NEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).

Transcription

1 EWTO S LAWS Consder two partcles If 1 0 then 0 wth p 1 m1v These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles as vewed from an nertal coordnate system Where R s the locaton of the th partcle relatve to the orgn of the nertal system, R s the vector from the orgn to a specal pont, to be chosen later, and r s the vector from the specal pont to the th partcle. break nto the ernal force on the th partcle (force due to objects other than the partcles) and the force due to the other -1 partcles j j j1

2 j j j1 There wll be of these equatons (one for each of the partcles). We add these equatons together to get d p, j 1 j,j 1 ow from ewton s 3 rd law j j. Hence j j,j d p 1 ow normally m s constant (total number of partcles may change but the ndvdual masses do not). Hence dv m ma Snce R R r V V v A A a m A a MA ma where M m We now choose the specal pont so that

3 mr0 mv 0 ma 0 MA Ths specal pont s called the center of mass. Roughly speakng, t s the geometrc center of the cluster of partcles. The equaton s the lnear form of ewton s 2 nd Law. We can also get an equvalent rotatonal form by defnng rotatonal analoges to momentum, velocty, force and mass. We start wth velocty. We defne angular velocty as a vector lyng along the axs of rotaton, pontng n the drecton gven by the rght hand rule: curl fngers of rght hand n drecton of rotaton thumb ponts n drecton of. The magntude of s the number of rad/sec. defne torque as R where R s the vector from the orgn of an nertal system to the pont where s appled. ow consder the system of partcles consdered above dv d m m V v j j j1

4 d R r j R r m V v j j1 dv dv dv dv R R r r mr m r R m m r j j j j j1 j1 As before, there wll be of these equatons. We add them up to get (1) R R r j r j j 1 j,j1,j1 dv dv dv dv MR mr R m mr 1 We now consder each term n turn. R s the torque about the nertal orgn due to the net ernal force on the system. R R 00 j j,j1 as before r 1 s the torque about the center of mass due to ernal forces. r j r1 2 1 r2 12 j,j1 r121r212 r r 1 1 Ths s not necessarly zero, but for central forces (force les along the lne between the masses) t s. ortunately, the most common forces, gravty and electrostatcs, are central. Hence, n most cases ths term s zero. ote, that magnetsm s not generally central.

5 ow defne the angular momentum of a partcle relatve to a pont to be J R P where R s the vector from the pont to the partcle, and P s the momentum of the partcle. Hence note that d R V V R V V R R dr dv dv dv dv d d d dj MR M R V R MV R P Where J s the angular momentum of the center of mass about the nertal orgn. dv d d dj mr rp J Where J s the angular momentum about the center of mass. The last two terms on the RHS of (1) are zero because of the defnton of the center of mass. Puttng t all together we fnd dj dj But Snce dj R R dv dv M M These two terms cancel and we are left wth

6 dj whch s the rotatonal form of ewton s 2 nd Law. consder the analogue of mass. We have p mv We would lke to get somethng lke J I Let s see how ths works out J rm v Suppose the relatve postons of the partcles are fxed (a rgd body). The velocty of the th partcle relatve to the center of mass s due exclusvely to the rotaton. Hence (from the dagram) V I (drected nto the board n the case). But I s gven by the drecton of magntude of r s r. Snce the rsn We have V r

7 Hence J m r r m rr r r Usng Cartesan coordnates ths s 2 m x y z xˆ yˆ zˆ x y z x xˆ y yˆ z ˆ x y z x y z z 2 ˆx x x y z x xx yy zz m yˆ x y z y x y z ẑz x y z z xx yy zz 2 y x y z 2 ˆxm x y z yxy zxz m ŷmyx z xyx zyz ẑmzx y xzy yzy We put ths n the form of a matrx product J I where J Jx x J J y y z z my z mxy mxz 1 I myx mx z myz mzx mzy m x y I P P P I P P P I xx xy xz yx yy yz zx zy zz

8 The I j are called moment of nerta, and the P j products of nerta. The result s that J s not n general parallel to. Ths has mportant practcal consequences as we shall see. However, f the object s symmetrc about the axs of rotaton, the P j wll be zero and we wll get where J I I m 2 We wll now apply these results to varous nterestng problems.