Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal MassSpring System


 Ambrose Stone
 4 years ago
 Views:
Transcription
1 Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal MassSpring System A Hrizntal MassSpring System is where a mass is attached t a spring, riented hrizntally, and then placed n a frictinless surface. When the mass is at rest, the spring will be at the equilibrium r rest psitin. This is the vertical line shwn in the diagram belw. We can then pull the mass t the right and hld it there. Let s call this psitin #1. Ntice at psitin #1 the frce f the spring, F s1, is t the left because the displacement f the spring frm equilibrium psitin, x 1, is t the right. We knw this because f Hke s Law, F s = k x. When we let g f the mass the spring frce will accelerate the mass t the left, the mass will pass thrugh the equilibrium psitin, which we can call psitin #2. After passing thrugh rest psitin, the mass will pause t the left f the equilibrium psitin. Let s call this psitin #3. Ntice psitin #3 is the same distance frm psitin #2 as psitin #1. At psitin #2, the displacement frm equilibrium psitin, x 2, is zer. Therefre, accrding t Hke s Law, the frce f the spring, F s2, is als equal t zer. At psitin #3, the displacement frm rest psitin, x 3, is t the left. Therefre, accrding t Hke s Law, the frce f the spring, F s3, is t the right. In the absence f frictin, the spring will cntinue t mve back and frth thrugh these psitins like this: 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1 and it will never stp. This is called Simple Harmnic Mtin. There are tw requirements fr the frce that causes simple harmnic mtin: 1) It must be a Restring Frce: A frce that is always twards equilibrium. a. The spring frce is a restring frce because it is always directed tward rest psitin and therefre will always accelerate the mass tward equilibrium psitin. 2) The frce must be prprtinal t displacement frm equilibrium psitin. a. Accrding t Hke s Law, F s = k x, the spring frce is prprtinal t displacement frm equilibrium psitin. In ther wrds, the larger the displacement frm equilibrium psitin, the larger the spring frce Lecture Ntes  Simple Harmnic Mtin Intrductin via a Hrizntal MassSpring System.dcx page 1 f 1
2 Flipping Physics Lecture Ntes: Simple Harmnic Mtin  Frce, Acceleratin, and Velcity at 3 Psitins We previusly defined three lcatins fr an bject in simple harmnic mtin. Psitins 1 and 3 are at the maximum displacement frm and n either side f equilibrium psitin. Psitin 2 is when the mass is at rest psitin. Nw let s determine sme basics abut the magnitudes f the velcities and acceleratins at thse psitins. Ntice that at psitins 1 and 3, the velcity f the mass changes directins. This means the velcities at 1 and 3 are zer. This is just like the velcity at the tp is zer fr an bject in free fall. This means the magnitude f the velcity halfway in between thse tw psitins, in ther wrds at psitin 2, will have a maximum value. At psitins 1 and 3, displacement frm equilibrium psitin, x, will have a maximum magnitude. That means, accrding t Hke s Law, psitins 1 and 3. If we sum the frces in the x directin, will als have its maximum magnitude at psitins 1 and 3. F s = k x, the spring frce will als have it s maximum magnitude at F x = F s = ma x, we can see the acceleratin Please realize that the spring frce changes as a functin f psitin, therefre, the net frce in the x directin changes as a functin f psitin, therefre the acceleratin f the mass changes as a functin f psitin, therefre simple harmnic mtin is nt unifrmly accelerated mtin. In simple harmnic mtin the acceleratin is nt cnstant, therefre, yu cannt use the unifrmly accelerated mtin equatins. F x = F s = ma x kx = ma x x cnstant therefre a cnstant. Yes, I am ignring whether the spring frce is t the left r right in this equatin. It des nt matter. I am simply shwing that simple harmnic mtin is nt unifrmly accelerated mtin Lecture Ntes  Simple Harmnic Mtin  Frce, Acceleratin, and Velcity at 3 Psitins.dcx page 1 f 1
3 Flipping Physics Lecture Ntes: Hrizntal vs. Vertical MassSpring System Hrizntal and vertical massspring systems are bth in simple harmnic mtin. A vertical mass spring system scillates arund the pint where the dwnward frce f gravity and the upward spring frce cancel ne anther ut. The restring frce fr a hrizntal massspring system is just the spring frce, because that is the net frce in the xdirectin. The restring frce fr a vertical massspring system is the net frce in the ydirectin which equals the spring frce minus the frce f gravity Lecture Ntes  Hrizntal vs Vertical MassSpring System.dcx page 1 f 1
4 Flipping Physics Lecture Ntes: When is a Pendulum in Simple Harmnic Mtin? Massspring systems and pendulums are bth in simple harmnic mtin. Bth scillate arund an equilibrium psitin and have a restring frce pinted twards the equilibrium psitin that increases prprtinally with displacement frm the equilibrium r rest psitin. The displacement frm equilibrium psitin fr a pendulum is an angular displacement. Units are in degrees r radians. Symbl is theta, θ. Maximum displacement frm equilibrium psitin is still Amplitude, A. The restring frce fr a pendulum is the frce f gravity tangential t the path f the pendulum. This frce is: Prprtinal t displacement frm equilibrium psitin and Directed tward equilibrium psitin. Actually, the frce f gravity tangential is nly cnsidered t be directed tward equilibrium r rest psitin fr small angles. Typically I cnsider this t be less than 15, hwever, sme surces require the angle t be less than 10. It depends n hw much errr yu are willing t allw. The larger the angle, the larger the errr. This is because f the small angle apprximatin Lecture Ntes  When is a Pendulum in Simple Harmnic Mtin.dcx page 1 f 1
