Section 35 SHM and Circular Motion

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1 Section 35 SHM nd Cicul Motion Phsics 204A Clss Notes Wht do objects do? nd Wh do the do it? Objects sometimes oscillte in simple hmonic motion. In the lst section we looed t mss ibting t the end of sping. We pplied the Second Lw to deie the equtions of motion fo SHM. In the pocess, we seemed to be using the ide of ngul fequenc just s we did when we looed t unifom cicul motion. In ddition, the equtions of motion fo SHM loo ee simil to the equtions of motion fo unifom cicul motion. In this section, we will inestigte the connection between SHM nd unifom cicul motion. Net, we ll continue to build ou undestnding of SHM b looing t the oscillto motion of simple pendulum. We ll discoe tht it lso is SHM unde cetin conditions. Section Outline 1. The Connection Between Unifom Cicul Motion nd SHM 2. Eneg in SHM 3. The Simple Pendulum 1. The Connection Between Unifom Cicul Motion nd SHM We he been using the ide of ngul fequenc s we did when we discussed cicul motion. Thee must be some connection, so let s inestigte. At the ight is n object going in cicle on otting tuntble. Just behind the object is sceen whee the shdow of the object cn be seen. The shdow moes bc nd foth s the object goes in cicle. The shdow ppes to be in SHM. At the ight is setch of the object in unifom cicul motion. It hs centipetl cceletion, tngentil elocit, nd position ecto ll shown. If we edw the thee ectos with thei tils t the oigin we cn imgine ll thee spinning s the object ottes. Finding the -components of ech the position, elocit, nd cceletion of the shdow, = cos, = sin, nd = cos. The tngentil elocit is elted to this ngul elocit, = ω. Also, the centipetl cceletion is elted to the ngul elocit, = 2 = ( ω )2 = ω 2. Substituting, we get, = cos, = ω sin, nd = ω 2 cos. Notice, just lie SHM we he = ω 2. The ngle chnges with time. We cn wite this using the definition of ngul fequenc, ω d dt d = ωdt = ωt + δ. Now we see nothe w of looing t the phse ngle, δ, s just n integtion constnt. Finll, we cn wite the -components of the, position, elocit nd cceletion fo the oscillting shdow s function of time, 35-1

2 Phsics 204A Clss Notes (t) = cos( ωt + δ ), (t) = ω sin( ωt + δ ), nd (t) = ω 2 cos( ωt + δ ). These e the sme s the SHM equtions of motion with A insted of. The component of the motion of n object in unifom cicul motion is SHM. Tht eplins wh we eep tling bout ngul fequencies! Emple 35.1: A 500g mss ests in equilibium t the end of hoizontl sping with sping constnt 9.80N/m. The mss is gien shp ic esulting in n initil elocit of 0.443m/s to the ight. ()Setch the initil position, elocit, nd cceletion ectos s if the object wee in cicul motion. Find (b)the loction of the equilent object in cicul motion, (c)the phse ngle, nd (d)the eqution fo (t). Gien: = 9.80N/m, m = 0.500g, (0) = 0.443m/s, nd (0) = 0. Find: =?, =?, =?, d =?, nd (t)=? ()We e gien tht the elocit ecto is to the ight nd the initil position is zeo. Theefoe, the -component of the elocit must be t mimum nd point to the ight. The - component of the position nd cceletion ectos must be zeo. The cceletion must point towd the cente of the cicle nd the position must point outwd. The nswe then is in the setch t the uppe ight. (b)the equilent object in cicul motion with the ectos pointing the ight diection must be s shown t the ight. (c)looing t the cicle, the phse ngle must be 270 o 3π 2. (d)using the ppopite eqution of motion fo cicul motion, the eqution fo the position s function of time is (t) = o sin(ωt + δ ). The ngul fequenc fo sping is, ω = m = = 4.43 d s. So, (t) = 0.443sin(4.43t + 3π ). 2 Note this esults in (0) = m/s s equied. 35-2

