For the plane motion of a rigid body, an additional equation is needed to specify the state of rotation of the body.

Transcription

1 The kecs of rgd bodes reas he relaoshps bewee he exeral forces acg o a body ad he correspodg raslaoal ad roaoal moos of he body. he kecs of he parcle, we foud ha wo force equaos of moo were requred o defe he plae moo of a parcle whose moo has wo lear compoes. For he plae moo of a rgd body, a addoal equao s eeded o specfy he sae of roao of he body. Thus, wo force ad oe mome equaos or her equvale are requred o deerme he sae of rgd-body plae moo.

2 ENERAL EQUATNS F MTN our sudy of Sacs, a geeral sysem of forces acg o a rgd body may be replaced by a resula force appled a a chose po ad a correspodg couple. By replacg he exeral forces by her equvale force-couple sysem whch he resula force acs hrough he mass ceer, we may vsualze he aco of he forces ad he correspodg dyamc respose. Dyamc respose

3 F ma M H a) Releva free-body dagram (FBD) b) Equvale force-couple sysem wh resula force appled hrough c) Kec dagram whch represes he resulg dyamc effecs

4 PLANE MTN EQUATNS Fgure shows a rgd body movg wh plae moo he x-y plae. The mass ceer has a accelerao ad he body has a agular velocy k ad a agular accelerao k. a The agular momeum abou he mass ceer for he represeave parcle m : H m v v : poso vecor relave o of parcle m cos s j H m k Velocy of parcle m

5 The agular momeum abou he mass ceer for he rgd body: s a cosa propery of he body ad s a measure of he roaoal era or ressace o chage roaoal velocy due o he radal dsrbuo of mass aroud he z-axs hrough. (MASS MMENT F NERTA of he body he abou z-axs hrough ) H k H k m k m H m j m j k m j m H H 1 s cos s cos s cos s cos 1 kgm dm m m d d d d d H M H M M

6 Aalyss Procedure he soluo of force-mass-accelerao problems for he plae moo of rgd bodes, he followg seps should be ake afer he codos ad requremes of he problem are clearly md. 1) Kemacs : Frs, defy he class of moo ad he solve ay eeded lear or agular acceleraos whch ca be deermed from gve kemac formao. ) Dagrams: Always draw he complee free-body dagram ad kec dagram. M 3) Apply he hree equaos of moo. ( F ma, )

7 Mass Momes of era Mass mome of era of dm abou he axs, d: r dm d r dm Toal mass mome of era of mass m : d r dm s always posve ad s us s kg. m.

8 Trasfer of Axes for Mass Mome of era: f he mome of era of a body s kow abou a axs passg hrough he mass ceer, may be deermed easly abou ay parallel axs. md d Mass Momes of era for Some Commo eomerc Shapes Th bar Th crcular plae Th recagular plae

9 Radus of yrao, k: The radus of gyrao k of a mass m abou a axs for whch he mome of era s s defed as k m k m Thus k s a measure of he dsrbuo of mass of a gve body abou he axs queso, ad s defo s aalogous o he defo of he radus of gyrao for area momes of era. The mome of era of a body abou a parcular axs s frequely dcaed by specfyg he mass of he body ad he radus of gyrao of he body abou he axs. Whe he expressos for he rad of gyrao are used, he equao becomes k k d

10 1) TRANSLATN a) Reclear Traslao: FBD 0 0 Kec Dagram F F 1 A m d P x F A m d P ma x x F 3 F exeral force Fx max e. f. ma M 0 e. f. M A 0 e. f. M P maxd e. f.

11 b) Curvlear Traslao: F 1 F F 3 FBD 0 0 m d A A F Kec Dagram B d B m a ma m m r mr F ma M e. f. e. f. F ma M A ma d e. f. e. f. M B mad B e. f. 0 ( A 0) + +

12 ) FXED-AXS RTATN For hs moo, all pos he body descrbe crcles abou he roao axs, ad all les of he body have he same agular velocy ad agular accelerao. The accelerao compoes of he mass ceer - coordaes: a r a r Equaos of Moo FBD Kec Dagram F F ma * mr F m r M **

13 For fxed-axs roao, s geerally useful o apply a mome equao drecly abou he roao axs. M mr a Usg rasfer-of-axs relao for mass momes of era; mr mr M mr mrr M o For he case of roao axs hrough s mass ceer : a 0 ad F 0 FBD Kec Dagram M

14 3) ENERAL PLANE MTN The dyamcs of geeral plae moo of a rgd body combes raslao ad roao. FBD Kec Dagram Equaos of moo: F ma M some cases, may be more covee o use he alerave mome relao abou ay po P. M P mad

