adius ; that is, κ=a n, whee A and n ae constants. You job is to detemine the exponent n. With the data fom the thee steel ods make a plot of κ vs. n whee n is some intege. Vay n until the gaph is a staight line though the oigin. Veify that you have the coect powe n by computing the atio κ fo the thee steel n ods. [f the gaph is eally a staight line though the oigin, then the atio should be a constant fo all thee ods.] On the same gaph, plot the point fo the bass od. You will have to ceate a second tace fo that single point, and, in the plot fomat window, set the tace type to points and symbol to box o diamond, etc. s bass stiffe than steel o vice-vesa? Fall 004

7 M4.7 PeLab Questions: 1. What is the definition of moment of inetia? (Wite down a geneal equation fo and explain what the symbols in the equation mean.) What ae the units of in the cgs system(cgs = centimete-gam-second)? What ae the units of in the MKS system? (MKS = mete-kilogam-second, also called the S system.). Use the definition of to show that the moment of inetia of a thin hoop of mass M, adius R, and negligible thickness is given by hoop = M R. 3. Use the definition of moment of inetia to show that MR integal fom of eq n (11), = ρ disk = 1. [Hint: Use the dv, whee ρ = M / V is the density of the disk.] 4. A solid disk of adius R and mass M is glued to a thin hoop of adius R and mass m (m M). The symmety axes of the disk and hoop ae aligned. What is total, the total moment of inetia of the disk + hoop about the axis of symmety? 5. The disk has a little hub at its cente (in the shape of an annulus) with a mass m = 70±1 gams and with inne and oute adii of 1 = 6.4 mm and = 16.8 mm. Use eq n (13) to compute the moment of inetia of the hub. (Caeful of you units! Ae you going to use kilogams o gams?, metes o centimetes o...?) 6. What ae the units of the tosion constant κ in the MKS system? n the cgs system? 7. What is the definition of toque? (Don t just give a vague, qualitative definition; give the pecise, technical definition, as given in physics text books. As usual, if you give an equation, explain what the tems in the equation mean.) What ae the units of toque in the MKS system? 8. Conside a tosion pendulum consisting of a solid disk of mass M and adius R, suspended fom a wie with tosion constant κ Wite an equation which shows how the peiod T depends on M, R, and κ What happens to the peiod T, if m, the amplitude of the angula motion, is deceased by a facto of two? 9. Conside two disks, labeled A and B, with disk A having two times the mass and thee times the adius of disk B (M A = M B, R A = 3R B ). Each disk is suspended fom a wie with same tosion constant κ, and the peiod of each (T A, T B ) is measued. What is the atio T A T? B 10. Conside a set of ods all of the same mateial and the same length, but with diffeing adii,. Assuming that you have the coect value of the exponent n, show with a sketch what a gaph of κ vs. n should look like. Fall 004

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

### Exam 3: Equation Summary

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Depatment of Physics Physics 8.1 TEAL Fall Tem 4 Momentum: p = mv, F t = p, Fext ave t= t f t= Exam 3: Equation Summay total = Impulse: I F( t ) = p Toque: τ = S S,P

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### Lab #7: Energy Conservation

Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### Displacement, Velocity And Acceleration

Displacement, Velocity And Acceleation Vectos and Scalas Position Vectos Displacement Speed and Velocity Acceleation Complete Motion Diagams Outline Scala vs. Vecto Scalas vs. vectos Scala : a eal numbe,

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

### 1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### Phys 2101 Gabriela González. cos. sin. sin

1 Phys 101 Gabiela González a m t t ma ma m m T α φ ω φ sin cos α τ α φ τ sin m m α τ I We know all of that aleady!! 3 The figue shows the massive shield doo at a neuton test facility at Lawence Livemoe