3 Phsics 204A Clss Notes 2. Eneg in SHM We he led looed t eneg in SHM. The esult ws tht the potentil eneg stoed in sping ws gien b, U s = So, let s loo t n oscillting mss t the end of sping. In the top imge t the ight the mss is t est t the equilibium position of the sping. In the middle, imge the mss hs been pulled to the ight distnce A. The sstem hs totl eneg equl to the potentil eneg in the sping, E o = U s = 1 2 A2. The lowe imge is fte the mss hs been elesed nd it is A heding bc to the left. It hs speed when it eches the position. Thee is still some potentil eneg in the sping plus some inetic eneg, E = K + U s = 1 2 m Appling the Lw of Consetion of Eneg, E o = E 1 2 A2 = 1 2 m , nd soling fo the speed, Recll fo sping, ω = m = ± m (A2 2 )., so the speed cn be witten s, = ±ω (A 2 2 ). This is the eqution of motion fo the speed s function of position. Elie we found this eqution b ppling the Second Lw nd the definitions of elocit nd cceletion. Now, we see it is just n epession of the Lw of Consetion of Eneg. 3. The Simple Pendulum A mss t the end of sting cn cetinl oscillte. The question is, is it SHM. Recll tht the condition fo SHM is, () = ω 2, whee ω is constnt. Fo mss t the end of sping, Newton s Second Lw ge us, () =. m Fom this eqution we found deduced tht the motion ws SHM with n ngul fequenc equl to the oot of the constnts on the ight hnd side, ω = m. If we ppl the Second Lw to othe sstems nd find tht the cceletion is equl to the negtie of some constnts multiplied b the position, then we cn follow the sme logic to deduce tht the motion will be SHM with n ngul fequenc equl to the oot of the constnts. Let s loo t some oscillto sstems nd see if the e, in fct, SHM. 35-3

4 Phsics 204A Clss Notes A simple pendulum consists of e light sting with concentted mss t the end. The foces on the mss t the end e git nd the tension. Howee, the tension eets no toque bout the top of the sting. Appling the Second Lw fo Rottion, Στ p = Iα mg sin = m 2 α α = g sin. Since we e looing t ottionl motion, we e checing to see if the ngul cceletion is equl to the negtie of some constnts multiplied b the ngul position. Sdl, this is not the cse fo the simple pendulum becuse we he sin insted of just. Howee, fo smll ngles, sin α g. This is the SHM eqution, the cceletion (ngul, in this cse) is equl to minus some constnts times the position (gin, ngul). This is the sme eqution we got fo the motion of the mss on the end of sping, ecept tht eplces. In othe wods, the equtions of motion fo the ngle,, will be the simple hmonic motion equtions with n ngul fequenc equl to the oot of the constnts so long s the ngle is smll. P F t F g Angul Fequenc of Simple Pendulum ω = g Emple 35.2: The pendulum in gndfthe cloc must he peiod of 2.00s so tht ech swing moes the second hnd twice. Find the length of the pendulum. Gien: T = 2.00s Find: =? The ngul fequenc of simple pendulum is, It is elted to the peiod, Plugging in the numbes, ω = g. ω = 2πf = 2π T T = 2π ω T = 2π g = gt2 4π 2. = (9.80)(2.00)2 4π 2 = 0.993m. This eplins wh ll pendulums in gndfthe clocs e bout this size. 35-4

5 Phsics 204A Clss Notes Section Summ Wh do objects do wht the do? We he been building ou undestnding of simple hmonic motion. We he lened tht n object with n cceletion tht is equl to minus the poduct of some constnt nd the position is in SHM n obes the SHM equtions of motion, () = ω 2 () = ±ω A 2 2 (t) = A cos( ωt + δ) (t) = ωasin( ωt + δ) (t) = ω 2 Acos ωt + δ ( ) whee A is the mplitude of the motion, ω is the ngul fequenc, nd δ is the phse ngle. The ngul fequenc will be equl to the oot of the constnts. We emined the connection between cicul motion nd SHM. SHM is the motion of the shdow of n object in unifom cicul motion. In othe wods, the equtions of motion fo the -component of unifom cicul motion e identicl to the equtions of motion fo SHM. With the nowledge boe, we loo t the oscilltions of simple pendulum nd found tht the e indeed SHM with n ngul fequenc gien b, ω = g. 35-5

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