5 Flipping Physics Lecture Ntes: Demnstrating What Changes the Perid f Simple Harmnic Mtin The perid f simple harmnic mtin is the time it takes t cmplete ne full cycle. The units fr perid are typically secnds r secnds per cycle, hwever, they culd als be in minutes, hurs, days, frtnights, decades, millenniums, etc. The symbl fr perid is T. Using ur previusly defined pstins f 1 and 3 where the bject is at its maximum displacement frm equilibirum psitin and psitin 2 is at the equilibrium psitin, recall that the simple harmnic mtin pattern is 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, 2, 1, 2, 3, One full cylce in terms f psitin culd be: 1, 2, 3, 2, 1 r 3, 2, 1, 2, 3 r 2, 1, 2, 3, 2 r 2, 3, 2, 1, 2 r starting and ending smewhere between ne f the psitins as lng as: 1. the bject starts and ends at the same lcatin 2. the bject is mving in the same directin at the end as at the start The equatins fr the perid f simple harmnic mtin are: Fr a massspring system: T = 2π m k m is the mass in the massspring system. k is the spring cnstant f the spring. Fr a pendulum: T = 2π L g L is the pendulum length which is the distance frm the center f suspensin t the center f mass f the pendulum bb. Center f suspensin is the tp, fixed end f the pendulum. Pendulum bb is the mass at the bttm f the pendulum. g is the acceleratin due t gravity in which the pendulum is lcated. On Earth that wuld be 9.81 m/s 2. This is called a simple pendulum. Meaning the rd/string is f negligible mass therefre the center f mass f a simple pendulum is the center f mass f the pendulum bb. What affects the perid f a pendulum and a massspring system? Amplitude is nt in either perid equatin. Amplitude des nt affect the perid f a pendulum r the perid f a massspring system. Acceleratin due t gravity is nt in the perid equatin fr a massspring system. g des nt affect the perid f a massspring system. Mass is nt in the perid equatin fr a pendulum. The mass f the pendulum bb des nt affect the perid f a pendulum. Increasing the mass in a massspring system increases its perid. m T fr a massspring system. Increasing the spring cnstant in a massspring system decreases its perid. k T fr a massspring system. Increasing the pendulum length increases its perid. L T fr a pendulum. Increasing the acceleratin due t gravity decreases the perid f a pendulum. g T fr a pendulum Lecture Ntes  Demnstrating What Changes the Perid f Simple Harmnic Mtin.dcx page 1 f 1
6 Flipping Physics Lecture Ntes: Triple the Mass in a MassSpring System. Hw des Perid Change? Example: If the mass in a massspring system is tripled, hw des the perid change? Knwns: m 2 = 3m 1 and T 2 =?T 1 We knw the equatin fr the perid f a massspring system: T = 2π m k S the perid f the riginal massspring system is: T 1 = 2π m 1 k And the perid f the new massspring system with three times the mass is: T 2 = 2π m 2 k = 2π 3m 1 k = 2π m 1 k 3 = T 1 3 T 2 = T 1 3 S tripling the mass, increases the perid by the square rt f 3: T 2 = T 1 3 Demnstratin: T 1 =1.67s & T 2 = T 1 3 = 1.67) 3 = s Hwever, the bserved value fr the perid with three times the mass is 2.83 secnds. E r = O A A 100 = I think we can cnfidently say, The physics wrks 100 = % 0256 Lecture Ntes  Triple the Mass in a MassSpring System. Hw des Perid Change.dcx page 1 f 1
7 Flipping Physics Lecture Ntes: Frequency vs. Perid in Simple Harmnic Mtin We have already defined the perid, T, f simple harmnic mtin as the time it takes fr ne full cycle r scillatin. Frequency, f, is defined as the number f cycles r scillatins per secnd. Hpefully yu recgnize then that frequency and perid are inverses f ne anther. The units fr frequency are cycles secnd T = 1 f which we call hertz Hz) after the 19th century German physicist Heinrich Hertz ) wh was the first t give cnclusive prf f the existence f electrmagnetic waves which were therized by James Clerk Maxwell's electrmagnetic thery f light which we will learn abut later. Fr example, if we have a vertical massspring system with a perid f 0.77 secnds, the frequency f that massspring system is: f = 1 T = 1 cycles = r1.3 Hz 0.77 secnd Which means the massspring system shuld g thrugh 1.3 scillatins every secnd. Anther example, if we have a pendulum which ges thrugh 15 cycles in 11 secnds, then the frequency f that pendulum is: 15 cycles f = = Hz 11 secnds Which we can cmpare t the perid f the pendulum: T = 1 f = 1 = sec Lecture Ntes  Frequency vs Perid in Simple Harmnic Mtin.dcx page 1 f 1
