Chapter 15 Oscillatory Motion I

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1 Chaper 15 Oscillaory Moion I Level : AP Physics Insrucor : Kim Inroducion A very special kind of moion occurs when he force acing on a body is proporional o he displacemen of he body from some equilibrium posiion. If his force is always direced oward he equilibrium posiion, repeiive back-and-force moion occurs abou his posiion. Such moion is called periodic moion, or harmonic moion, oscillaion or vibraion. Eamples of periodic moion; a block aached o a spring, he swinging of a child on a playground, he moion of a pendulum, and he vibraions of a sringed musical insrumen, ec... The ideal moion is called simple harmonic moion, where here is no loss in energy Ideal Spring When he block is displaced a small disance from equilibrium, he spring eers on he block a force ha is proporional o he displacemen and given by Hook s law F s = k : Hook s Law We call his a resoring force because i is always direced oward he equilibrium posiion herefore opposie he displacemen. Tha is, if a force is applied o a spring, he spring eers an resoring force ha is always opposie in direcion F = F s + F app = 0 => F app = -F s => F app = -(-k) F app =k F app=k Q1) An objec of mass 9kg is hung on a spring, causing he spring o srech by =0.0191m. Then deermine he force needed o srech he spring by m. Fs Fapp a) 194N b) 235N c) 276N d) 310N (a) Fg =mg (b) Q2) The graph shows he force F ha an archer applies o he spring of a long bow versus he sring s displacemen. Drawing back his bow is analogous o sreching a spring. From he daa in he graph deermine he effecive spring consan of he bow a) 324.5N b) 397.1N c) 533.6N d) 666.7N F(N)

2 Simple Harmonic Moion and Posiion(or displacemen) Funcion over ime When a block of mass m is aached o a spring ha is FP neiher sreched nor compressed, he block is a he posiion =0, called he equilibrium posiion of he sysem. If he block is disurbed from is equilibrium posiion, he block will oscillae back and forh. =0 For eample, assuming he surface is fricionless, if he block is given a 'insananeous' push(f P) o he righ, hen he block will oscillae. If we record he posiion of he block hrough ime and plo in on a - graph, we can see he following; v=0 Fs (s) =0 =1m -1 Assume he block momenarily sops a posiion =1m and he ime i akes o reach ha posiion is =1.57s vma Fs= (s) =0-1 The block reurns o he origin, reaching maimum speed he momen i passes he origin v=0 Fs (s) = 1m =0-1 If he surface is fricionless, hen he block will be compressed unil i reaches = 1m. (maimum compression) vma Fs= (s) =0-1

3 Q4) A block is aached o spring and se in simple harmonic moion. Surface is fricionless. Which of he following funcion bes describes he posiion() funcion of he block as a funcion of ime? a) () = 2 b) () = 2 (c) () = 3 d) () = sin =0 ma Hence, he displacemen funcion ha bes describes he moion over ime can be epressed as () = sin where he maimum srech or compression is = ±1m. If he block is sreched or compressed more, like =2m, hen () = 2sin However, since our observaion begins according o our choice and if we happened o sar observing when he block is a he maimum srech from he equilibrium posiion, hen he displacemen funcion can be epressed as () = 2sin( + π ), where π/2=1.57 or () = 2cos 2 Hence, choosing cosine or sine funcion depends on when we sar our observaion =0 ma Q5) Which of he following funcion is equal o () = cos( - π)? There are wo answers a) () = sin b) () = sin( - π ) c) () = cos( - π ) d) () = cos e) () = cos 2 2 Q6) Which graph represens he posiion () funcion? Connec by drawing a line () = cos(+3π/2) () = cos(+π/2) () = cos( - π)

4 Newon's 2nd law and Posiion Funcion for Simple Harmonic Moion Applying Newon s 2 nd law o he moion of he block, we obain solving he above equaion gives F s = k= ma a = k m d 2 d 2 = k m ()=Acos(w+φ) where w 2 =k/m, k is he spring consan, m is mass of he block ()=Acos(w+φ) is he displacemen funcion of a paricle ehibiing simple harmonic moion. Tha is, he funcion provides he posiion of he paricle from he equilibrium a any ime - A is he ampliude, which represens he maimum displacemen of he paricle eiher he posiive or negaive direcion. - w is he angular frequency of he moion and has unis of rads /s - The consan angle φ is he phase consan. I is deermined by iniial displacemen and velociy of he paricle The quaniy (w + π) is called he phase of he moion and is useful in comparing he moion of wo oscillaors. Noe ha he rigonomeric funcion Acos(w+φ) is periodic and repeas iself every ime w increases by 2πrads. Here we can define he period T, since period T of he moion is he ime i akes for he paricle o go hrough one full cycle or one oscillaion. Hence, he value of a ime equals he value of a ime +T. Hence T = 2π w w + φ + 2π = w( +T) + φ In roaional moion, recall ha he angular speed w was defined as w=δθ/δ. If an objec complees one cycle, hen Δθ=2π and Δ=T. So w=δθ/δ=2π/t, hen angular speed, which is also ofen called angular frequency can be wrien as w = 2π T

