The Components of Vector B. The Components of Vector B. Vector Components. Component Method of Vector Addition. Vector Components

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1 Upcming eens in PY05 Due ASAP: PY05 prees n WebCT. Submiing i ges yu pin ward yur 5-pin Lecure grade. Please ake i seriusly, bu wha cuns is wheher r n yu submi i, n wheher yu ge hings righ r wrng. Due his Wed.0pm: he firs WebAssign assignmen. Read he -page descripin f WebAssign in he syllabus. Discussins sar his week. Labs sar his week. D he pre-lab assignmen, which is n WebCT, befre ging lab. The Cmpnens f Vecr B Hw much f B is in he -direcin? Hw much is in he y- direcin? Adaped frm Andrew Duffy s The Cmpnens f Vecr B B is -3 unis in he -direcin, while B y is + unis in he y-direcin. Wha is he ecr sum f B and B y? Vecr Cmpnens In general, he cmpnens f a ecr are fund using he gemery f he righ-angled riangle. A ecr pins dwn and he righ. Make he hypenuse f a righ-angled riangle, wih he her w sides parallel he crdinae aes. Use gemery find he magniude f each cmpnen Use he diagram ell yu he sign Vecr Cmpnens Here we use uni ecr nain, where a uni ecr such as ˆ (-ha) has a lengh f ne uni in he -direcin. csθ s, + csθ ˆ Cmpnen Mehd f Vecr Addiin R C sinθ y s, sinθ yˆ y B The enire ecr can be wrien u as: + + csθ ˆ sinθ yˆ y B + C R B y + C y R y

2 Kinemaics and Dynamics Kinemaics deals wih he cnceps ha are needed describe min. Min in One-Dimensin Dynamics deals wih he effec ha frces hae n min. Tgeher, kinemaics and dynamics frm he branch f physics knwn as Mechanics. Disance and Displacemen Eample Prblem Disance is a scalar represening he lengh f sme pah. If yu me 5 meers nrh, Δ + 5 meers nrh. Displacemen is a ecr represening a change in psiin. Is magniude is he sraigh-line disance beween he sar and end pins, while is direcin is he direcin f he sraigh line frm he sar pin he end pin. If yu sar a an iniial psiin and me a final psiin i f, yur displacemen Δ is defined as: Δ f i Nw g he her direcin, wih a displacemen f 3 m suh. Wha is he al disance raeled? Wha is yur ne displacemen? Adaped frm Andrew Duffy s Speed and Velciy Speed is a scalar represening hw fas an bjec is raeling. Velciy is a ecr represening hw fas he displacemen f an bjec is changing wih ime. Ofen ime we nly wan knw he aerage alues (aeraged er ime) f he speed r elciy. al disance aerage speed al ime ne displacemen Δ aerage elciy, r, al ime Δ Insananeus s. Aerage alues Bu smeimes we are ineresed in knwing he insananeus speed r insananeus elciy, i.e., he alues f he speed r elciy a a paricular insan. When driing, wha, in yur car, wuld yu use find yur insananeus speed? If yu drie frm Bsn New Yrk Ciy, wha, in yur car, wuld yu use find yur aerage speed fr he rip? When yu pass he sae rper n he Mass Pike, is he rper ineresed in yur aerage speed r yur insananeus speed?

3 Insananeus s. Aerage alues Smeimes we are ineresed in insananeus speed r insananeus elciy, he alues f he speed r elciy a a paricular insan. When driing, wha, in yur car, wuld yu use find yur insananeus speed? The speedmeer. If yu drie frm Bsn New Yrk Ciy, wha, in yur car, wuld yu use find yur aerage speed fr he rip? When yu pass he sae rper n he Mass Pike, is he rper ineresed in yur aerage speed r yur insananeus speed? Insananeus s. Aerage alues Smeimes we are ineresed in insananeus speed r insananeus elciy, he alues f he speed r elciy a a paricular insan. When driing, wha, in yur car, wuld yu use find yur insananeus speed? The speedmeer. If yu drie frm Bsn New Yrk Ciy, wha, in yur car, wuld yu use find yur aerage speed fr he rip? The dmeer and he clck. When yu pass he sae rper n he Mass Pike, is he rper ineresed in yur aerage speed r yur insananeus speed? Insananeus s. Aerage alues Smeimes we are ineresed in insananeus speed r insananeus elciy, he alues f he speed r elciy a a paricular insan. When driing, wha, in yur car, wuld yu use find yur insananeus speed? The speedmeer. If yu drie frm Bsn New Yrk Ciy, wha, in yur car, wuld yu use find yur aerage speed fr he rip? The dmeer and he clck. When yu pass he sae rper n he Mass Pike, is he rper ineresed in yur aerage speed r yur insananeus speed? Yur insananeus speed. Insananeus s. Aerage elciy ne displacemen Δ aerage elciy, r, al ime Δ lim insananeus elciy 0 Δ This is an inimidaing definiin. I s fen easier, and mre inuiie, find insananeus elciy frm a graph. Cmmn ways represen a min There are many ways represen min. A min diagram recrds he displacemen f an bjec a regular ime inerals. The insananeus elciy f a nedimensinal min (i.e., min alng a sraigh line nly) a ime is deermined by he slpe f he min diagram a ime. Cmmn ways represen a min Anher eample f a min diagram: When s. is n a sraigh line, is slpe a ime is deermined by he slpe f is angen line a ime. Since he slpe f he angen line is differen a differen imes, he elciy f he min represened by a nn-linear (i.e., cured) min diagram is arying wih ime. 3

