IX Mechanics of Rigid Bodies: Planar Motion

Transcription

1 X Mechancs of Rd Bodes: Panar Moton Center of Mass of a Rd Bod Rotaton of a Rd Bod About a Fed As Moent of nerta Penduu, A Genera heore Concernn Anuar Moentu puse and Coson nvovn Rd Bodes. Rd bod: dea s a sste of partces whose reatve postons are fed. or the dstance between an two partces s constant. rue partce? Strct rd? o Stud the case of the rd bod oton that the drecton of the as of rotaton does not chane. o Conservaton Laws: 1. Lnear oentu. Anuar oentu 3. Mechanca ener hod for rd bod. 1

2 Center of ass of a rd bod: For a sste of N partces: For a rd bod: v v dv dv,,, v where: denst dv eeent of voue dv v dv, v dv v dv Setr: f a bod has a pane of setr, the center of ass es n that pane. f a bod has a ne of setr, the center of ass es on that ne. Fndn center of ass: Sod Hesphere a radus r denst d d a For a thn she (ds) s d s s d s For a thn wre, ds surface eeent Setr center of ass on the radus, nora to the Pane face. Eeents: r 1 ( a ) & d d v r d ( a ) d d d, d near eeent a ( a ) d 0 a 0 ( a ) d 3 8 a 3 4

3 Rotatn about a fed as: A rd bod rotates about a fed as: v Anuar Moentu about the as of rotaton: L r v L, r r L L, ( r ) tae as an as of rotaton. v r L orque: Rate of chane of anuar oentu: Knetc ener of rotaton: rot v ( r ) N d L d ( ) d where: r Moentu of nerta Moentu of nerta about as he quantt s constant for a ven bod rotatn about a ven as ( as here). N torque, tota oent of a forces about the as of rotaton. N d 5 6

4 Anao between Rectnear Moton and Rotatona Moton about an as Rectnear Rotatona Poston: Anuar poston: d d Veoct: v Anuar veoct: dv d Acceeraton: a Anuar acceeraton: a dv d d d v v 0 at t 1 0 v0t at t t r Mass: Moent of nerta: Lnear oentu: p v Anuar Moentu: L Force: F orque: N ( rfsn ) F av N d p d L F N Knetc Ener: 1 1 v rot Potenta Ener: V ( ) F( ) d V ( ) N( ) d dv ( ) dv ( ) F ( ) N( ) d d Cacuaton of Moent of nerta: For dscontnuous, scattered sste: r For contnuous bod: r d where: r the perpendcuar dstance of the ass eeent d fro the rotatona as. d eeent of ass (denst factor).(approprate dfferenta) r d r d A r dv For coposte bod (or sste): ( ass per unt enth) ( ass per unt area) ( ass per unt voue) 7 8

5 Perpendcuar-As heore for a Pane Lana: Pane Lana: a bod whose ass s concentrated n a sne pane. he perpendcuar as theore s appcabe to a pane ana of an shape. he Moent of nerta of an pane ana about an as nora to the pane of ana s equa to the su of the oents of nerta about an two utua perpendcuar aes passn throuh the ven as and n n the pane of ana. Parae-As heore: oent of nerta about an as oent of nerta about a parae as passn throuh the center of ass oent of nerta of center of ass about ven as ( = product of ass and the square of the dstance between two aes) o r r r CM r 9 10

6 Radus of Graton: Epressn the oent of nerta of a rd bod n ters of a dstance, defned as:, Radus of Graton dstance fro the as of rotaton where a the ass of the bod s assued to be concentrated. Spe Penduu: Mass A and assess strn reated as rd bod sn A cos oton of penduu rotatona oton n a vertca pane about as throuh o N N ( ) N ( sn ) [ sn orque decrease the ane] N ( ) ( sn ) sn 0 For sa,, sn 0 (S.H.M) cos( t 0 0 ) f, 11 1

7 Phsca Penduu: A rd bod suspended and free to swn under ts own weht about a fed as of rotaton. (Phsca or copound penduu) N N F r sn N o sn sn 0 O As O hs s dentca to the equaton of oton of a spe penduu. For sa,, sn A 1 f 1 f where: the radus of raton about an as c the radus of raton about center of ass Usn parae theore: ( ) Chann as of rotaton fro O to O at a dstance fro center of ass: o 0 (S.H.M.) f cos( t 0 ) 13 14

8 pont O reated to O b ths equaton s caed center of oscaton for O. Sar, O s center of oscaton for O. o or 4 herefore: Known & and easurn vaue of can be easured. Don t need to now CM. (dstance between O & O = ) 15