8 Flipping Physics Lecture Ntes: Cmparing Simple Harmnic Mtin t Circular Mtin Circular mtin, when viewed frm the side, is simple harmnic mtin. It is difficult t see n paper, which is why I make the vides. If yu just lk at the x r y directin mtin f the yellw marker cap n tp f the rtating turntable, the cap is mving in simple harmnic mtin. Here is a picture, hwever, really, yu shuld g watch the vide: If yu lk at the lcatin f the cap in ne dimensin as a functin f time, then yu end up with a sine/csine curve. Again, g watch the vide: 0260 Lecture Ntes  Cmparing Simple Harmnic Mtin t Circular Mtin.dcx page 1 f 1
9 Flipping Physics Lecture Ntes: Simple Harmnic Mtin Psitin Equatin Derivatin Circular mtin, when viewed frm the side, is simple harmnic mtin. We can use this fact t derive an equatin fr the psitin f an bject in simple harmnic mtin. r is radius f the circular mtin. x is the psitin f the cap in the xdirectin, assuming the center f the turntable is the center f ur crdinate system. θ is the angular displacement f the cap frm an initial psitin where the cap was at its extreme psitin t the right. A x = x = r cs θ & H r Δθ θ f θ i θ 0 θ ω= = = = θ = ωt Δt tf ti t 0 t cs θ = Assuming we let θ i = 0; ti = 0; θ f = θ ; tf = t Δθ 2π 1 = = 2π f remember: f = Δt T T Therefre θ = ω t = 2π ft and x = r cs θ = r cs ω t = r cs 2π ft ) x = r cs θ = r cs ω t & ω = ) ) Identifying that the maximum displacement frm equilibrium psitin is the amplitude, A, is als r in the psitin equatin. Therefre: ) x t = Acs 2π ft ) Ntice this is an equatin which can be used t describe an bject scillating in simple harmnic mtin. The equatin culd als be: ) ) ) ) x t = Asin 2π ft r even x t = Acs 2π ft + φ. φ is the phase cnstant and phase shifts the sine and csine wave alng the hrizntal axis. Realize φ is nt in the AP Physics 1 curriculum, hwever, it is very useful. π Fr example: x t = Acs 2π ft = Asin 2π ft + 2 ) ) Sme useful pints: θ was in radians in ur derivatin, therefre angles in the equatins fr simple harmnic mtin are in radians and yur calculatr needs t be in radians when using these equatins. ω is angular frequency which is nt the same as frequency, f. Yes, ω= T= Δθ 2π = = 2π f Δt T 2π, is n the AP Physics equatin sheets, hwever, yu are much better served t ω remember and understand its derivatin Lecture Ntes  Simple Harmnic Mtin  Psitin Equatin Derivatin.dcx page 1 f 1
10 Flipping Physics Lecture Ntes: Simple Harmnic Mtin  Velcity and Acceleratin Equatin Derivatins Previusly we derived the equatin n the AP Physics 1 equatin sheet fr an bject mving in simple harmnic mtin: x t ) = Acs 2π ft). In rder t derive the equatins fr velcity and acceleratin, let s get psitin in terms f angular frequency: f = 1 T & ω = Δθ Δt = 2π T = 2π f therefre x t ) = Acs 2π ft) = Acs ωt). ) = Acs ωt +φ). Let s add the phase cnstant that shifts the wave alng the hrizntal axis: x t Velcity is the derivative f psitin as a functin f time. I knw sme f yu might nt have taken calculus yet and might nt understand derivatives. Realize yu need derivatives t derive velcity and acceleratin simple harmnic mtin equatins. Sme f this math might g ver yur heads, hwever, it is still useful t get sme expsure t. J Uses chain rule. ) ) ) ) v = dx dt = d Acs ωt +φ dt = A d cs ωt +φ dt = A sin ωt +φ d ωt +φ dt v t) = Asin ωt +φ)ω v t) = Aω sin ωt +φ) Nte: Because 1 sinθ 1 v max = Aω The derivatin f acceleratin is very similar: Again, uses chain rule. ) ) ) ) a = dv dt = d Aω sin ωt +φ dt = Aω d sin ωt +φ dt = Aω cs ωt +φ d ωt +φ dt ) a = Aω cs ωt +φ ω ) a t) = Aω 2 cs ωt +φ) Again nte: Because 1 csθ 1 a max = Aω 2 Remember: Because the derivatin f these equatins requires theta t be in radians, all angles in these equatins need t be in radians and yur calculatr needs t be in radian mde when using equatins fr psitin, velcity, and acceleratin as a functin f time in simple harmnic mtin Lecture Ntes  Simple Harmnic Mtin  Velcity and Acceleratin Equatin Derivatins.dcx page 1 f 1
11 Flipping Physics Lecture Ntes: Simple Harmnic Mtin Graphs f Psitin, Velcity and Acceleratin Previusly we derived equatins fr psitin, velcity, and acceleratin f an bject in simple harmnic mtin: ) ) ) ) ) x t = Acs ω t + φ ; v t = Aω sin ω t + φ ; a t = Aω 2 cs ω t + φ Angular frequency, ω, derivatin: f= ) 1 Δθ 2π &ω = = = 2π f T Δt T Fr ur graphs, we are ging t assume the phase cnstant, ɸ, is zer. In ther wrds the graphs will nt be phase shifted n the hrizntal axis Lecture Ntes  Simple Harmnic Mtin  Graphs f Psitin, Velcity, and Acceleratin.dcx page 1 f 2
12 dx v= velcity = slpe f psitin vs. time dt dv a= acceleratin = slpe f velcity vs. time dt 0263 Lecture Ntes  Simple Harmnic Mtin  Graphs f Psitin, Velcity, and Acceleratin.dcx page 2 f 2
13 Flipping Physics Lecture Ntes: Simple Harmnic Mtin Graphs f Mechanical Energies As previusly determined, an bject in simple harmnic mtin has the fllwing values: Psitins) x Velcity Acceleratin 2 1&3 Maximum magnitude, A 0 Maximum magnitude, Aω 2 0 Maximum magnitude, Aω Lecture Ntes  Simple Harmnic Mtin  Graphs f Mechanical Energies.dcx page 1 f 2
14 In the cmplete absence f frictin, mechanical energy is cnserved: ME ttal = 1 2 m v max ) 2 = 1 2 ka2 Hwever, the reality is that sme energy will be cnverted t internal energy f the spring via wrk dne by frictin: 0264 Lecture Ntes  Simple Harmnic Mtin  Graphs f Mechanical Energies.dcx page 2 f 2
