Lecture 4 Kinetics of a particle Part 3: Impulse and Momentum

Transcription

1 MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an arbirary force Σ F where a and v are he paricle s and, respecively. Boh are measured in an frame of reference. Rearranging he erms and inegrae he equaion of moion using he limis: v = v a = v = v a = o obain he following equaions Σ d= v F m dv v (EQN. ) The erm is referred o as he paricle s. The vecor L has he same direcion as he velociy of he paricle and is magniude is mv. Is uni is given by. Recall Newon s second law of moion which saes ha he force acing on a body is equal o he rae of change of is momenum. The erm is referred o as he. I is a vecor quaniy which quanifies he effecs of a force during he ime he force acs. I has he same direcion as he force, and is magniude has unis of. / 7

2 MEE Engineering Mechanics II Lecure 4 Graphically, he impulse is deermined by he measuring he area under he force-ime graph, beween specific limis (See figure ) Equaion also represens he principle of linear impulse and momenum. Rearranging equaion o obain (EQN. ) Figure Equaion saes ha he iniial momenum of he paricle a ime plus he sum of all impulses applied o he paricle beween ime and mus be equivalen o he final momenum of he paricle a ime. Equaion is in he vecor form, which can be resolved ino hree componens o obain he scalar equaions in x-, y-, and z-direcions, given by (EQN. A) The principle of linear impulse and momenum will also apply o a. In he case of muliple paricles in he sysem, we can derive he principle of linear impulse and momenum using he of he momena / 7

3 MEE Engineering Mechanics II Lecure 4 and impulses from all paricles. Using he same derivaion echnique as before, we can obain (EQN. 3) From equaion3, we can derive anoher imporan relaionship beween impulses and momenum. When he sum of he exernal impulses acing on he sysem of paricle is zero, we obain (EQN. 4) which saes ha he oal and final momena are equal. This equaion is referred o as he conservaion of linear momenum. Impac Examples of impac loadings include he sriking of a hammer on nail, or a golf club on a ball. Impac occurs when. There are generally wo ypes of impac, namely impac and impac. Figure. Cenral impac The cenral impac is characerised by he of he direcion of moion of he paricles mass cenres wih he line of impac. (See figure a) 3 / 7

4 MEE Engineering Mechanics II Lecure 4 There are several key poins abou he cenral impac. A cenral impac will occur if velociies of wo paricles are, and heir magniudes are no equal During he collision, he paricles are hough o be (nonrigid). During his period of deformaion, hey exer deformaion impulse on each oher. I is also during he maximum deformaion when boh paricles move wih a. Afer he maximum deformaion has occurred, he paricles ener he period of where hey reurn o heir original shape (or remain permanenly deformed). This resiuion impulse causes he paricles o depar from each oher. Normally, he deformaion impulse is always greaer han ha of resiuion, i.e. coefficien of resiuion is. The coefficien of resiuion is he of he relaive velociy of he paricles afer he impac o he relaive velociy of he paricles approaches before he impac. I is given by (EQN. 5) where subscrips, denoe insances before and afer impac, respecively subscrips v A, v B denoe velociies of paricle A and paricle B, respecively A collision is said o be when he associaed coefficien of resiuion is e=, i.e. he relaive separaion velociy is he same as he relaive approach velociy of he paricles before and afer he collision. This canno be achieved in realiy. 4 / 7

5 MEE Engineering Mechanics II Lecure 4 A is opposie o he perfecly elasic impac. The coefficien of resiuion is e=0 in his case. This ype of impac is characerised by he paricles sharing a common velociy afer he impac has occurred, i.e. he wo paricles are suck ogeher during he collision.. Oblique impac An oblique impac is characerised by he moion of one or boh of he paricles is a an angle wih he line of impac. See figure b. In his case, we have o deermine boh x- and y- componens of he velociy of he paricles afer he impac. Angular momenum The angular momenum H O of a paricle abou poin O is defined as he momen of he paricle s linear momenum abou O. This quaniy is someimes referred o as he momen of momenum. Using vecor, he angular momenum of a paricle of mass m is given by where (EQN. 6) r= r i+ r j+ r k denoes a posiion vecor from poin O o he paricle P x y z v= v i+ v j+ v k denoes he paricle s velociy x y z The angular momenum can also be compued using scalar formulaion. Suppose ha he paricle s moion lies in he x-y plane. The magniude of he angular momenum abou he z-axis abou poin O is, herefore, given by where d denoes he perpendicular disance from he origin o he line of acion of mv. See figure 3b for illusraion. 5 / 7

6 MEE Engineering Mechanics II Lecure 4 Figure 3 Recall ha he relaionship beween he resulan force Σ F and momenum of a paricle is given by ΣF= L & = d d ( mv) where L & denoes he ime rae of change of paricle s linear momenum. The relaionship of he angular momenum H O and he resulan momen poin O of a paricle akes a similar form, which is given by ΣM O abou (EQN. 7) Noe ha equaion 7 is also applicable o sysems of muliple paricles Angular impulse and momenum principles Le us rewrie and inegrae equaion 7 beween he limis H O = ( H O ) o obain he relaionship =, O ( H O ) H = and =, 6 / 7

7 MEE Engineering Mechanics II Lecure 4 Σ MO d = ( HO) ( HO) or (EQN. 8) Equaion 8 is referred o as he principle of angular impulse and momenum. The erm Σ M O d is called he angular impulse, hence equaion 8 may be inerpreed as he iniial angular momenum plus he angular impulse mus equal he final angular momenum. This form is again similar o he linear momenum counerpar. Noe ha he angular impulse is given by Σ M d O = (EQN. 9) where r is a posiion vecor which exends from poin O o any poin on he line of acion of force F. Finally, when he angular impulses acing on a paricle are zero during he ime inerval < <, equaion 8 reduces o (EQN.0) This equaion is known as he conservaion of angular momenum. 7 / 7