### Experiment MF Magnetic Force

Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

### Gravitation and Kepler s Laws Newton s Law of Universal Gravitation in vectorial. Gm 1 m 2. r 2

F Gm Gavitation and Keple s Laws Newton s Law of Univesal Gavitation in vectoial fom: F 12 21 Gm 1 m 2 12 2 ˆ 12 whee the hat (ˆ) denotes a unit vecto as usual. Gavity obeys the supeposition pinciple,

### F G r. Don't confuse G with g: "Big G" and "little g" are totally different things.

G-1 Gavity Newton's Univesal Law of Gavitation (fist stated by Newton): any two masses m 1 and m exet an attactive gavitational foce on each othe accoding to m m G 1 This applies to all masses, not just

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

### Lesson 7 Gauss s Law and Electric Fields

Lesson 7 Gauss s Law and Electic Fields Lawence B. Rees 7. You may make a single copy of this document fo pesonal use without witten pemission. 7. Intoduction While it is impotant to gain a solid conceptual

### Introduction to Fluid Mechanics

Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### TECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications

JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### Physics HSC Course Stage 6. Space. Part 1: Earth s gravitational field

Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Voltage ( = Electric Potential )

V-1 of 9 Voltage ( = lectic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage

### Model Question Paper Mathematics Class XII

Model Question Pape Mathematics Class XII Time Allowed : 3 hous Maks: 100 Ma: Geneal Instuctions (i) The question pape consists of thee pats A, B and C. Each question of each pat is compulsoy. (ii) Pat

### The force between electric charges. Comparing gravity and the interaction between charges. Coulomb s Law. Forces between two charges

The foce between electic chages Coulomb s Law Two chaged objects, of chage q and Q, sepaated by a distance, exet a foce on one anothe. The magnitude of this foce is given by: kqq Coulomb s Law: F whee

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### Multiple choice questions [60 points]

1 Multiple choice questions [60 points] Answe all o the ollowing questions. Read each question caeully. Fill the coect bubble on you scanton sheet. Each question has exactly one coect answe. All questions

### Symmetric polynomials and partitions Eugene Mukhin

Symmetic polynomials and patitions Eugene Mukhin. Symmetic polynomials.. Definition. We will conside polynomials in n vaiables x,..., x n and use the shotcut p(x) instead of p(x,..., x n ). A pemutation

### YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

### An Introduction to Omega

An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

### PHYSICS 111 HOMEWORK SOLUTION #13. May 1, 2013

PHYSICS 111 HOMEWORK SOLUTION #13 May 1, 2013 0.1 In intoductoy physics laboatoies, a typical Cavendish balance fo measuing the gavitational constant G uses lead sphees with masses of 2.10 kg and 21.0

### 10. Collisions. Before During After

10. Collisions Use conseation of momentum and enegy and the cente of mass to undestand collisions between two objects. Duing a collision, two o moe objects exet a foce on one anothe fo a shot time: -F(t)

### Motion Control Formulas

ems: A = acceleation ate {in/sec } C = caiage thust foce {oz} D = deceleation ate {in/sec } d = lead of scew {in/ev} e = lead scew efficiency ball scew 90% F = total fictional foce {oz} GR = gea atio J

### AP Physics Electromagnetic Wrap Up

AP Physics Electomagnetic Wap Up Hee ae the gloious equations fo this wondeful section. F qsin This is the equation fo the magnetic foce acting on a moing chaged paticle in a magnetic field. The angle

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### Multiple choice questions [70 points]

Multiple choice questions [70 points] Answe all of the following questions. Read each question caefull. Fill the coect bubble on ou scanton sheet. Each question has exactl one coect answe. All questions

### Semipartial (Part) and Partial Correlation

Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

### Problems of the 2 nd and 9 th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)

Poblems of the nd and 9 th Intenational Physics Olympiads (Budapest Hungay 968 and 976) Péte Vankó Institute of Physics Budapest Univesity of Technology and Economics Budapest Hungay Abstact Afte a shot

DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation

### 4a 4ab b 4 2 4 2 5 5 16 40 25. 5.6 10 6 (count number of places from first non-zero digit to

. Simplify: 0 4 ( 8) 0 64 ( 8) 0 ( 8) = (Ode of opeations fom left to ight: Paenthesis, Exponents, Multiplication, Division, Addition Subtaction). Simplify: (a 4) + (a ) (a+) = a 4 + a 0 a = a 7. Evaluate

### Quantity Formula Meaning of variables. 5 C 1 32 F 5 degrees Fahrenheit, 1 bh A 5 area, b 5 base, h 5 height. P 5 2l 1 2w

1.4 Rewite Fomulas and Equations Befoe You solved equations. Now You will ewite and evaluate fomulas and equations. Why? So you can apply geometic fomulas, as in Ex. 36. Key Vocabulay fomula solve fo a

### 7 Circular Motion. 7-1 Centripetal Acceleration and Force. Period, Frequency, and Speed. Vocabulary

7 Cicula Motion 7-1 Centipetal Acceleation and Foce Peiod, Fequency, and Speed Vocabulay Vocabulay Peiod: he time it takes fo one full otation o evolution of an object. Fequency: he numbe of otations o

### Chapter 30: Magnetic Fields Due to Currents

d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.

### Solution Derivations for Capa #8

Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

### (a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of

Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the

### Strength Analysis and Optimization Design about the key parts of the Robot

Intenational Jounal of Reseach in Engineeing and Science (IJRES) ISSN (Online): 2320-9364, ISSN (Pint): 2320-9356 www.ijes.og Volume 3 Issue 3 ǁ Mach 2015 ǁ PP.25-29 Stength Analysis and Optimization Design

### Skills Needed for Success in Calculus 1

Skills Needed fo Success in Calculus Thee is much appehension fom students taking Calculus. It seems that fo man people, "Calculus" is snonmous with "difficult." Howeve, an teache of Calculus will tell

### Thank you for participating in Teach It First!

Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident

### Do Vibrations Make Sound?

Do Vibations Make Sound? Gade 1: Sound Pobe Aligned with National Standads oveview Students will lean about sound and vibations. This activity will allow students to see and hea how vibations do in fact

### 2. Orbital dynamics and tides

2. Obital dynamics and tides 2.1 The two-body poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### 4.1 - Trigonometric Functions of Acute Angles

4.1 - Tigonometic Functions of cute ngles a is a half-line that begins at a point and etends indefinitel in some diection. Two as that shae a common endpoint (o vete) fom an angle. If we designate one

### SELF-INDUCTANCE AND INDUCTORS

MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................

### TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION

MISN-0-34 TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION shaft TORQUE AND ANGULAR MOMENTUM IN CIRCULAR MOTION by Kiby Mogan, Chalotte, Michigan 1. Intoduction..............................................

### 2. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES

. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an

### NUCLEAR MAGNETIC RESONANCE

19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic

### Mechanics 1: Work, Power and Kinetic Energy

Mechanics 1: Wok, Powe and Kinetic Eneg We fist intoduce the ideas of wok and powe. The notion of wok can be viewed as the bidge between Newton s second law, and eneg (which we have et to define and discuss).

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### 12.1. FÖRSTER RESONANCE ENERGY TRANSFER

ndei Tokmakoff, MIT epatment of Chemisty, 3/5/8 1-1 1.1. FÖRSTER RESONNCE ENERGY TRNSFER Föste esonance enegy tansfe (FR) efes to the nonadiative tansfe of an electonic excitation fom a dono molecule to

### Gravity. A. Law of Gravity. Gravity. Physics: Mechanics. A. The Law of Gravity. Dr. Bill Pezzaglia. B. Gravitational Field. C.

