+ve 10 N. Note we must be careful about writing If mass is not constant: dt dt dt

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Force /N Moet is defied as the prodct of ass ad elocity. It is therefore a ector qatity. A ore geeral ersio of Newto s Secod Law is that force is the rate of chage of oet.

1 Force /N Moet is defied as the prodct of ass ad elocity. It is therefore a ector qatity. A ore geeral ersio of Newto s Secod Law is that force is the rate of chage of oet. I the absece of ay exteral force, the total oet i a syste is therefore costat. The coseratio of oet i a force-etral syste is oe of the ost basic laws of Physics. p dp f dt Moet Newto II dp d d f a dt dt dt If ass is costat i.e. force = ass x acceleratio +e Total oet is cosered i this collisio If a tie-aryig force is actig po a object (i a particlar directio) the area der the (tie, force) graph will correspod to a oet chage. The iplse p is the area der the tie, force graph tie /s A oet chage cased by the applicatio of a force is called a iplse I the exaple of the left, the force f is related to tie t by f ( t) t 4t p 4 t 4t dt 0 p t t p p 0 N Note we st be carefl abot writig If ass is ot costat: d d f d dt dt dt d f p dt wold see like a atral thig to write, bt it is i fact ot correct, sice trasforig to a oig (bit ot acceleratig) frae of referece will chage the force. This iolates the priciple of (Galilea) relatiity. i.e. appropriate whe elocities are ch less tha the speed of light. p fdt The correct extesio of Newto s Secod Law whe ass is aryig is: d d f R R dt dt is the relatie elocity of the ejected ass For a space-rocket, there is o exteral force actig. If the relatie elocity of propellat ejected ot the back of the rocket is ad ass ejectio rate is, Newto s Secod Law becoes: d r f t dt t 0 dt t r l r f t r f l r f t f Rocket ass Propellat ass whe t = 0 Matheatics topic hadot: Moet Dr Adrew Frech. PAGE t 0 r f Note axi br tie is ax / t f This is called the Tsiolkosky rocket eqatio

2 I a collisio, althogh oet will always be cosered, the kietic eergy of the collidig objects ay ot. Frictioal losses, the deforatio of oe of the bodies followig collisio etc will extract eergy. A elastic collisio is defied as whe o kietic eergy is lost. To odel this, ad also ielastic collisios (where kietic eergy is lost) we shall defie a paraeter called the coefficiet of restittio. This is defied as the speed of separatio / speed of approach. ollisios i a straight lie +e By coseratio of oet Defie the coefficiet of restittio Hece We ca ow sole for the elocities post-collisio: Special case: Elastic ollisios = It is possible, if soewhat tedios, to show that kietic eergy is cosered i.e. Special case: Ielastic ollisios = 0 i.e. both collidig bodies oe with the sae elocity. They are stck together! I this case the total kietic eergy post-collisio is The kietic eergy loss is E Matheatics topic hadot: Moet Dr Adrew Frech. PAGE

3 Exaple : Fid the ass M, ad the calclate the aot of kietic eergy lost i the collisio. M Exaple : Fid the elocities post-collisio Asse the collisio is elastic. Masses are i kg 4 +e Note the coefficiet of restittio is = 0.5 i this case. +e M 4 By coseratio of oet M M M 5 The aot of kietic eergy lost is E (5)( ) ()( ) (5)() ()( ) E E E 5J By coseratio of oet 4 4() ()() 4 Sice collisio is elastic Sbtractig these eqatios eliiates 5 5 Hece Matheatics topic hadot: Moet Dr Adrew Frech. PAGE

4 Iterestig sceario: two balls dropped together g +e H h g gh Both balls are dropped fro height h. The lower (ad ore assie oe) collides elastically with a hard floor The balls the collide Upper ball rises to height H By coseratio of oet oefficiet of restittio is defied i this case as: Hece: If the collisios are elastic ad Hece H 9h is the axi height the pper ball will rise. This ca be qite a startlig deostratio! For best classroo reslts, se a basketball ad a teis ball. If the collisios are elastic: If the collisios are elastic ad Hece H h Matheatics topic hadot: Moet Dr Adrew Frech. PAGE 4

