What is the length of the pendulum? Period = T = 12.0 s

Size: px

Start display at page:

Download "What is the length of the pendulum? Period = T = 12.0 s"

Sara Norton
5 years ago
Views:

1 Eaple: hen visitin an office buildin you notice that it has an atiu with an enoous pendulu with one end attached to the oof of the buildin and the othe end nealy eachin the floo. You obseve that the pendulu takes 6.0 seconds to swin as the floo. How tall is the buildin? hat is the lenth of the pendulu? Peiod = T =.0 s Use dynaics to et euation of otion of pendulu bob. Use toues : ais of otation is oof suspension t = Ia a = dw dt = d d dtł dtł = d dt ssue: No toues due to fiction in suspension No ai esistance Bob is a point object, assless stin Fee body diaa of bob T I = Taet : Find -sin = d dt -sin = d dt Toue diaa +y t t = F + t = -sin = I d dt Difficult to uess solution to this eu. Make an appoiation If anle is sall sin = (in adians) d DC = DC sin = d fo sall DC fi d so sin fi -sin = d dt - = d dt uess peiodic solution as = ( pft + f) d = - sin pft + f dt ( )pf fo sall anles becoes d = - ( pft + f )( dt pf) - ( pft +f) = - ( pft +f) ( pf) = ( pf ) f = /T = p Ł T ł T = Ł pł T = Ł pł check units: Ø ø º Œ s ß œ [ s] = [ ] 3 ft.0s s = Ł p ł 7 ft = Is this uneasonable? ok lone lenth takes oe tie which is easonable. t 0 ft pe stoy, this is a stoy buildin Tallest buildins in Minneapolis ae about 50 floos hih. This is not uneasonable.

2 Note that feuency of pendulu fo toue euation = ( pf ) p feuency of oscillation of pendulu If you ecall Independent of ass Depends on feuency of oscillation of spin p k Independent of Depends on ass Choose the appoach to the poble Seveal ae possible Consevation ppoach eney oentu anula oentu Dynaics ppoach How to choose foces toues Do you have enouh infoation to use the appoach? Does the appoach ive you what you want to know? Do the sae poble but now use a consevation of eney appoach. Eaple: hen visitin an office buildin you notice that it has an atiu with an enoous pendulu with one end attached to the oof of the buildin and the othe end nealy eachin the floo. You obseve that the pendulu takes 6.0 seconds to swin as the floo. How tall is the buildin? hat is the lenth of the pendulu? Peiod = T =.0 s Use consevation of eney to et euation of otion of pendulu bob. Syste: bob & Eath Initial tie: bob at hihest point Final tie: duin swin Types of eney to conside Gavitational potential eney otational kinetic eney ssue: No eney tansfe due to fiction in suspension o ai esistance Massless stin Syste: bob & Eath Initial Eney v=0 -h h +y y o =0 E i =h E tansfe = 0 Consevation of Eney: E f - E i = E tansfe w = d dt Iw + y- h= 0 need y in tes of = - y I= y = - Final Eney w y -y E f = Iw + y Consevation of eney: Iw + y- h= 0 d + ( - ) -h= 0 d h = 0 Ł dt ł» - d - - Ł ł +-h= 0 d + - h= 0 d + -h = 0 Fo sall anles