15 1. The uform 30-kg bar B s secured o he accelerag frame he 30 o poso from he horzoal by he hge a ad roller a A. f he horzoal accelerao of he frame s a=0 m/s, compue he force F A o he roller ad he x- ad y- compoes of he force suppored by he p a. SLUTN FBD mg=30(9.81) N x Kec Dagram d ma=30(0)=60 N F A y M ex. f. mad FA cos s 30 F ma 1110 s x F cos N y + ex. f. ex. f. x y x 45 N y F A 1110 N

16 . The block A ad aached rod have a combed mass of 60 kg ad are cofed o move alog he 60 o gude uder he aco of he 800 N appled force. The uform horzoal rod has a mass of 0 kg ad s welded o he block a B. Frco he gude s eglgble. Compue he bedg mome M exered by he weld o he rod a B.

17 SLUTN FBD x Kec Dagram m T a x =60a x x combed mass of 60 kg, uform horzoal rod has a mass of 0 kg. Compue he bedg mome M exered by he weld o he rod a B. N 60 o W=60(9.81) N a x F x ma x m / s ( 9. 81)s 60 60a x B y FBD of rod KD of rod m 1 a x =0a x B x M W 1 =0(9.81) N M M B ma x d N m M ( s 60

18 3. The parallelogram lkage show moves he vercal plae wh he uform 8 kg bar EF aached o he plae a E by a p whch s welded boh o he plae ad o he bar. A orque (o show) s appled o lk AB hrough s lower p o drve he lks a clockwse dreco. Whe q reaches 60 o, he lks have a agular accelerao ad agular velocy of 6 rad/s ad 3 rad/s, respecvely. For hs sa calculae he magudes of he force F ad orque M suppored by he p a E.

19 8 kg bar, q = 60 o, agular accelerao 6 rad/s ad agular velocy 3 rad/s. calculae he force F ad orque M a E. FBD M E E F 60 o 30o F Kec Dagram + ma ma + + mg=30(9.81) N + a E a ac aa AB CD 3 rad / s AB CD 6 rad / s a r m / s a r E ma N ma N E 7. m / s mad M s cos 30 M 8. N m ccw 7 M E ex. f. + F ma F s F 77. N ex. f. 6 F ma F cos F 10. N 37 ex. f. FE F F N

20 4. The uform 100 kg log s suppored by he wo cables ad used as a baerg ram. f he log s released from res he poso show, calculae he al eso duced each cable mmedaely afer release ad he correspodg agular accelerao of he cables.

21 SLUTN FBD + KD + T A T B ma W=100(9.81) N + ma Whe sars o move, v=0, =0 bu 0 0 a r F 0 T cos A TB mg TA TB d k F ma mg s30 ma a m / s a T A r M d. k. N T B The moo of he log s curvlear raslao. d. k T A N.45 s 60(1.5) T rad / s Legh of he cables B s 60(0.5) 0 3T A + T B * *

22 5. A 18 kg ragular plae s suppored by cables AB ad CD. Whe he plae s he poso show, he agular velocy of he cables s 4 rad/s ccw. A hs sa, calculae he accelerao of he mass ceer of he plae ad he eso each of he cables. A C 4 cm 60 B 60 D 10 cm 0 cm 0 cm Aswer: a 6.3 m / s TAB N TCD N

23 6. The uform 8 kg sleder bar s hged abou a horzoal axs hrough ad released from res he horzoal poso. Deerme he dsace b from he mass ceer o whch wll resul a al agular accelerao of 16 rad/s, ad fd he force R o he bar a jus afer release. SLUTN FBD 18b 1 1 M F F o ml 1 1 mg 80.6 ma b 78.48b 3.84 ma ma KD ma ma bb 9.81b kgm Whe s released, =0 (v=0, a = r=0) bu 0. b a b m N

24 7. The sprg s ucompressed whe he uform sleder bar s he vercal poso show. Deerme he al agular accelerao a of he bar whe s released from res a poso where he bar has bee roaed 30 o clockwse from he poso show. Neglec ay sag of he sprg, whose mass s eglgble.

25 SLUTN Usreched legh of he sprg: 5 l o (l / 4) l l Whe q=30 o, legh of he sprg: Whe q=30 o, sprg force: F sprg 3 l sprg 5 k l l 3 l kl 5 3 ( compresso) W + 60 o + 30 o 60 o. l F sprg l sprg 30 o + ma + ma m M mg cos l 0 4 ma k m l F 4 sprg g l l 1 1 ml ma l 4 l 4