Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Transcription

1 Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring 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 Mass-Spring 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 Mass-Spring System Hrizntal and vertical mass-spring 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 mass-spring system is just the spring frce, because that is the net frce in the x-directin. The restring frce fr a vertical mass-spring system is the net frce in the y-directin which equals the spring frce minus the frce f gravity Lecture Ntes - Hrizntal vs Vertical Mass-Spring System.dcx page 1 f 1

4 Flipping Physics Lecture Ntes: When is a Pendulum in Simple Harmnic Mtin? Mass-spring 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 mass-spring system: T = 2π m k m is the mass in the mass-spring 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 mass-spring system? Amplitude is nt in either perid equatin. Amplitude des nt affect the perid f a pendulum r the perid f a mass-spring system. Acceleratin due t gravity is nt in the perid equatin fr a mass-spring system. g des nt affect the perid f a mass-spring 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 mass-spring system increases its perid. m T fr a mass-spring system. Increasing the spring cnstant in a mass-spring system decreases its perid. k T fr a mass-spring 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 Mass-Spring System. Hw des Perid Change? Example: If the mass in a mass-spring 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 mass-spring system: T = 2π m k S the perid f the riginal mass-spring system is: T 1 = 2π m 1 k And the perid f the new mass-spring 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 Mass-Spring 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 mass-spring system with a perid f 0.77 secnds, the frequency f that mass-spring system is: f = 1 T = 1 cycles = r1.3 Hz 0.77 secnd Which means the mass-spring 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 x-directin, 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