3 Fo a peiodic solution Guess = sin(pft +f) Plu it in to check d + - h = 0 Ł dt ł d = ( pft+ f)b dt ( b ( pft+ f) ) + ( sin ( pft+f) ) -h= 0 ( pf ) ( pft+f) + sin ( pft+f) -h = 0 ( pf ) ( pft+f) + sin ( pft+f) = h Only possible if ( pf ) ( pft+f) + sin ( pft+ f) Does not depend on tie ( pft+ f) + sin ( pft+ f) = This is independent of t ill esult fo ( pf ) ( pft +f) + sin ( pft + f) if ( pf ) = f = p ( pft+f) + sin ( pft+f) = h = h = h Check a = h = is a v=0 -h h Fo sall anles, sin = -( -h) h-h h( -h) = = Fo sall, >>h h( - h) h( ) = = h This aees with the solution = h = sin(pft +f) f = p Eaple You fiend is desinin a decoative clock fo a class poject. The plan calls fo the tiin of the clock to be eulated by a swinin in. The in is suspended at its top so that it swins back and foth. You fiend asks you to deteine how the size and ass of the in affects the tiin of the clock. pendulu M Find the feuency of a in of ass M and adius. Use toues Fee body diaa of in t = F t = -Msin = I d dt -M = I d dt t = Ia a = dw dt = d d dtł dtł = d dt fo sall anles uess peiodic solution as = ( pft +f) Find I -M = I d dt d = - sin pft + f dt ( )pf d = - ( pft+ f )( dt pf) -M( pft + f)= -I( pft +f) ( pf ) M = I( pf ) p M I f = /T Find I fo in about cente of ass Use that to find I about ais of otation

4 I = Md + Ico Moent of inetia aound the cente of ass plus oent inetia of cente of ass d = Fo in pendulu p M I I = M + M = M p Paallel ais theoe p M M I co = M p = T Copae stin pendulu with in pendulu Fo what lenth of stin pendulu do you et the sae peiod as a in pendulu? Stin -sin = d dt p p = T in -M = I d dt p Sae peiod if = p = T s in lab deteine the otion of the object Glide on aitack connected to two spins as an eaple t euilibiu Displace fo euilibiu and elease Know: ass of lide : spin constant of each spin : k, k Question: hat is position of lide as a function of tie? Use: consevation of eney: Initial tie Syste: lide & spins Initial tie: just afte elease Final tie: any tie afte that ssue: no fiction assless spins v = 0 k k o Ei = PE (spin ) + PE (spin ) E i = k o + k o Final tie v k k E f = PE (spin ) + PE (spin ) + KE E f = k + k + v Consevation of Eney E f - E i = E tansfe = 0 k + k + v - k o + k o Ł ł = 0 ( k +k ) + v -( k +k ) o = 0 ( k +k ) + d -( k +k ) o = 0 Peiodic solution - not obvious ( k +k ) + d - k Ł dt ł ( +k ) o = 0 = bt+ ( f) (bt+f) epeats in one peiod (bt+ f) = (b(t+t)+ f) bt + f + p = bt + bt + f p = bt b = p T b = pf d dt = -sinbt+f ( )b ( k + k )( bt+ ( f) ) +( - sin( bt+ f)b) -( k +k ) o = 0

5 ( k + k ) ( bt + f) + b sin ( bt + f) -( k + k ) o = 0 ( k + k ) ( bt + f) + b ( - ( bt + f) )- ( k + k ) o = 0 ( k + k ) ( bt + f) - b ( bt + f) + b - ( k + k ) o = 0 Pat of euation vaies with tie that pat = 0 Pat of euation constant that pat = 0 Tie dependent pat ( k +k ) - b = 0 ( k +k ) = b ( k +k ) = b p ( k + k ) sae as with foce appoach!! Tie independent pat of euation b = ( k + k ) o ( k +k ) = k ( +k ) o = o solution to eney euation: = ( pft+ f) o hat about f? t t=0, = o o = o 0 ( + f) = o pft ( ) You knew that f =0

SOLUTIONS TO CONCEPTS CHAPTER 12

SOLUTIONS TO CONCEPTS CHAPTER 12 SOLUTONS TO CONCEPTS CHPTE. Given, 0c. t t 0, 5 c. T 6 sec. So, w sec T 6 t, t 0, 5 c. So, 5 0 sin (w 0 + ) 0 sin Sin / 6 [y sin wt] Equation of displaceent (0c) sin (ii) t t 4 second 8 0 sin 4 0 sin 6