Kinemaic f Paricle Chaper Inrducin Kinemaic: i he branch f dynamic which decribe he min f bdie wihu reference he frce ha eiher caue he min r are generaed a a reul f he min. Kinemaic i fen referred a he gemery f min Example f kinemaic prblem ha engage he aenin f engineer. The deign f cam, gear, linkage, and her machine elemen cnrl r prduce cerain deired min, and The calculain f fligh rajecry fr aircraf, rcke and pacecraf. If he paricle i cnfined a pecified pah, a wih a bead liding alng a fixed wire, i min i aid be Cnrained. Example 1. - A mall rck ied he end f a ring and whirled in a circle underge cnrained min unil he ring break If here are n phyical guide, he min i aid be uncnrained. Example. - Airplane, rcke 1
Le cnider a paricle and i pah f rael The piin f paricle P a any ime can be decribed by pecifying i: Recangular crdinae; X,Y,Z Cylindrical crdinae; r,θ,z pherical crdinae; R, θ,ф Al decribed by meauremen alng he angen and nrmal n he cure(pah ariable). The min f paricle(r rigid bdie) may be decribed by uing crdinae meaured frm fixed reference axi (ablue min analyi) r by uing crdinae meaured frm ming reference axi (relaie min analyi). Recilinear Min I a min in which a paricle ming alng a raigh line(ne-dimeninal min) Cnider a paricle P ming alng a raigh line. The piin f P a any inan f ime i. The piin f P a ime i Diplacemen f P Change in piin f he paricle during he ineral f. ie. ( ) The diplacemen wuld be negaie if he paricle med in he negaie -direcin Diance The al lengh f he pah raced by he paricle (i i alway piie) Aerage elciy: Fr he ime ineral Δ, i i defined a he rai f he diplacemen Δ he ime ineral Δ. V a = A Δ becme maller and apprache zer in he limi, he aerage elciy apprache he inananeu elciy f he paricle. d V lim Va lim S (1) Aerage Accelerain Fr he ime ineral Δ, i i defined a he rai f he change in elciy Δ he ime ineral Δ. a a A Δ becme maller and apprache zer in he limi, he aerage accelerain apprache he inananeu accelerain f he paricle. OR d a lim (a) a lim d d d d (b) Ne:-The accelerain i piie r negaie depending n wheher he elciy increaing r decreaing.
Cnidering equain 1 and a, we ge he fllwing d d and a and d by equaing he w equain a d d a ad d (3a) OR d d (3b) Example Equain 1, and 3 are he differenial equain fr he recilinear min f a paricle. Recilinear min are led by inegrain f hee baic differenial relain. 1. A paricle a paricle ming in a raigh line, and auming ha i piin i defined by he equain. 3 6 - Where, i expre in ecnd and i in meer. Deermine he elciy and accelerain f he paricle a any ime.. The accelerain f a paricle i gien by, a 4 3 where a i in meer per ecnd quared and i in ecnd. Deermine he elciy and diplacemen a funcin ime. The iniial diplacemen a = i =-5m, and he iniial elciy i =3m/. 3
Graphical Repreenain f Relainhip Amng,, and Graph f V By cnrucing angen he cure a any ime, we bain he lpe, he lpe f he - cure a any ime inan gie he inananeu elciy =d/ Graph f V The area under he - cure i he ne diplacemen f he paricle during he ineral frm 1. The lpe he - cure a any inan gie he inananeu accelerain. The area under he - cure i he ne diplacemen f he paricle during he ineral frm 1. da d, d A 1 d 1 1 Equaing he w equain gie 1 A d, A 1 Graph f a V The area under he a- cure during ime i he ne change in elciy f he paricle beween 1 and. The area under he a- cure i he ne change in elciy f he paricle 1. da a A a 1 d a, a d a d 1 1 Equaing he w equain gie A d A 1 1 4
Graph f a V The ne area under he a- cure can be fund da ad A ad 1 d ad d 1 1 ad Equaing he w equain gie A 1 d V V1 A d ince, an d 1 Similar Triangle CB an d CB d CBd, d ad d Equaing he w equain gie CBd ad CB a Frm he graph diance CB i accelerain The graphical repreenain decribed are ueful fr:- iualizing he relainhip amng he eeral min quaniie. apprximaing reul by graphical inegrain r differeniain. experimenal daa and min ha inle dicninuu relainhip b/n ariable are frequenly analyzed graphically. Mehd fr deermining he elciy and diplacemen Funcin 5
Cnan Accelerain A he beginning f he ineral,, d a a d d a a a uing d ad d a ad a a( ) Uing d d d ( a) a 1 a Thee relain are necearily rericed he pecial cae where he accelerain i cnan. The inegrain limi depend n he iniial and final cndiin and fr a gien prblem may be differen frm he ued here. Typically, cndiin f min are pecified by he ype f accelerain experienced by he paricle. Deerminain f elciy and piin require w ucceie inegrain. Three clae f min may be defined fr: - accelerain gien a a funcin f ime, a = f() - accelerain gien a a funcin f piin, a = f(x) - accelerain gien a a funcin f elciy, a = f() 6
Accelerain gien a a funcin f ime, a=f() d a f d d d f f d 1 1 f d Accelerain gien a a funcin f diplacemen, a = f() d ad d f d d f d f d 1 1 f d Accelerain gien a a funcin f elciy, a=f() d d f a f d f d f d d ad d f () d d f d f d d f Example 1. The piin f a paricle which me alng a raigh line i defined by he relain 3 S 6 15 4 Where S i expreed in mand in ecnd. Deermine: a) The ime which he elciy will be zer. b) The piin and diance raelled by he paricle a ha ime. c) The accelerain f he paricle a ha ime. d) The diance raelled by he paricle beween 4 ec and 6 ec. 7
. A girl rll a ball up an incline and allw i reurn her. Fr he angle and ball inled, he accelerain f he ball alng he incline i cnan a.5g, direced dwn he incline. If he ball i releaed wih a peed f 4 m/, deermine he diance i me up he incline befre reering i direcin and he al ime required fr he ball reurn he child hand. 3.The main elear A f he CN Twer in Trn rie abu 35 m and fr m f i run ha a cnan peed f km/h. Aume ha bh he accelerain and decelerain hae a cnan 1 magniude f and deermine he ime durain 4 g f he elear run. 4. A priner reache hi p peed f 1.6m/ in ecnd frm re wih eenially cnan accelerain. If he mainain hi peed and cer he 1m diance in 1.5, find he accelerain ineral and hi aerage aring accelerain a. 5. A mrcycle parlman ar frm re a A w ecnd afer a car, peeding a he cnan rae f 1 km/h, pae pin A. If he parlman accelerae a he rae f 6 m/ unil he reache hi maximum permiible peed f 15 km/h, which he mainain, i calculae l he diance frm pin A he pin a which be erake he car. 8
6. When he effec f aerdynamic drag i included, he y-accelerain f a baeball ming erically upward i, while he accelerain when he ball i ming dwnward i, where kia piie cnan and i he peed in fee per ecnd. If he ball i hrwn upward a 3 m/ec frm eenially grund leel, cmpue i maximum heigh h and i peed upn impac wih he grund. Take k be.66m -1 and aume ha g i cnan. 9