5 The inverse of he period is called he frequency f of he moion. The frequency represens he number of oscillaions ha he paricle makes per uni ime: f = 1 T = w 2π and w = 2πf The uni of f are cycles per second, ha is s -1 or herz(hz). Since w= k m, he period can also be epressed as T=2π m k and f = 1 2π k m Summary F s= k and F app=k The posiion of a simple harmonic moion: ()=Acos(w+φ) A is he ampliude, w is he angular frequency where w= k, φ is he phase consan m T is he period where T = 2π w =2π m k f is he frequency where f = 1 T = w 2π = 1 2π k m Q7) An objec undergoes simple harmonic moion of ampliude A. Through wha oal disance does he objec move during one complee cycle of is moion? (a) A/2 (b) A (c) 2A (d) 4A Q8) If an objec akes 2seconds o complee one cycle, wha are he period, frequency and angular frequency? Check your answer below Ans) 2s, 0.5Hz, 3.14rad/s Velociy and Acceleraion Funcion for Simple Harmonic Moion We can obain he linear velociy of a paricle undergoing simple harmonic moion by differeniaing equaion ()=Acos(w+φ) The speed of he paricle is v()= d = -wasin(w + φ ), where vma=wa (compare wih v=rw) d The acceleraion of he paricle is a() = dv d = -w2 Acos(w + φ ), where a ma=w 2 A(compare wih a r=v 2 /r=w 2 r)

6 Displacemen vs Velociy vs Acceleraion A block aached o a spring on a fricionless surface is pulled a cerain disance away from he equilibrium and hen released. The block has maimum displacemen he insan i was released. A ha momen, he speed is minimum and acceleraion is maimum. (a) Displacemen vs ime graph is shown. () is maimum he momen he block is being released. () is minimum he momen he block passes he equilibrium posiion ()=Acos(w) (b) Velociy vs ime graph is shown. v()=0 he momen he block is being released. v() is maimum in he negaive direcion he momen he block passes he equilibrium posiion () A -A v() wa v()= d d = -Awsin(w) (c) Acceleraion vs ime graph is shown. a() is maimum in negaive direcion he momen he block is being released. a()=0 he momen he block passes he equilibrium posiion a()= dv d = -Aw2 cos(w) -wa -w 2 A a() w 2 A Calculaors mus be in radians!!!!!! Or plunge ino forever darkness~ Q9) The displacemen of a paricle a =0.25s is given by he epression ()=(4.00m)cos(3π + π), where is in meers and is in seconds. Deermine meers (a) he frequency and period of he moion Ans) 1.5Hz, 0.67s (b) he ampliude of he moion (c) he phase consan Ans) 4m Ans) π (d) he displacemen of he paricle a =0.25s. Ans) 2.83m (e) Plo a displacemen vs ime graph of he moion below. Plo your graph manually and hen check calculaor (s)

7 Q10) An objec oscillaes wih simple harmonic moion along he ais. Is displacemen from he origin varies wih ime according o he equaion = (4.00m) cos( π + π 4 ) where is in seconds and he angles in he parenheses are in radians. (a) Deermine he ampliude, frequency and period of he moion Ans) 4m, 0.5Hz, 2s (b) Calculae he speed and acceleraion of he objec a ime =1s. Ans) 8.89m/s (righ) 27.9m/s 2 (c) Deermine he maimum speed and a wha ime does he objec have maimum speed? Ans) 12.6m/s, v ma a = 0.25s, 1.25s, 2.25s,.... (d) Plo a displacemen vs ime graph of he moion below. (s) (e) Find he displacemen of he objec beween = 0 and =1s and he disance raveled during hose ime. Ans) = 5.66m, d=8m

8 Q11) A 0.8kg objec is aached o one end of a spring and he sysem is se ino simple harmonic moion. The displacemen of he objec as a funcion of ime is shown. Wih he aid of his daa, deermine he (a) ampliude A of he moion Ans) 0.08m (b) he angular frequency w and frequency f ime(s) Ans) 1.57rad/s, 0.25Hz (c) he spring consan k Ans)1.97N/m (d) he maimum displacemen from he equilibrium posiion ma Ans) 0.08m (e) A wha ime(s) is he objec a he equilibrium posiion? Ans) =0,2,4,.. (f) Epress he displacemen of he moion as a funcion of ime () Ans) ()=0.08cos( π 2 π 2 ) or 0.08sin( π 2 ) (g) Find he posiion of he objec a =0.2, 0.5, 2.4, 3.0 seconds Ans) m, m, 0.047m, 0.08m (h) Wha is he maimum speed of he objec? Also find he epression for speed as a funcion of ime v() Ans) m/s, v()= 0.04πsin( π 2 π 2 ) (i) Find he speed of he objec a =0.4, 1.5 and 3.2 seconds. Ans) 0.102m/s, 0.089m/s, 0.039m/s (j) Find he an epression for acceleraion as a funcion of ime a() and find he acceleraion a =0.4, 1.5, 3.2, 3.9, 3.99, seconds Ans) a()= 0.02π 2 cos( π 2 π 2 ), 0.116m/s2, 0.14m/s 2, 0.188m/s 2, m/s 2, m/s 2, approach 0

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