We ma wrie he basic equaion of moion for he paricle, as or F m dg F F linear impulse G dg G G G G change in linear F momenum dg The produc of force and ime is defined as he linear impulse of he force, and his equaion saes ha he oal linear impulse on m equals he corresponding change in linear momenum ofm. d F ma m m G m F G
Alernaiel, we ma wrie G F G I The iniial linear momenum of he bod plus he linear impulse applied o i equals is final linear momenum. m + = G m G m F F m m G G G F m m G G G Fz mz mz Gz Gz Gz
Conseraion of Linear Momenum If he resulan force on a paricle is zero during an ineral of ime, is linear momenum G remains consan. In his case, he linear momenum of he paricle is said o be consered. Linear momenum ma be consered in one direcion, such as, bu no necessaril in he - or z- direcion. G G G m m This equaion epresses he principle of conseraion of linear momenum.
PROBLEMS. The kg lunar lander is descending ono he moon s surface wih a eloci of 6 m/s when is rero-engine is fired. If he engine produces a hrus T for 4 s which aries wih he ime as shown and hen cus off, calculae he eloci of he lander when =5 s, assuming ha i has no e landed. Graiaional acceleraion a he moon s surface is.6 m/s.
SOLUTION
PROBLEMS. The 9-kg block is moing o he righ wih a eloci of.6 m/s on a horizonal surface when a force P is applied o i a ime =. Calculae he eloci of he block when =.4 s. The kineic coefficien of fricion is m k =.3.
SOLUTION.3(88.3) 88.3 9(9.8) N F N N mg N F k f m moion P W=mg N F f =m k N s m m m F direcion in /.83 5.4 9 6.49(.4) 36(.) 7(.).6) 9(.3(88.3) 36 7.4.4..
PROBLEMS 3. A ennis plaer srikes he ennis ball wih her racke while he ball is sill rising. The ball speed before impac wih he racke is =5 m/s and afer impac is speed is = m/s, wih direcions as shown in he figure. If he 6-g ball is in conac wih he racke for.5 s, deermine he magniude of he aerage force R eered b he racke on he ball. Find he angle b made b R wih he horizonal.
SOLUTION =5 m/s = m/s in direcion F R.5.5R m m.6.7 cos.65cos R 4.53 N W=mg in direcion R F.5.5R m m.6(9.8).35.5 R.6 6.49 N sin.65sin R 43. N R an b b 8. 68 R R R R R R b R
PROBLEMS 4. The 4-kg bo has aken a running jump from he upper surface and lands on his 5- kg skaeboard wih a eloci of 5 m/s in he plane of he figure as shown. If his impac wih he skaeboard has a ime duraion of.5 s, deermine he final speed along he horizonal surface and he oal normal force N eered b he surface on he skaeboard wheels during he impac.
(m B +m S )g m B N Linear momenum is consered in -direcion; B mss mb ms 5 cos 3 4 5 3. 85 m / s 4 m in direcion B B 4 m S S.5 N m 5sin 3 N.5 459.8.5 m N 44 N or N.44 B S g kn
H o r m r G
Momen of he resulan force M o r We now differeniae F r H o of a cross produc and obain F m r m abou he origin O is he ecor cross produc wih ime, using he rule for he differeniaion H o d r m r m r m r mr ma M o H o M o M o change in angular momenum oal angularimpulse r m r m H o H o M o Ho F
Conseraion of Angular Momenum H o HO HO This equaion epresses he principle of conseraion of angular momenum.
PROBLEMS. The assembl sars from res and reaches an angular speed of 5 re/min under he acion of a N force T applied o he sring for seconds. Deermine. Neglec fricion and all masses ecep hose of he four 3-kg spheres, which ma be reaed as paricles. SOLUTION M H z z H z. 4 3 T r pulle 5.8 s m.4 5.4 6 sphere rlink sphere z
PROBLEMS. A pendulum consiss of wo 3. kg concenraed masses posiioned as shown on a ligh bu rigid bar. The pendulum is swinging hrough he erical posiion wih a clockwise angular eloci w=6 rad/s when a 5-g bulle raeling wih eloci =3 m/s in he direcion shown srikes he lower mass and becomes embedded in i. Calculae he angular eloci w which he pendulum has immediael afer impac and find he maimum deflecion q of he pendulum.
SOLUTION Angular momenum is consered during impac; M O H O H O, HO H O (3) B B ()() M O r m r m.5 3cos.4 3.6.. 3.6.4.4 B A A ()() A (3).5 3.w.4.4 3.w.. A B B ()() B w.77 rad / s ( ccw) O w w A () () A
(3) () B w q O q Reference line Energ consideraions afer impac; A () (3) T Vg T V (Daum a O) g. 5 3.. 4. 77 3... 77 3.. 9. 8 3.. 5. 49. 8 3.. 9. 8cos q 3.. 5. 49. 8 cos q q 5. o