15 Flipping Physics Lecture Ntes: Demnstrating Psitin, Velcity, and Acceleratin f a MassSpring System The basic equatins f simple harmnic mtin are: ) ) ) ) ) y t = Acs ω t + φ ; v t = Aω sin ω t + φ ; a t = Aω 2 cs ω t + φ Fr ur demnstratins we are ging t assume the phase cnstant, phase shift in ur demnstratin. ) φ, is zer. This means there is n The psitin, velcity, and acceleratin as a functin f time graphs: Determining the perid using the time fr tw cycles: 2T = 2.72sec T = 2.72 = 1.36sec Lecture Ntes  Demnstrating Psitin, Velcity, and Acceleratin f a MassSpring System.dcx page 1 f 2
16 Determining the spring cnstant using the perid and the mass: ) 4π m 4π 2 m N 2 2 m T = 2π T = 4π k = k= = k m T 1.36 k m = 305g 1kg = 0.305kg 1000g Just s yu knw, the 305 grams includes the mass f the mass hanging. The bestfit line equatin fr psitin as a functin f time: Determining perid using angular frequency, ω: ω= Δθ 2π 2π 2π = T = = = sec Δt T ω Lecture Ntes  Demnstrating Psitin, Velcity, and Acceleratin f a MassSpring System.dcx page 2 f 2
17 Hnestly, yu ve gt t watch the vide: Flipping Physics Lecture Ntes: Creating Circular Mtin frm Sine and Csine Curves It s why I make them. Yu will understand this cncept better if yu watch the bjects in simple harmnic mtin creating the circular mtin Lecture Ntes  Creating Circular Mtin frm Sine and Csine Curves.dcx page 1 f 1
Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal MassSpring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal MassSpring System A Hrizntal MassSpring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationPhys101 Final Code: 1 Term: 132 Wednesday, May 21, 2014 Page: 1
Phys101 Final Cde: 1 Term: 1 Wednesday, May 1, 014 Page: 1 Q1. A car accelerates at.0 m/s alng a straight rad. It passes tw marks that are 0 m apart at times t = 4.0 s and t = 5.0 s. Find the car s velcity
More informationSolution to HW14 Fall2002
Slutin t HW14 Fall2002 CJ5 10.CQ.003. REASONING AND SOLUTION Figures 10.11 and 10.14 shw the velcity and the acceleratin, respectively, the shadw a ball that underges unirm circular mtin. The shadw underges
More information1 Course Notes in Introductory Physics Jeffrey Seguritan
Intrductin & Kinematics I Intrductin Quickie Cncepts Units SI is standard system f units used t measure physical quantities. Base units that we use: meter (m) is standard unit f length kilgram (kg) is
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationAP Physics Kinematic Wrap Up
AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in twdimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x
More informationSPH3U1 Lesson 06 Kinematics
PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems.
More informationWork, Energy, and Power
rk, Energy, and Pwer Physics 1 There are many different TYPES f Energy. Energy is expressed in JOULES (J 419J 4.19 1 calrie Energy can be expressed mre specifically by using the term ORK( rk The Scalar
More informationCLASS XI SET A PHYSICS
PHYSIS. If the acceleratin f wedge in the shwn arrangement is a twards left then at this instant acceleratin f the blck wuld be, (assume all surfaces t be frictinless) a () ( cs )a () a () cs a If the
More informationCHAPTER 6  ENERGY. Approach #2: Using the component of mg along the line of d:
SlutinsCh. 6 (Energy) CHAPTER 6  ENERGY 6.) The f.b.d. shwn t the right has been prvided t identify all the frces acting n the bdy as it mves up the incline. a.) T determine the wrk dne by gravity
More informationYeuSheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN
YeuSheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN 38104 Office: (901) 3213424 Rm: N110 Fax : (901) 3213402
More informationExample 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W
Eample 1 rbt has a mass f 60 kg. Hw much des that rbt weigh sitting n the earth at sea level? Given: m Rbt = 60 kg ind: Rbt Relatinships: Slutin: Rbt =589 N = mg, g = 9.81 m/s Rbt = mrbt g = 60 9. 81 =
More informationIntroduction to Spacetime Geometry
Intrductin t Spacetime Gemetry Let s start with a review f a basic feature f Euclidean gemetry, the Pythagrean therem. In a twdimensinal crdinate system we can relate the length f a line segment t the
More informationQ x = cos 1 30 = 53.1 South
Crdinatr: Dr. G. Khattak Thursday, August 0, 01 Page 1 Q1. A particle mves in ne dimensin such that its psitin x(t) as a functin f time t is given by x(t) =.0 + 7 t t, where t is in secnds and x(t) is
More informationPhysics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1
Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs
More informationEXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017
OBJECTIVES 1. Ft Pressure EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017 Determine the surface area f an bject. Given the weight and surface area, calculate the pressure. 2. Measuring Vlume & Mass Prvided a
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationCHAPTER 8b Static Equilibrium Units
CHAPTER 8b Static Equilibrium Units The Cnditins fr Equilibrium Slving Statics Prblems Stability and Balance Elasticity; Stress and Strain The Cnditins fr Equilibrium An bject with frces acting n it, but
More information14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?