5 +e Now iagie a stack of N balls dropped together. For breity, let s asse all collisios are elastic. Let the ball asses be i a geoetric ratio By coseratio of oet w Note w w k w Sice collisio is elastic w w k k k k g k k k k k a b k k k a b a b a a b b a ab b 4 a b a a b ab b... N N i N a b a i a N N N a b a k a k k b k k k k a k k k b a N N N a a The recoil elocity of the th ass ca ow be deteried N N k k Usig the s of a geoetric series If oe repeats the aboe aalysis takig ito accot a coefficiet of restittio N N k k The Irish Mooshot (!) A rather f extesio to this is to calclate how ay balls are reqired to case the pper oe to escape fro Earth (!). Let s asse k = ad all collisios are elastic, i.e. = k 4 GM R gh 4 GM log log 4 log log 6 GM R gh R gh log 4 log The latter step asses the 6 ball syste is dropped fro etre. Realistic? Well if the top ball is kg, the botto ball is 5 kg, i.e..6 illio toes! So perhaps the elastic collisio assptio ay ot be a good oe! Matheatics topic hadot: Moet Dr Adrew Frech. PAGE 5

6 Obliqe collisios D Exaple ollisio lie ollisio lie I a obliqe collisio betwee particles, the oly actal collisio is alog a collisio lie joiig the particle cetres. Velocity copoets perpediclar to this lie are chaged. Perpediclar elocities chaged cos cos cos () cos oseratio of oet alog collisio lie si si Restittio (alog collisio lie) si si si si ( ) si ( ) si () () () + () si si ( ) si cos si si cos ( ) ta ta ( ) ta ta ( ) si ( ) ta ta cos cos Exaple: 60 o o 60 cos cos ta 0 o ( ) si o 0 Matheatics topic hadot: Moet Dr Adrew Frech. PAGE 6

7 Obliqe collisios D Geeralized! ollisio lie ollisio lie I a obliqe collisio betwee particles, the oly actal collisio is alog a collisio lie joiig the particle cetres. Velocity copoets perpediclar to this lie are chaged. () Perpediclar elocities chaged cos cos cos cos oseratio of oet alog collisio lie si si si si Restittio (alog collisio lie) si si si si cos cos cos cos si si cos ta cos ta si si cos ta cos ta si si si si cos ta () () () () i () si si ta cos si si cos ta si si cos ta cos ta si si cos ta cos ta si si si si ( ) cos ta si si ( ) cos ta si si ta ( ) cos cos cos cos cos () i () () Matheatics topic hadot: Moet Dr Adrew Frech. PAGE 7

Obliqe collisios D If collisios are ot i a straight lie, a ector aalysis is reqired to work ot the post-collisio elocities To trasfor to the ZMF, we shall sbtract a elocity V fro both asses sch that

8 Obliqe collisios D If collisios are ot i a straight lie, a ector aalysis is reqired to work ot the post-collisio elocities To trasfor to the ZMF, we shall sbtract a elocity V fro both asses sch that the total oet is zero V V 0 V V I the ZMF, the asses are ow collidig i a straight lie. Hece we ca ow write dow the post-collisio elocities i ters of the origial elocities ad the coefficiet of restittio, V V V V I ters of ad the asses, this becoes By coseratio of oet Defie the coefficiet of restittio If = 0, i.e. a ielastic collisio, oe ca show the loss i kietic eergy is gie by E E Howeer, this does t help s isolate ad to do this reqires the Zero Moet Frae (ZMF) Matheatics topic hadot: Moet Dr Adrew Frech. PAGE 8

9 Ball bocig o a horizotal srface A ball is dropped fro rest fro ertical height h oto a horizotal floor. The ipact elocity is (ia coseratio of eergy) gh The ball-floor collisio has a coefficiet of restittio of. The ball therefore leaes the floor with elocity gh g h gh h gh 4 h By coseratio of eergy, the ball rises to ew height h gh ' gh h' Ad therefore the ipact elocity is gh' gh Ad the rebod elocity is gh To geeralize, the distace traelled after boces is 4 D h h h... h D... h D h D h D h Geoetric progressio The fall tie before the first boce is h t gt h g Betwee the first ad secod boce the tie iteral is h' h t g g The tie traelled after boces is h h h h T... g g g g T g... h T g h Geoetric progressio h h T T g g Matheatics topic hadot: Moet Dr Adrew Frech. PAGE 9

Engineering Mechanics Dynamics & Vibrations. Engineering Mechanics Dynamics & Vibrations Plane Motion of a Rigid Body: Equations of Motion

Engineering Mechanics Dynamics & Vibrations. Engineering Mechanics Dynamics & Vibrations Plane Motion of a Rigid Body: Equations of Motion 1/5/013 Egieerig Mechaics Dyaics ad Vibratios Egieerig Mechaics Dyaics & Vibratios Egieerig Mechaics Dyaics & Vibratios Plae Motio of a Rigid Body: Equatios of Motio Motio of a rigid body i plae otio is