4 Cmmn ways represen a min - I is als cmmn use a elciy diagram, which recrds he elciy f an bjec a regular ime inerals represen a min. Wrkshee, par - We can als simply describe he min in wrds.. Wha is he elciy a 0 s?. Wha is he elciy a 5 s? 3. Wha is he displacemen during he 0-secnd ineral frm 0 s? 4. Wha is he aerage elciy er he 50-secnd perid? 5. Wha is he aerage speed er he 50-secnd perid? Wrkshee, par Wrkshee, par. The insananeus elciy a 0 s is: Δ + 00 m ( + 50 m) ˆ + (.5 m/s ) ˆ 0 s. The insananeus elciy a 5 s is: zer 3. The displacemen fr ha ineral is: zer Wrkshee, par A Fllw-up eercise Draw he elciy diagram fr he preius prblem. (m/s) The aerage elciy fr he 50 s ineral is: Δ (0 m 50 m) ˆ (.0 m/s ) ˆ 50 s 5. The aerage speed fr he 50 s ineral is: al disance 50 m aerage speed 3.0 m/s al ime 50 s Analysis: 0 0s, (00 50) m/0s +.5 m/s 0 30s, 0 m/s 30 50s, (0 00)m/0s -5 m/s (s) Q. Can ne deduce he displacemen a ime frm he elciy diagram? A. Yes. I is he area under he - graph frm ime 0 up ime. 4

Cnsider anher min wih he frward min, bu he reurn min is changed as shwn belw. (m/s) +.5 X Min in he + 0.0 X las eample. -.5 X Min in his eample. -5.

5 Cnsider anher min wih he frward min, bu he reurn min is changed as shwn belw. (m/s) +.5 X Min in he X las eample. -.5 X Min in his eample Aerage Speed (m) (s) Analysis: 0 0s, (00 50) m/0s +.5 m/s 0 30s, (0 00)m/0s -0 m/s (s) Nice ha he ne displacemen f he w mins are he same s he area under he w - graphs mus be he same. Aerage speed Tal disance raeled Tal elapsed ime 50.0 m m 50.0 m/s 5 m/s Aerage Velciy 00 (m) Aerage Velciy 00 (m) (s) (s) ne displacemen aerage elciy al ime -50 m Why can yu jus aerage he +.5 m/s and he -0.0 m/s, ge m/s?.67 m/s Reasn: The bjec spends differen amun f ime raeling a +.5 m/s han a -0.0 m/s The prper apprach is d a ime aerage, in which ne weighs he elciies accrding he ime he bjec spen a each elciy alue ha has been adped in he min. (+.5 m/s) 0 s + (-0.0 m/s) 0 s 5.0 m/s.67 m/s 3 Aerage elciy Fr he elciy diagram shwn belw, wha is aerage elciy f he bjec beween 0 and 5 s? Whiebard Fr he elciy alues lying beween 5 and 36 m/s, he bjec spends an equal amun f ime raeling a each elciy alue. Therefre, he aerage elciy is simply he mean alue f he iniial and final elciy (36 + 5)/ m/s 0.5 m/s. 5

6 Accelerain Accelerain is a ecr represening hw fas, and in wha direcin, an bjec's elciy is changing. Accelerain is he rae f change f elciy. Δ Aerage accelerain: a Δ In he limi ha he ime ineral appraches zer, he aerage accelerain equain gies he insananeus accelerain. If he accelerain is cnsan, he insananeus accelerain is equal he aerage accelerain. Accelerain Insananeus accelerain can be deermined frm a elciy diagram as he slpe he cure a any gien ime. Fr elciy diagrams shwing a sraigh line as abe, he slpe is he same a all imes. Therefre, i represens a min ha has cnsan accelerain. Adaped frm Andrew Duffy s Min under Cnsan Accelerain Min under Cnsan Accelerain 0 a + a The slpes in he abe w diagrams hae he same magniude bu ppsie signs. 0 a a When he sign f and a are he same, he bjec speeds up. When he sign f and a hae he ppsie sign, he bjec slws up. a a a + a N displacemen,, is inled. Min under Cnsan Accelerain Fie kinemaic ariables:. displacemen,. accelerain (cnsan), a 3. final elciy (a ime ), 4. iniial elciy, 5. elapsed ime, Min under Cnsan Accelerain + a ( + ) ( + a) + Aerage elciy + a N final elciy () a ime is inled.

7 Min under Cnsan Accelerain a a ( ) ( ) ( ) + + N elasped ime is inled. a a? Eample : Min f a speed ba + a ( 6.0m s)( 8.0 s) + (.0m s )( 8.0 s) + 0 m? Eample : Launching a Je Knwn parameers: 0m s a a +3m s ( 6 m s) ( 0m s) ( ) 3m s +6m + 6 m s

Kinematics Review Outline

Kinematics Review Outline Kinemaics Review Ouline 1.1.0 Vecrs and Scalars 1.1 One Dimensinal Kinemaics Vecrs have magniude and direcin lacemen; velciy; accelerain sign indicaes direcin + is nrh; eas; up; he righ - is suh; wes;