Physics 0 Public Exam Questins Unit 1: Circular Mtin NAME: August 009 1. Which describes
More information14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?
Physics 3204 Public Exam Questins Unit 1: Circular Mtin NAME: August 2009 12. Which
More informationf = µ mg = kg 9.8m/s = 15.7N. Since this is more than the applied
Phsics 141H lutins r Hmewrk et #5 Chapter 5: Multiple chice: 8) (a) he maimum rce eerted b static rictin is µ N. ince the blck is resting n a level surace, N = mg. the maimum rictinal rce is ( ) ( ) (
More informationPhys101 First Major131 Zero Version Coordinator: Dr. A. A. Naqvi Wednesday, September 25, 2013 Page: 1
Phys11 First Majr11 Zer Versin Crdinatr: Dr. A. A. Naqvi Wednesday, September 5, 1 Page: 1 Q1. Cnsider tw unifrm slid spheres A and B made f the same material and having radii r A and r B, respectively.
More informationEXAM #1 PHYSICAL SCIENCE 103 Spring, 2016
OBJECTIVES 1. Ft Pressure EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016 Determine the surface area f an bject. Given the weight and surface area, calculate the pressure. 2. Measuring Vlume & Mass Prvided a
More informationCorrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec
U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.
More informationPHYSICS 151 Notes for Online Lecture #23
PHYSICS 5 Ntes fr Online Lecture #3 Peridicity Peridic eans that sething repeats itself. r exaple, eery twentyfur hurs, the Earth aes a cplete rtatin. Heartbeats are an exaple f peridic behair. If yu
More informationPhys101 Second Major061 Zero Version Coordinator: AbdelMonem Saturday, December 09, 2006 Page: 1
Crdinatr: AbdelMnem Saturday, December 09, 006 Page: Q. A 6 kg crate falls frm rest frm a height f.0 m nt a spring scale with a spring cnstant f.74 0 3 N/m. Find the maximum distance the spring is cmpressed.
More informationPHYS 314 HOMEWORK #3
PHYS 34 HOMEWORK #3 Due : 8 Feb. 07. A unifrm chain f mass M, lenth L and density λ (measured in k/m) hans s that its bttm link is just tuchin a scale. The chain is drpped frm rest nt the scale. What des
More informationChapter 5: Force and Motion Ia
Chapter 5: rce and Mtin Ia rce is the interactin between bjects is a vectr causes acceleratin Net frce: vectr sum f all the frces n an bject. v v N v v v v v ttal net = i = + + 3 + 4 i= Envirnment respnse
More informationmaking triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=
Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents
More informationENGI 4430 Parametric Vector Functions Page 201
ENGI 4430 Parametric Vectr Functins Page 01. Parametric Vectr Functins (cntinued) Any nnzer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
More informationInformation for Physics 1201 Midterm I Wednesday, February 20
My lecture slides are psted at http://www.physics.histate.edu/~humanic/ Infrmatin fr Physics 1201 Midterm I Wednesday, February 20 1) Frmat: 10 multiple chice questins (each wrth 5 pints) and tw shwwrk
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
More information20 Faraday s Law and Maxwell s Extension to Ampere s Law
Chapter 20 Faraday s Law and Maxwell s Extensin t Ampere s Law 20 Faraday s Law and Maxwell s Extensin t Ampere s Law Cnsider the case f a charged particle that is ming in the icinity f a ming bar magnet
More informationFigure 1a. A planar mechanism.
ME 5  Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary,
More information= m. Suppose the speed of a wave on a string is given by v = Κ τμ
Phys101 First Majr11 Zer Versin Sunday, Octber 07, 01 Page: 1 Q1. Find the mass f a slid cylinder f cpper with a radius f 5.00 cm and a height f 10.0 inches if the density f cpper is 8.90 g/cm 3 (1 inch
More informationFaculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination February 27, 2006; 7:00 pm 8:30 pm
Faculty f Engineering and Department f Physics Engineering Physics 131 Midterm Examinatin February 27, 2006; 7:00 pm 8:30 pm N ntes r textbks allwed. Frmula sheet is n the last page (may be remved). Calculatrs
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (RobbinsMiller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (RbbinsMiller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More informationPlan o o. I(t) Divide problem into subproblems Modify schematic and coordinate system (if needed) Write general equations
STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.
More informationLecture 7: Damped and Driven Oscillations
Lecture 7: Damped and Driven Oscillatins Last time, we fund fr underdamped scillatrs: βt x t = e A1 + A csω1t + i A1 A sinω1t A 1 and A are cmplex numbers, but ur answer must be real Implies that A 1 and
More informationPhysics 2B Chapter 23 Notes  Faraday s Law & Inductors Spring 2018
Michael Faraday lived in the Lndn area frm 1791 t 1867. He was 29 years ld when Hand Oersted, in 1820, accidentally discvered that electric current creates magnetic field. Thrugh empirical bservatin and
More informationChapter 2. Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion.
Chapter Kinematics in One Dimensin Kinematics deals with the cncepts that are needed t describe mtin. Dynamics deals with the effect that frces have n mtin. Tgether, kinematics and dynamics frm the branch
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationLecture 6: Phase Space and Damped Oscillations
Lecture 6: Phase Space and Damped Oscillatins Oscillatins in Multiple Dimensins The preius discussin was fine fr scillatin in a single dimensin In general, thugh, we want t deal with the situatin where:
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
More informationStudy Guide Physics PreComp 2013
I. Scientific Measurement Metric Units S.I. English Length Meter (m) Feet (ft.) Mass Kilgram (kg) Pund (lb.) Weight Newtn (N) Ounce (z.) r pund (lb.) Time Secnds (s) Secnds (s) Vlume Liter (L) Galln (gal)
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3.5 Learning Objectives:.3.
More information37 Maxwell s Equations
37 Maxwell s quatins In this chapter, the plan is t summarize much f what we knw abut electricity and magnetism in a manner similar t the way in which James Clerk Maxwell summarized what was knwn abut
More informationPhysics 101 Math Review. Solutions
Physics 0 Math eview Slutins . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t
More informationPhysics 321 Solutions for Final Exam
Page f 8 Physics 3 Slutins fr inal Exa ) A sall blb f clay with ass is drpped fr a height h abve a thin rd f length L and ass M which can pivt frictinlessly abut its center. The initial situatin is shwn
More informationLab 11 LRC Circuits, Damped Forced Harmonic Motion
Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether
More informationPHYSICS LAB Experiment 10 Fall 2004 ROTATIONAL DYNAMICS VARIABLE I, FIXED
ROTATIONAL DYNAMICS VARIABLE I, FIXED In this experiment we will test Newtn s Secnd Law r rtatinal mtin and examine hw the mment inertia depends n the prperties a rtating bject. THE THEORY There is a crrespndence
More informationChapter 10. Simple Harmonic Motion and Elasticity. Example 1 A Tire Pressure Gauge
0. he Ideal Spring and Simple Harmnic Mtin Chapter 0 Simple Harmnic Mtin and Elasticity F Applied x k x spring cnstant Units: N/m 0. he Ideal Spring and Simple Harmnic Mtin 0. he Ideal Spring and Simple
More informationAP Physics Laboratory #4.1: Projectile Launcher
AP Physics Labratry #4.1: Prjectile Launcher Name: Date: Lab Partners: EQUIPMENT NEEDED PASCO Prjectile Launcher, Timer, Phtgates, Time f Flight Accessry PURPOSE The purpse f this Labratry is t use the
More informationCHAPTER 4 Dynamics: Newton s Laws of Motion /newtlaws/newtltoc.html
CHAPTER 4 Dynamics: Newtn s Laws f Mtin http://www.physicsclassrm.cm/class /newtlaws/newtltc.html Frce Newtn s First Law f Mtin Mass Newtn s Secnd Law f Mtin Newtn s Third Law f Mtin Weight the Frce f
More informationPhys102 Final061 Zero Version Coordinator: Nasser Wednesday, January 24, 2007 Page: 1
Crdinatr: Nasser Wednesday, January 4, 007 Page: 1 Q1. Tw transmitters, S 1 and S shwn in the figure, emit identical sund waves f wavelength λ. The transmitters are separated by a distance λ /. Cnsider
More informationLecture 2: Singleparticle Motion
Lecture : Singleparticle Mtin Befre we start, let s l at Newtn s 3 rd Law Iagine a situatin where frces are nt transitted instantly between tw bdies, but rather prpagate at se velcity c This is true fr
More informationPhy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1
Ph 13: General Phsics III 6/14/007 Chapter 8 Wrksheet 1 Magnetic Fields & Frce 1. A pint charge, q= 510 C and m=1103 m kg, travels with a velcit f: v = 30 ˆ s i then enters a magnetic field: = 110 T ˆj.
More informationThree charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter).
Three charges, all with a charge f 0 are situated as shwn (each grid line is separated by meter). ) What is the net wrk needed t assemble this charge distributin? a) +0.5 J b) +0.8 J c) 0 J d) 0.8 J e)
More informationLab #3: Pendulum Period and Proportionalities
Physics 144 Chwdary Hw Things Wrk Spring 2006 Name: Partners Name(s): Intrductin Lab #3: Pendulum Perid and Prprtinalities Smetimes, it is useful t knw the dependence f ne quantity n anther, like hw the
More information. (7.1.1) This centripetal acceleration is provided by centripetal force. It is directed towards the center of the circle and has a magnitude
Lecture #71 Dynamics f Rtatin, Trque, Static Equilirium We have already studied kinematics f rtatinal mtin We discussed unifrm as well as nnunifrm rtatin Hwever, when we mved n dynamics f rtatin, the
More informationFinding the Earth s magnetic field
Labratry #6 Name: Phys 1402  Dr. Cristian Bahrim Finding the Earth s magnetic field The thery accepted tday fr the rigin f the Earth s magnetic field is based n the mtin f the plasma (a miture f electrns
More informationAP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) =
P Physics Name Summer ssignment 0 Date I. The P curriculum is extensive!! This means we have t wrk at a fast pace. This summer hmewrk will allw us t start n new Physics subject matter immediately when
More informationI understand the new topics for this unit if I can do the practice questions in the textbook/handouts
1 U n i t 6 11U Date: Name: Sinusidals Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will learn hw trignmetry can be used t mdel wavelike relatinships. These wavelike functins are called
More informationHubble s Law PHYS 1301
1 PHYS 1301 Hubble s Law Why: The lab will verify Hubble s law fr the expansin f the universe which is ne f the imprtant cnsequences f general relativity. What: Frm measurements f the angular size and
More informationHooke s Law (Springs) DAVISSON. F A Deformed. F S is the spring force, in newtons (N) k is the spring constant, in N/m
HYIC 534 XRCI4 ANWR Hke s Law (prings) DAVION Clintn Davissn was awarded the Nbel prize fr physics in 1937 fr his wrk n the diffractin f electrns. A spring is a device that stres ptential energy. When
More informationChapter 32. Maxwell s Equations and Electromagnetic Waves
Chapter 32 Maxwell s Equatins and Electrmagnetic Waves Maxwell s Equatins and EM Waves Maxwell s Displacement Current Maxwell s Equatins The EM Wave Equatin Electrmagnetic Radiatin MFMcGrawPHY 2426 Chap32Maxwell's
More informationCHAPTER 6 WORK AND ENERGY
CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS 16. REASONING AND SOLUTION A trapeze artist, starting rm rest, swings dwnward n the bar, lets g at the bttm the swing, and alls reely t the net. An assistant,
More informationThis section is primarily focused on tools to aid us in finding roots/zeros/ intercepts of polynomials. Essentially, our focus turns to solving.
Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ intercepts f plynmials. Essentially, ur fcus
More informationKinetics of Particles. Chapter 3
Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between
More informationiclicker iclicker Newton s Laws of Motion First Exam Coming Up! Components of Equation of Motion
First Eam Cming Up! Sunda, 1 Octber 6:10 7:30 PM. Lcatins t be psted nline. Yes this is a Sunda! There will be 17 questins n eam. If u have a legitimate cnflict, u must ask Prf. Shapir b Oct. 8 fr permissin
More informationPrecalculus A. Semester Exam Review
Precalculus A 015016 MCPS 015 016 1 The semester A eaminatin fr Precalculus cnsists f tw parts. Part 1 is selected respnse n which a calculatr will NOT be allwed. Part is shrt answer n which a calculatr
More informationEinstein's special relativity the essentials
VCE Physics Unit 3: Detailed study Einstein's special relativity the essentials Key knwledge and skills (frm Study Design) describe the predictin frm Maxwell equatins that the speed f light depends nly
More informationQ1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.
Phys10 Final133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed
More informationLHS Mathematics Department Honors PreCalculus Final Exam 2002 Answers
LHS Mathematics Department Hnrs Prealculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the
More information1 PreCalculus AP Unit G Rotational Trig (MCR) Name:
1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will
More informationSAFE HANDS & IITian's PACE EDT04 (JEE) Solutions
ED (JEE) Slutins Answer : Optin () ass f the remved part will be / I Answer : Optin () r L m (u csθ) (H) Answer : Optin () P 5 rad/s ms  because f translatin ωr ms  because f rtatin Cnsider a thin shell
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More information1.2.1 Vectors. 1 P age. Examples What is the reference vector angle for a vector that points 50 degrees east of south?
1.2.1 Vectrs Definitins Vectrs are represented n paper by arrws directin = magnitude = Examples f vectrs: Examples What is the reference vectr angle fr a vectr that pints 50 degrees east f suth? What is
More informationSubject: KINEMATICS OF MACHINES Topic: VELOCITY AND ACCELERATION Session I
Subject: KINEMTIS OF MHINES Tpic: VELOITY ND ELERTION Sessin I Intrductin Kinematics deals with study f relative mtin between the varius parts f the machines. Kinematics des nt invlve study f frces. Thus
More informationMathacle PSet  Algebra, Trigonometry Functions Level Number Name: Date:
PSet  Algebra, Trignmetry Functins I. DEFINITIONS OF THE SIX TRIG FUNCTIONS. Find the value f the trig functin indicated 1 PSet  Algebra, Trignmetry Functins Find the value f each trig functin.
More informationPages with the symbol indicate that a student should be prepared to complete items like these with or without a calculator. tan 2.
Semester Eam Review The semester A eaminatin fr Hnrs Precalculus cnsists f tw parts. Part 1 is selected respnse n which a calculatr will NOT be allwed. Part is shrt answer n which a calculatr will be allwed.
More informationChapter 9 Vector Differential Calculus, Grad, Div, Curl
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2Space and 3Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13 Mdern systems generally DSP Scenari get a cntinuustime signal frm a sensr a cnt.time
More informationENGINEERING COUNCIL CERTIFICATE LEVEL THERMODYNAMIC, FLUID AND PROCESS ENGINEERING C106 TUTORIAL 5 THE VISCOUS NATURE OF FLUIDS
ENGINEERING COUNCIL CERTIFICATE LEVEL THERMODYNAMIC, FLUID AND PROCESS ENGINEERING C106 TUTORIAL 5 THE VISCOUS NATURE OF FLUIDS On cmpletin f this tutrial yu shuld be able t d the fllwing. Define viscsity
More informationECE 2100 Circuit Analysis
ECE 2100 Circuit Analysis Lessn 25 Chapter 9 & App B: Passive circuit elements in the phasr representatin Daniel M. Litynski, Ph.D. http://hmepages.wmich.edu/~dlitynsk/ ECE 2100 Circuit Analysis Lessn
More informationNGSS High School Physics Domain Model
NGSS High Schl Physics Dmain Mdel Mtin and Stability: Frces and Interactins HSPS21: Students will be able t analyze data t supprt the claim that Newtn s secnd law f mtin describes the mathematical relatinship
More informationPhysics 1200 Mechanics, Kinematics, Fluids, Waves
Physics 100 Mechanics, Kinematics, Fluids, Waes Lecturer: Tm Humanic Cntact inf: Office: Physics Research Building, Rm. 144 Email: humanic@mps.histate.edu Phne: 614 47 8950 Office hurs: Tuesday 3:00 pm,
More informationCalculus Placement Review. x x. =. Find each of the following. 9 = 4 ( )
Calculus Placement Review I. Finding dmain, intercepts, and asympttes f ratinal functins 9 Eample Cnsider the functin f ( ). Find each f the fllwing. (a) What is the dmain f f ( )? Write yur answer in
More informationComputational modeling techniques
Cmputatinal mdeling techniques Lecture 2: Mdeling change. In Petre Department f IT, Åb Akademi http://users.ab.fi/ipetre/cmpmd/ Cntent f the lecture Basic paradigm f mdeling change Examples Linear dynamical
More informationPRECALCULUS B FINAL REVIEW NAME Work out problems in your notebook or on a separate piece of paper.
PRECALCULUS B FINAL REVIEW NAME Wrk ut prblems in yur ntebk r n a separate piece f paper. CHAPTER 5 Simplify each t ne trig wrd r a number: tan (1) sec sec () sin ct + 1 + cs + ( sin cs ) () sin  cs
More informationMatter Content from State Frameworks and Other State Documents
Atms and Mlecules Mlecules are made f smaller entities (atms) which are bnded tgether. Therefre mlecules are divisible. Miscnceptin: Element and atm are synnyms. Prper cnceptin: Elements are atms with
More information39th International Physics Olympiad  Hanoi  Vietnam Theoretical Problem No. 1 /Solution. Solution
39th Internatinal Physics Olympiad  Hani  Vietnam  8 Theretical Prblem N. /Slutin Slutin. The structure f the mrtar.. Calculating the distance TG The vlume f water in the bucket is V = = 3 3 3 cm m.
More informationES201  Examination 2 Winter Adams and Richards NAME BOX NUMBER
ES201  Examinatin 2 Winter 20032004 Adams and Richards NAME BOX NUMBER Please Circle One : Richards (Perid 4) ES20101 Adams (Perid 4) ES20102 Adams (Perid 6) ES20103 Prblem 1 ( 12 ) Prblem 2 ( 24
More information"1 O O O. U 7 P fl> 1 3. jff. (t) o o 17 PAa s: A  o 0'»«« "Tf O ") ftt Ti 0  CO 1 O. fa n. i,, I. n F en 2.0» 4 T2. 5 Ut.
crv/ 3 P U 7 P fl> 1 3 (t) jff? "1 s P 917 ~* PAa s: A  "C '»««in i,, I ftt Ti  c 4 T25 Ut j 3 C 1 p fa n l> n F en 2.» "Tf ") r . x 2 "Z * "! t :.!, 21,, V\ C fn
More information( ) + θ θ. ω rotation rate. θ g geographic latitude   θ geocentric latitude   Reference Earth Model  WGS84 (Copyright 2002, David T.
1 Reference Earth Mdel  WGS84 (Cpyright, David T. Sandwell) ω spherid c θ θ g a parameter descriptin frmula value/unit GM e (WGS84) 3.9864418 x 1 14 m 3 s M e mass f earth  5.98 x 1 4 kg G gravitatinal
More informationSection 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law
Sectin 5.8 Ntes Page 1 5.8 Expnential Grwth and Decay Mdels; Newtn s Law There are many applicatins t expnential functins that we will fcus n in this sectin. First let s lk at the expnential mdel. Expnential
More informationLab 1 The Scientific Method
INTRODUCTION The fllwing labratry exercise is designed t give yu, the student, an pprtunity t explre unknwn systems, r universes, and hypthesize pssible rules which may gvern the behavir within them. Scientific
More informationLEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot
Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES
More information