Physics: Mechanics 1 Gavity D. Bill Pezzaglia A. The Law of Gavity Gavity B. Gavitational Field C. Tides Updated: 01Jul09 A. Law of Gavity 3 1a. Invese Squae Law 4 1. Invese Squae Law. Newton s 4 th law

### Classical Mechanics (CM):

Classical Mechanics (CM): We ought to have some backgound to aeciate that QM eally does just use CM and makes one slight modification that then changes the natue of the oblem we need to solve but much

### Solutions for Physics 1301 Course Review (Problems 10 through 18)

Solutions fo Physics 1301 Couse Review (Poblems 10 though 18) 10) a) When the bicycle wheel comes into contact with the step, thee ae fou foces acting on it at that moment: its own weight, Mg ; the nomal

### AP Physics C. Oscillations/SHM Review Packet

AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete

### Oscillations: Mass on a Spring and Pendulums

Chapter 3 Oscillations: Mass on a Spring and Pendulums 3.1 Purpose 3.2 Introduction Galileo is said to have been sitting in church watching the large chandelier swinging to and fro when he decided that

### AP1 Oscillations. 1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false?

1. Which of the following statements about a spring-block oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The

### The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

### Fluids Lecture 15 Notes

Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

### Uniform Rectilinear Motion

Engineeing Mechanics : Dynamics Unifom Rectilinea Motion Fo paticle in unifom ectilinea motion, the acceleation is zeo and the elocity is constant. d d t constant t t 11-1 Engineeing Mechanics : Dynamics

### Prelab Exercises: Hooke's Law and the Behavior of Springs

59 Prelab Exercises: Hooke's Law and the Behavior of Springs Study the description of the experiment that follows and answer the following questions.. (3 marks) Explain why a mass suspended vertically

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Supplementary Material for EpiDiff

Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module

### Experiment 9. The Pendulum

Experiment 9 The Pendulum 9.1 Objectives Investigate the functional dependence of the period (τ) 1 of a pendulum on its length (L), the mass of its bob (m), and the starting angle (θ 0 ). Use a pendulum

### Simple Harmonic Motion

Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

### YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE

YIELD TO MATURITY ACCRUED INTEREST QUOTED PRICE INVOICE PRICE Septembe 1999 Quoted Rate Teasuy Bills [Called Banke's Discount Rate] d = [ P 1 - P 1 P 0 ] * 360 [ N ] d = Bankes discount yield P 1 = face

### The Detection of Obstacles Using Features by the Horizon View Camera

The Detection of Obstacles Using Featues b the Hoizon View Camea Aami Iwata, Kunihito Kato, Kazuhiko Yamamoto Depatment of Infomation Science, Facult of Engineeing, Gifu Univesit aa@am.info.gifu-u.ac.jp

### CF CFS. MINICAM Series. The world-smallest MINICAM with a stud diameter of 2 mm is newly introduced! CAT-57115B

poducts ae avaiable fom: MARYLAD METRIS phones: ()- (00)- faxes: ()-12 (00)72-2 P.O.ox 21 Owings Mills, MD 21117 USA URL: http://mdmetic.com E-mail: sales@mdmetic.com MIIAM Seies FFS AT-711 The wold-smallest

### Practice Exam Three Solutions

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,

### Physics 1120: Simple Harmonic Motion Solutions

Questions: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Physics 1120: Simple Harmonic Motion Solutions 1. A 1.75 kg particle moves as function of time as follows: x = 4cos(1.33t+π/5) where distance is measured

### Chapter 19: Electric Charges, Forces, and Fields ( ) ( 6 )( 6

Chapte 9 lectic Chages, Foces, an Fiels 6 9. One in a million (0 ) ogen molecules in a containe has lost an electon. We assume that the lost electons have been emove fom the gas altogethe. Fin the numbe

### Charges, Coulomb s Law, and Electric Fields

Q&E -1 Chages, Coulomb s Law, and Electic ields Some expeimental facts: Expeimental fact 1: Electic chage comes in two types, which we call (+) and ( ). An atom consists of a heavy (+) chaged nucleus suounded

### An Epidemic Model of Mobile Phone Virus

An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity