INDEX:assign 2---2.11,14,17,22,29,31,34 assign 3---3.2,7assign 4---3.6,10,14,15,19 assign 4--- 4.4assign 5--- 4.3,11,13,21,23assign 6--- 4.27 assign 12---5.2 assign 6---6.9assign 7---6.11,18,20,21assign 8---6.24,26,32,36 assign 8---7.1assign 9---7.2,4,5assign 10---7.5 assign 10---8.2,5,9,10 assign 11---9.6,8,10 assign 12---10.3,5
Assignment 1. 2-Sept-2005
Please note that this problem set assumes that you remember somestuff from your cegep/U0 courses in mechanics and vector algebra
1. Kleppner and Kolenkow, chapt.1, problem 1.5.
2. Kleppner and Kolenkow, chapt.1, problem 1.11.
3. Kleppner and Kolenkow, chapt.1, problem 1.12.
4. Kleppner and Kolenkow, chapt.1, problem 1.16.
5. Kleppner and Kolenkow, chapt.1, problem 1.19. The answers areto be expressed in terms of their x and y components. Sketch theposition as a function of time.
Assignment 2. 9-Sept-2005
1. Kleppner and Kolenkow, chapt.2, problem 2.11. In what way would thephysics be different if the bottom string is replaced by a rod?
2. Kleppner and Kolenkow, chapt.2, problem 2.14.
3. Kleppner and Kolenkow, chapt.2, problem 2.17.
4. Kleppner and Kolenkow, chapt.2, problem 2.22.
5. Kleppner and Kolenkow, chapt.2, problem 2.29.
6. Kleppner and Kolenkow, chapt.2, problem 2.31.
����������� �������������������������
"!$#&%('�)+*,#&)+-/.0)21� "34365
798�:<;�=�>@? A�BDCFE�G�A�AIHJLKM�NFO�PQO�RSPQO<TUNFO�V+PUR&WYXZTUR[TUXZ\+]F]^RS_4`a\SPcb�O<TUN^R[Ted^fF_Z_hgeiF\[jk]2j�RlPci6mnTUNFO�TUO�]^gQXo\S]p`a\SPcb�Oqg@infFOTQ\�O�RSbcNrgsTQPUXZ]^V+g�mSRl]^itTUNFOeb�O�]2TQPUXZd"O�TcRl_^`a\SPcb�O@infFO@TQ\<TUNFO@PU\lTURlTQXo\S]�\l`4TUNFO@urRSgUgvRlPU\Sf^]^iTQN^O�R[wnXog�xy Kz,XZPcgsT�mYTQN^O�b�O�]2TUPQXod$O�TURS_4`a\SPcb�OIXog@VSXoWSO�]�{Y|
}�~L�����q�0���fnTL�sP���Xhg�VSXoWSO�]�{Y| �<���l�s�o���� \�j�O�V+O�Tq� }�~L���p�[�s�o�k�[�0���O�jkXo_o_"f$gsO�TQN^OIb�|Y_ZXo]^inPUXob�Rl_4gs|ngsTQO�u�\l`�b�\Y\SPcinXZ]$R[TQOqg0jkN^O�PUOeTUNFOI����R[wnXogLXhg�TQNFOIR[wnXhgL\S`PU\lTURlTQXo\S]�x�M�NFOI`a\+PUb�O�gkRlPUO�O�wndFPUO�gUgsOqi6mnXZ]�TQN^Xog@b�\Y\SPcinXZ]$R[TQO<gQ|ng�TUO�u�R+g
�} ~ ���p�[�s�o�k�[� �^���}��I����� �
�¡4¢�£<�¥¤k¡6¢�£,¦�§2�n� ��e¨©¡6¢�£��Q�o��� � �¡4ªZ«Q¬��¥¤k¡4ªZ«Q¬�¦�§2�n� ��I¤¡4ªZ«Q¬��Q�o�k� �
M�NFO�]�m�gsXo]^b�O�TUNFOurR+gQg�Xog�®FwnOqi°¯±R+b�b�O�_oO�PcR[TQXo\S]²XogI³+´eXo]��sP��RS]^i��s�µ�b�\2\+PUiFXZ]^RlTQOqg�m"jLON^R&W+O ¶·} ·Z¸ ��¹
¶·} ·Zº ��¹
»
jkNFXhbcN<Xog���f^gsT,R@gsTURlTQO�u�O�]+T,\l`��@O�j�TQ\+]�� g�_hR&j<x���]<TQN^OvPs��b�\Y\+PUinXo]^RlTQOvRl]$iI����b�\Y\SPcinXo]^R[TUO�g�mj�O�N$R&WSO } ~ ¤�¡4¢�£ ¦�§+�F�<¤¡ ªZ«Q¬ ¦�§+�n��� ¹���<¨©¡6¢�£��Q�Z�k�<¤�¡ ªZ«Q¬ �Q�Z�k��� ¹
��O�N^R&WSO��rO�2f^R[TUXZ\+]^g@RS]^i�tfF]��Y]F\[jk]^g�m$gQ\rj�O�b�RS] gs\+_ZW+O�TUNFOgQ|ng�TUO�u \l` O�2f^RlTQXo\S]�x� _hgQ\^mFgQXZ]^b�O�TQNFO�RS]FVS_oOIXog� ��iFO�VSPUO�Oqg�mYj�O<N^R&WSOI�Q�Z�k��� ¦�§+�F����� �� xvM�NYf^g�j�O<VSO�T
¡"ªo«s¬/���� �p� �0��¤�� �� ���
¡ ¢�£ � �� �p� � � ¨�� �� ���
��K��] TUNFXhg��2fFOqg�TUXZ\+]�m2jLO�j�RS]+TLTQ\��Y]F\[j�jkN^R[T�jL\+fF_hi N^RSdFd$O�] XZ` TQN^O�_o\[jLO�PLgsTQPUXZ]^V�j�\SfF_hi{"O�bcN^RS]FVSOqi/{Y|/R²PQ\ni6x/M�NFO�TQO�]$gsXo\S] Xo] TUNFO�fFdFd"O�P�g�TUPQXo]FV²jkXo_o_L{"O�TUNFOpgQRSutO+m�TQNFOurR[TUNFO�urR[TUXob�Rl_FO�wndFPQOqgQgQXo\S]�jkXZ_o_^{"O@TUNFOegURlu�OSx M�NFOeu�RSXZ] iFX��4O�PUO�]^b�O@Xhg TQN$R[TvTUNFO@`a\+PUb�OO�wnO�PsTUO�i{2|²TQNFO�PQ\ni/b�RS]�{"Ord"\+gQX TUXZW+Or\SP<]FO�V+R[TUXZW+OSx�M�NYf^g�m Xo]/TQN^O b�RSgQOr\l`�TQN^O PQ\ni6mTQN^O<O�wndFPUO�gUgsXo\S]pRlPUO<Rl_oj�R&|ng�W[Rl_oXoi�`a\+P@Rl_o_�W[Rl_ofFO�gk\S`��@x���]�TQNFO�b�RSgQOIjkNFO�PUOITQNFO�PQO<j�RSg�tg�TUPQXo]FV2g�mYTUNFO�PUOIj�R+g�R�u�Xo]FXou�fFuDW&RS_Zf^O�\S`�� `a\SPkjkNFXhbcNp\SfFP�O�wYd^PQOqgQgQXZ\+]�j�RSg�W[Rl_oXhi6x798�:<;�=�>@? E�BDCFE�G�A �rHM�\egs\+_ZW+O�TQNFXhg�d^PQ\+{F_ZO�upmqj�O0gsTURSPsT {Y|Ib�\+]^gsXhinO�PQXo]FV�TQNFO0`a\SPcb�Oqg6\S]�OqRSbcNIurRSgUg gQO�d^RSPURlTQO�_Z|+mRl]$iRSdFdF_o|!�@O�j�TQ\+]�� gI_hR&jD¯#" } ���%$´�\S]/O�R+bcNurRSgUg�xt��OrVSO�T<TQNFO�`a\+_Z_o\[jkXo]FV%&O�2f^RlTQXo\S]^g�x �(')$*'I� ¡�'
�(+,$*+0��¡-+��~,$n~�� ¤k¡�~ ¨ �p~s�
��O�N$R&WSO�]F\[j.&O�2f^RlTQXo\S]^gkRS]^i0/tfF]��Y]F\[jk]$g�x���O<]FO�Oqi1&t\lTUNFO�P�O2�+f$R[TQXo\S]$g0TU\rgs\+_ZW+OTQN^O gs|ngsTQO�u \l`IO�2f^R[TUXZ\+]^g�x M�N^O�gQO�\STQNFO�PrO�2f^RlTQXo\S]^grb�Rl] {$Op`a\+fF]^i {Y|°_o\2\3�YXZ]FV/R[Tb�\+]^gsTQPcRlXo]+Tcg�\S]pTUNFO�gQ|YgsTQO�upx�z,XoPUgsT�m"{$Oqb�RSf^gsOITUNFO�urRSgUgQO�gk� ' Rl]^ip� + RSPQO<_oXZ]4�SO�i�{Y|TQN^O�gQRSutO<gsTQPUXZ]^V^mYTQN^OITQO�]$gsXo\S]^gkRSPQO�TQN^O�gQRSutO+x¡�'I��¡�+
M�NFOqgsO<b�RS]�RS_ogQ\�{$O<PUO�_hR[TUO�i TQ\�TQN^OITQO�]$gsXo\S]�¡�~I�¡ ~ � ¡�'k¨ ¡-+
z,XZ]$Rl_o_Z|+mYTQNFO<_oO�]FVSTQN��,\l`�TQNFO�gsTQPUXZ]^VtXhg���65 ~798;: ¤<5=' 798;: ¨?>�@ ¨?5 ~)7A8;: ¤<5=+ 7A8;:
�
¯A5 · Xhg6TUNFO0d"\+gQXZTQXo\S]�\S`YurR+gQg"��X �cm������vXog�TQNFOLPUR+inXof^g�\S`2TUNFO0d^fF_Z_oO�|F´s�L|�TcR �YXZ]^VkTQNFO�gsOqb�\+]^iinO�PQXoW[R[TQXoWSO�jkXZTQNpPQOqgsd"O�b�TkTQ\tTQXou�O<\l`,TQNFXhgkO�wYd^PQOqgQgQXZ\+]�mYjLOIV+O�T�TQNFOI®$]^Rl_ b�\S]^gsTQPcRlXo]2T�x$n~L� $ ' ¨ $ +
��@\[j<mnj�OIN^R&WSO /�O�2f^RlTQXo\S]^g�RS]^i(/f^]��Y]F\[jk]^g�x�TUNFO�gQ|YgsTQO�u�b�RS]�{"O�gQ\S_oWSO�i�x,��O<VSO�T
$*'I� � �2� +�� ~�+� ' � + ¨ � + ��~�¨ � ' ��~
$ + � � �Y�('�� ~�S� ' � + ¨ � + ��~�¨ � ' ��~
$n~L� �Y� + ��~�¨©�2� ' �p~�S� '�� +�¨ � +�� ~ ¨ � 'q� ~
798�:<;�=�>@? � BDCFE�G�A���HJLK��O@N^R&WSOkRI{F_o\nb ��\+]�RS]�XZ]^b�_ZXo]FOqi�dF_oRS]FOSx � g0f^gQf^Rl_ mSj�OkdFfnTvTUNFOkwY��R[wnXog�\S`6\SfFPvgQ|ng�TUO�u\l`eb�\Y\SPcinXZ]$R[TQOqg�Xo] TUNFO�gURlu�O�iFXZPUO�b�TQXo\S]°R+g�TUNFO�Xo]^b�_ZXo]FO�i°dF_hRl]FO+x��O�N$R&WSO0& `a\SPcb�O�gRSb�TQXo]FV°\+]�TQN^O²{F_o\nb �"�©]^\SPUu�RS_@`a\+PUb�OSmkTUNFOV+PUR&WYXZTURlTQXo\S]F]^RS_k`a\SPcb�ORl]^i�TQNFO²`aPQXhbµTUXZ\+]`a\SPcb�O+x M�NFOIVSPcR&W2XZTURlTQXo\S]F]$Rl_"`a\SPcb�O�Xog@VSXoWSO�]�{Y|
} ����������s�o� 7 � :} ��� � ¤k���L¦�§2� 7 � :
M�NYf^g�mYTQNFO<]F\+PQurRS_4`a\SPcb�O�Xog � � ����¦�§+� 7 � :
Rl]$i TUNFOI`aPQXhbµTUXZ\+]�`a\SPcb�O�Xog }�����¥¤� 6����¦�§+� 7 � :zF\+P�TQNFO<{^_Z\nb �rTU\t{"O�gsTUR[TUXobSmFj�O�]^O�O�i ¶
·�} · � ¹
��]/TQN^O |2� inXoPUO�bµTUXZ\+]�m�TUNFO�]^\SPUu�RS_�`a\SPcb�O�RS]^iTQNFO�VSPcR&W2XZTURlTQXo\S]F]$Rl_�`a\SPcb�O b�Rl]^b�O�_hg<O�R+bcN\lTUNFO�Pqx ��]�TQNFOIwY��inXZPUO�b�TQXo\S]�mYj�OIN^R&WSO�����s�o� 7 � : ¤� 6���L¦�§+� 7 � : � ¹
��O�b�Rl]pgQ\S_oWSO�`a\SP�TQN^O�Rl]FV+_ZOI�rRl]^i�j�O�V+O�T�������c¦����l� 7 :
&
y K��O j�RS]2T<TQ\�®^]$i�TQNFO u�Xo]FXou�fFu RSb�b�O�_ZO�PURlTQXo\S]�PQO2�+f^XZPUO�iTQ\R[TsTcRlXo]�TQNFO�g�TcRl{FXo_ZXZT�|²\S`TQN^O�{F_o\nb �4x�M�\�in\�TUN^R[TqmnjLO�dFfnTk\+fFPUgQO�_oWSOqgvXo]�TUNFOIRSb�b�O�_oO�PcR[TUO�i�`aPcRlu�O�\l`�TUNFOIXZ]^b�_ZXo]FOqidF_hRl]FOr¯ � � ´�x �kO�u�O�u�{"O�P�TUN^R[T@Xo]�TQN^RlT�`aPURSutO+mFj�O�N$R&WSO
�} � � �}(¤�� �$���� �$ �
jkNFO�PQO �} � Xhg,TQNFOk]FO�T `a\SPcb�O�XZ]tTUNFO@RSb�b�O�_oO�PcR[TUO�iI`aPcRlu�O<¯ � � ´�m �$ � Xog,TUNFO@RSb�b�O�_ZO�PURlTQXo\S]�gQO�O�]Xo] � �Zm �} Xog TQN^O0gsf^u \S`nPQOqRl_[`a\+PUb�O�g RS]^iI� �$
�Xhg Rk®$bµTUXZ\+f^g6`a\+PUb�OvinfFOvTU\kTQN^OLRSb�b�O�_ZO�PURlTQXo\S]�$
�\S` � �6jkXZTQN�PQOqgsd"O�b�T�TU\ � x�M \�N^R&W+OtR�g�TcR[TUXobt{F_o\nb ��jkX TUNPUO�gQd$OqbµTITQ\�TUNFO�XZ]^b�_ZXo]FOqidF_hRl]FO ¯±\SPeTQ\ � � ´µm"jLO�]FO�Oqi �$ � � ¹Fx � _hgQ\^m^j�O�gQfFdFd"\+gQOITQN^RlT@TQN^ORSb�b�O�_ZO�PURlTQXo\S]�\l`�TQNFOXo]^b�_oXZ]^O�idF_oRS]FO�XogIN^\SPUXZ��\S]2TURS_ x ��]²TUNFO�|+��inXoPQOqbµTUXZ\+]�m4TQNFXhg<_hRSgsT<O�2f^R[TUXZ\+]²{"O�b�\Su�O�g�m�Xo]\Sf^P@b�RSgQO ¨ }����I¤�($ ��� ¹
M�NYf^g�mYTQNFO<]F\+PQurRS_4`a\SPcb�O�Xog �¥¤@} ��� ¨ �%$ ������L¦�§+� 7 � : ¨ �($L�Q�Z� 7 � :
��]�TQNFOIwY��inXZPUO�b�TQXo\S]�mnj�OIN^R&WSO¤@} ��e¨ }����<¤�($ ����¹
��fnT�m^jLO �Y]F\[j�TUN^R[T@} � mYTQNYf^g@jLOIVSO�T
¤� 7 ���L¦�§+� 7 � : ¨©�($L�Q�o� 7 � :,: ¨ ���L�Q�o� 7 � : ¤�($�¦�§+� 7 � : � ¹��O�b�Rl]pgQ\S_oWSO�`a\SP0�QR&�IRS]^i�TQNFOI®^]$Rl_6PUO�gQfF_ T@Xhg
$� ·�� � � 7 �s�o� 7 � : ¤ I¦�§+� 7 � :,:7 ¦�§2� 7 � : ¨ I�Q�o� 7 � :,:
��K�@\[j<mnj�O<j�Rl]2TkTQ\�®^]$i�TUNFO<urR[wnXou�fFu RSb�b�O�_oO�PcR[TUXZ\+]�x �O<PUO�d"O�RlTkTQNFO�gURlu�O�gsTQO�d^g@RSgXo]²d^RSPsT�{$´µx��e\[jLO�WSO�PqmFTUNFO�gsXoVS]\l` TQN^O`aPUXhbµTQXo\S] `a\SPcb�O�Xhg�]F\[j¥bcN^RS]FVSOqip{$Oqb�Rlf$gsOTQNFO{F_o\nb �"m$jkNFO�]pTUNFO�R+b�b�O�_ZO�PURlTQXo\S]p\l`�TUNFO�j�O�inV+O<XZ]^b�PQOqRSgQO�g�mFj�Rl]2TUg�TU\ VS\�fFd�xkM�NYf^g�m$jLOjkXo_Z_�N$R&WSO }����¨°} ���<¤��($ ��� ¹� ]$i�f$gsXo]FVr}�t�
mnj�O<N^R&WSO 7 ���L¦�§2� 7 � : ¨ �%$��Q�Z� 7 � :;: ¨©���L�Q�Z� 7 � : ¤��%$�¦�§2� 7 � : � ¹
��O�b�Rl]pgQ\S_oWSO�`a\SP0�QR&�IRS]^i�TQNFOI®^]$Rl_6PUO�gQfF_ T@Xhg$�' ����� 7 �s�o� 7 � : ¨ I¦�§+� 7 � :,:
7 ¦�§2� 7 � : ¤� <�Q�Z� 7 � :,:
798�:<;�=�>@? ��BDCFE�GUE�EHJLKz,XZPcgsT�m j�O�j�RS]2T<TQ\�®^]^i�TUNFO�TQO�]^gsXo\S]/R[T<TUNFOrO�]$iFg<\l`�TQN^OrPQ\+d$O+x � XZ]$b�Orj�O RlPUO�b�\S]n�gQXoinO�PQXo]FV�TUNFO�O�]^iFg@\S`�TQNFOPQ\+d$O+m^j�O�b�RS]pTUR �+O<TQN^O�PQ\+d$ORSgeR d$\+XZ]2TeurRSgUgegQf^gQd$O�]^inOqi{Y|%�turRSgUgQ_ZOqgQg@gsTQPUXo]FV+g�mFurR �YXo]FV Rl]�Rl]^VS_oO<�rjkX TUN�PQOqgsd"O�b�TkTQ\rTUNFO�j�Rl_o_ x�M�NFO�]�m^TQNFO�PQORlPUO &�`a\+PUb�O�geR+bµTUXZ]FV�\S]pTQN^O�urRSgUg�m^TQNFOVSPcR&WYX TcR[TQXo\S]^]^Rl_4`a\+PUb�O�� Rl]^ipTUNFO �rTQO�]$gsXo\S]^g¡����9¡ � x�� Xhg@\S]^_Z|�XZ] TQNFO|+��inXoPQOqbµTUXZ\+]�m$RS]^ipTQN^OgQ|YgsTQO�uDXhg�g�TcR[TQXhblx�M�NFO�PUO�`a\+PQO+mFjLON^R&W+OS�
� ¤�¡�� ��¤¡ � �I��¹� ¤ � ¡�¦�§+� 7 � : ��¹
M�NFO�PQO�`a\SPUOSm2TQNFOITUO�]^gQXo\S]�Xog¡�� �
� ¦�§+� 7 � :y K��]�TUNFOb�O�]2TQO�Pe\l` TUNFO�PU\Sd"OSm$TQN^O�TQO�]^gQXZ\+] \S]F_o|�N^RSg�Rl]pwY��b�\Su�d"\S]FO�]2T�xe��fnTIgsXo]^b�O�TQNFOgQfFu \l`<Rl_o_�`a\SPcb�OqgtXogr��O�PU\^m TUNFOpwY� b�\Su�d"\S]FO�]+T RlTtTUNFO�O�]$i°\S`�TUNFO�PQ\+d$O�gQNF\Sf^_oi {$Ob�RS]^b�O�_Z_oO�i°{Y|°Rl] O�2fFXoW[Rl_oO�]2T�TQO�]$gsXo\S] R[T�TQNFO b�O�]2TQO�P�\l`eTUNFO�PU\Sd"OSx M�NFO�PUO�`a\+PQO+m TQNFOTQO�]^gQXZ\+]�RlTkTQNFO�b�O�]2TQO�P@jkXZ_o_�\+]F_Z|�{$OITUNFOIwY� b�\Su�d$\+]FO�]2Tk\l`,TQN^OITQO�]$gsXo\S]�R[TkTUNFO<O�]$i�\S`TQN^O<PQ\+d$O+x ¡ ����¡p�Q�Z� 7 � :
� �� � �S� 7 � :
798�:<;�=�>@? ��BDCFE�GUE���HJLK��]�d$\+_oRSP@b�\Y\SPcinXo]^R[TUO�g�m+TUNFO�RSb�b�O�_ZO�PURlTQXo\S]�Xhg�VSXoWSO�]�{Y|
�$�� 7���¤����S� :l��@¨ 7 � ��I¨ � ��� : ����]�\SfFP@b�RSgQOSmnj�O�N$R&WSO ��I� �����¹�m �����(m ��� ����RS]^i��<����� 8 x M�N2f$gkjLOIV+O�T
�$�¥¤�� � 8 �0� ��k¨ � � � � ��y KM�NFO b�RlPIjkXZ_o_0gsTURSPsT<TU\pgQ_oXoinOrjkN^O�]�� } �������� $���� �� jkX TUN/� ��� � x z^PQ\+u TQN^Xogb�\+]^inXZTQXo\S]�mnj�OIN^R&WSO
�� � �� ����� 8 � ���L¨ �S� �
�
M�NFO�]�mFj�OIb�Rl]pgs\+_ZW+Oe`a\+P�TQNFOITUXZu�O+m^Rl]^i�j�OIVSO�T8 � � � ��¤���� �� � �
� � � ���KM�NFOt`aPUXob�TQXo\S] `a\+PUb�OtgQNF\+fF_oi{"OtXo]TQNFO�\Sd^d$\2gsXZTQOrinXoPQOqbµTUXZ\+]RSg�TUNFOrRSb�b�O�_oO�PcR[TUXZ\+] �}��¤k� �$798�:<;�=�>@? � BDCFE�G � AIHJLKM�NFO�TQ\STURS_6`a\SPcb�O�O�wnO�PQTQOqi�\+]�TQNFO<urRSgUg�{Y|rTQN^O�gsdFPUXo]FVXhg
} �9¤���� 5M�NFO�TQO�]^gQXZ\+]piFfFO�TQ\�TQN^O��gQdFPUXZ]^V�gQO�d$RlPcR[TQO�_Z|rXhg
¡ � �9¤�� � 5 �¡ � �9¤�� � 5 �M�NFO<inXhgsd^_oR+b�O�u�O�]+T@Xhg 5p�65 � ¨?5 �M�NFO�PQO�`a\SPUOSmYjLOIN^R&W+O
} ����� ¡ �� � ¨���� ¡ �� ���fnTegQXZ]^b�O�TQNFO�gQdFPUXZ]FV2g�RlPUOIRS_Z_6_oXo]��SOqi�TU\SV+O�TQN^O�Pqm2j�OIN^R&WSO�TUN^R[T@¡ � ��¡ � � }�xvM�NYf^g�� � � �� � ¨ �� ��kO�TQfFPU]FXo]FVtTQ\r\Sf^PL®$PUgsT@O�2f^RlTQXo\S]�mnXZTeb�RS]�{"O<jkPQXZTsTUO�]pRSg
� �5 7A8;: ¨���� 5 7A8;: � ¹��|b�\Su�d^RlPUXo]FV�jkXZTQN/TUNFOrO�2f^RlTQXo\S]/\l`�u�\lTUXZ\+]\S`�RS]�N$RlPUut\+]FXhb\2gQb�XZ_o_hR[TQ\+P�m jLO�®^]$iTQN$R[T
�©��� � �� �� 7 � � ¨�� � :y K��]�TQN^RlTeb�RSgQOSm �e\Y\ �SO3� gL`a\+PUb�O�Xhg@gQXZu�dF_o|
}� � ¤ 7 � �v¨�� � : 5M�NYf^g�m� � � � �v¨�� ��
/
Assignment 3. 16-Sept-2005
1. Kleppner and Kolenkow, chapt.2, problem 2.34.
2. Kleppner and Kolenkow, chapt.3, problem 3.2
3. Find the position of the center of mass of a right circular cone of height H
and base radius R and uniform mass density λ.
4. Consider the cone of the previous problem with a hole or radius R0 = 0.5R
drilled vertically along its axis. Find the position of the center of mass of
this object.
5. Kleppner and Kolenkow, chapt.3, problem 3.7
Assignment 4. 23-Sept-2005
1. Kleppner and Kolenkow, chapt.3, problem 3.6
2. Kleppner and Kolenkow, chapt.3, problem 3.10
3. Kleppner and Kolenkow, chapt.3, problem 3.14
4. Kleppner and Kolenkow, chapt.3, problem 3.15
5. Kleppner and Kolenkow, chapt.3, problem 3.19
6. Kleppner and Kolenkow, chapt.4, problem 4.4.
����������� ������������������������ !#"%$�&('*),+� -),./.,),'%$10�2�'*3�!54�67"%3
879%:<;>= & : '*?@6A!�B%C%CEDFHG�IKJ�L�MON P QSR7TVUXWZY[]\5^Z\5^�_%\�`Oab\�c�dfe(gih j]kml#cKn o pHqsr�tvuwtyxz�{|\5`f\V}~j�����e(`���\5j�l(��d�}�g|��e�g�d�{|\���kml�g|\���z�{|\�_|`�l��7}�g|����e�`���\ p�� l�g���df{|\���`f}~��d�}�e(g��e�`���\�e���df{�\ jfl�g���_�l�������\K��\�d
� p � � r��7��q¡ �r£¢¥¤¡¦Xx§ x�q � p�� � r��A�r¨¢©¤
ªyg�d�{�l«d���l�jf\�¬ ¤ }~j d�{|\�^�l(jfj�e��]df{�\��|k~l�g|\�l�g�� r }~j d�{|\�^�l(jfj�e��]df{�\�j�l�g��_�l����z�{|\5gi¬|c®\��5l�g�¯�jf\°d�{|\���e�kme«c]}mg|���A}mg|\5^�l�df}~�-\5±A¯�l«d�}�e(g²7³´ q ²A³µ
¢¥¶�x% v· ´ � · µ ¦z�{|\�¸�g�l�k�¹(\�kme��}�dwº�l�g���d�{|\ }mg|}�df}~l�ki�%e(jf}»d�}�e(g�l�`�\�¼�\�`�e��®z�{|\�g�¬�c®\ ��l�g�¸�g���df{|\K¸�g�l�k�%e(jf}»d�}�e(g · ´ q ² ³½ �r¨¢©¤¡¦¶| vp � ¢�r��A��¦
z�{|\Kg|\�¾7dbjwd�\���}mj]dfe�¸�g���d�{|\K}mg|}»d�}ml�k�¹(\�kme7��}»dwº ² ½ �¿z�{�\K¹�\�kme��}�dwºVe���df{�\ �|k~l�g|\ ²<À�Á�Â�Ã�Äc]}�df{V^�l�j�j ¤ c]}mk�k%_%\°��{�l�g|��\<��_%\5�5l�¯�jf\]d�{|\b^�l�j�jÅe��Edf{|\-jXºjXdf\5^Æ��{�l�g|��\<jc]{|\5g�df{|\�|k~l�g|\�jfg�l��Aj°df{|\V�5l�_|km\�����\V��l�g�¯�jf\V��e(g�jf\�`�¹#l�df}me�g�e��¿^�e(^Z\5gAdf¯|^Çdfe���e(^��|¯df\ ² ½_%\5��l�¯�jf\°d�{|\�`�\�}mjOg|e�\�¾7df\5`fg�l�k*��e�`���\<j��¤ ² À�Á�Â�Ã�Ä q� /¤È¢�r�¦ ² ½§ ² ½ q
¤¤É¢�r ²<À�Á�Â�Ã�ÄÊ�}�g�l�kmk�º(¬7df{|\�¸�g�l�ki�%e(jf}»d�}�e(g�e��1d�{|\K�|k~l�g|\�c]}mk�k�_�\· ´ q ² ³À�Á�Â�Ã�Ä ¤ ³¶| vp��1¢�r��A��¦� �r¨¢©¤¡¦ªË��c¿\�¯�jf\-df{|\ �|l�d�lZ��}m¹�\5g�}mg�df{|\K��`fe(_|k�\5^�¬c¿\���\�d· ´�ÌÎÍ(Ï�Ï rÐq�Ñ<¶AÒ�¶®Ó%Ô
Õ
FHG�IKJ�L�MON ��QSR7TVU�P���Yz1eZjfe�km¹�\°df{|}~j��|`�e�_|km\�^�¬c®\°c]}mkmkE¯�jX\-d�{|\�}�^��|¯|k~jf\��Åz�{|\K�\�¸�g|}�df}me�g�e���d�{|\�}�^��|¯�kmjf\�}mj
� ´ � � µ q�� t��t� p� �ÔX¦��Ôe�` r ´ ² ´ � r µ ² µ q � t��t � p� vÔX¦�(Ôªyg�e(¯|`-�5l�jf\�¬�df{|\Z}mg|}»d�}ml�k�¹�\�kme��}�dwº�}~j� |¬EjfeV}mj°d�{|\�}mg|}�df}~l�k1df}m^�\����Ok~jXe�¬%df{|\���e�`���\ p }mj��e(g�jXd�l�gAd�df{�`fe(¯|��{�df}m^Z\(�@z�{|\5`f\���e�`�\�¬Adf{|\Kk~l�jXd]\5±A¯�l�df}me�g�_�\<��e�^�\<j
r ´ ² ´ q p �t��½
(Ô
² ´ qp�Ô ´r ´z�{|\�^�l�j�j�¬|l�d�df{|\K\5g���c]}�kmkE_�\ r ´ q>¤ ¢�r ���}�g���\-df{|\ j�l�g�����\�d�j�}mgAdfeZdf{|\ �5l�`]l«dlZ`�l�df\��°l�g���jf}�g���\-df{|\5`f\�}~j]g|e�jfl�g���}�g�df{�\ ��l�`Ol�d�df{|\K_%\��(}�g|g�}�g|��¬df{�\�g rÐq � Ô ´ �]�Ê�}�g�l�kmk�º(¬c¿\K�(\�d
² ´ qp�Ô ´¤É¢ � Ô ´
FHG�IKJ�L�MON TVQSR7TVU�P���Y����Î{|\5g�l�k�k7d�{|\]a ^�\�g�jwd�l�g���e�g�df{�\���l«dÅ��l�`5¬�df{|\®dfe�d�l�k^Ze(^�\�gAdf¯|^ }mj�¼5\�`�e�� � Ä ´ u�� Ä q� ��z�{|\5`f\]}~jÅg|e�\�¾7df\5`fg�l�k��e�`���\<jÅl����|k�}m\5��e(g�df{|\°jXºjXdf\�^�¬AjXe c®\O��l�g�¯�jf\b��e(g�jX\5`f¹«l�df}me�ge���^�e�^�\�gAdf¯�^�� � � Ä ´ u!� Ä q �  ´ t Ä �
§ � q �#" Äyà ¢ � ´ Á� t%$  �§ � q ��& r('�¢¥¤ ²c]{|\5`f\*)X^,+�}~j]d�{|\ ^�l�j�j]e��\<l���{�^�\�gi¬-)w¯-+K}~j]df{�\�}m`b¹�\5k�e��}»dwº�l�g��.) ² +�}~j]d�{|\ ¹�\5k�e��}»dwºe��1df{|\��%l«d��5l�`<����\K¹«l�g�jfe�km¹�\-��e�`�d�{|\K¹�\5k�e��}�dwº�e���df{�\���l«d���l�`<�² q & r/'¤
�
� �z�{|\��|`�e�_|km\�^£_%\5��e�^�\5j�^�e(`f\���e(^Z��k�}~��l�df\5��c]{|\5g�d�{|\�^Z\5g��w¯|^���e���e�g|\Kl«dOl�df}m^�\��z�{|\O^�\5g�{�l#¹�\Ol�¹(\�kme7��}»dwº�) ' +]`�\�k~l«df}m¹�\�d�e�df{|\-��l�`���g|e�dÅdf{�\O��`�e�¯|g���������\°��l�gV�|\�¸�g|\lK¹(\�kme7��}»dwº ² µ c]{�}m��{�}~jÅd�{|\b¹(\�kme��}�dwº�e��Ed�{|\°�5l�`Å`�\�k~l«d�}�¹(\]dfe df{|\°��`�e�¯|g���l�� df\�` )�+]^�\5g{�l#¹(\��w¯|^���\<�E� z�{|\�^�l(jfj°e��@df{|\Z��l�`<¬*}»�¿g|e�_%e�º��w¯|^���\<� }~j��w¯�jwd ¤ ½ q�¤ ¢ & r � Oe«c®\�¹(\�`<¬�}�� )�+Å^Z\5g��w¯|^��%\5�E¬«}�d�}~j ¤ µ q ¤¨¢� &�� ¦wr ��z�{�\�`�\���e�`�\�¬<d�{|\@^�e�^�\5g(d�¯|^e��1df{|\ �5l�`]e(g���\�)X} +�^Z\5g��w¯|^���\<��}~j��w¯�jXd
� µ qs¤ µ ² µ q�� ¤ ¢ & � ¦wr�� ² µz�{|\5gi¬i¯�jf}�g�����e�g�jf\�`�¹«l«d�}�e(g e��®^�e�^�\�gAdf¯�^�¬1c¿\��5l�g�¸�g���l�g�}»d�\�`�l«d�}�¹(\�\<±(¯�l«df}me�g���e�`df{�\�¹(\�kme7��}»d�}�\<j°jf}mg���\Zc¿\Z�7g|e«cHdf{�\�}mg|}�df}~l�k�¹�\5k�e��}�dwº�¬*c]{|\5g g|e(_�e�º��w¯�^Z�%\5� ² ½ q � ��i\�db¯�j°��e�g�jX}~�\�`Odf{|\Z��l�jf\Kc]{|\5g�df{|\5`f\ }~j�)X}�� Õ +K^Z\5g�}�g�d�{|\��5l�`bl�g���e�g�k�º�e�g|\K^�l�g�w¯|^���j]e(¯d]e���df{|\ �5l�`<���¿e�g�jf\�`�¹«l«df}me�g�e���^Ze(^�\�gAdf¯|^Ðjfl#ºj¿df{�l«d
� µ���� q � µ ¢ � " Â�ç ¤ µ���� ² µ���� q ¤ µ ² µ � ² " Â�à r
O\5`f\(¬ ² " Â�à }~j�df{|\�¹(\�kme7��}»dwº�e��Odf{|\�^�l�g©`f\5kml�df}m¹�\�dfe�df{|\���`�e�¯�g��E��ªËd�}mjZ�(}�¹(\�g�_7º² " Â�à q ² µ � ' ���\ ��l�g��|¯dbdf{|\K\�¾7��km��}»d]\�¾�|`�\5j�jf}�e(g�}mg�d�{|\Kk~l�jXd�\<±A¯�l«d�}�e(gi�� ¤ ¢ &Ð� � Ñ#¦yr�� ² µ���� q � ¤ ¢ &Ð� ¦wr�� ² µ ¢�r/' � r ² µ��\ ��l�g�jf}�^��|km}»��º�l�g�����\�d² µ���� q ² µ ¢
r('¤É¢ & � � Ñ<¦wrz�{|}~j-\5±A¯�l«d�}�e(g���}m¹�\<jb¯�jbd�{|\�}�g���`�\�^�\�gAd°}mg�¹�\�kme��}�dwº�l«d�\5l���{!�w¯|^��i��"°¯|`���eAl�k1}mj°dfe��e(^��|¯df\ ²$# c]{|}~��{�}mj df{|\V¹(\�kme��}�dwº c]{|\5g�a£^�\�g�{�l#¹(\%�w¯�^Z�%\5�i¬}�g�d�\�`�^�j e���df{|\}mg|}»d�}ml�ki¹�\5k�e��}»dwº ² ½ �vc]{|\�g�g|e(_�e�º&�w¯|^���\<�������°jOl�¸�`�jwdOjXdf\���¬|k�\�d' ¢sÑbq & �²$# q ²$#%( � ¢
r('¤)®¯d ²$#%( � q ²$#%( ³
¢ r/'¤É¢�rjf}�g���\�c¿\K��l�g�l�k�c�l#ºj®c]`f}�df\�e�g�\K¹�\�kme��}�dwº�}�g�d�\�`�^�j�e��1d�{|\Ke�g�\-_%\���e(`f\(�z�{A¯�j
²$# q ²$#%( ³¢ r/'¤ ¢ r('¤É¢�r
��\ ��l�g���e(gAdf}mgA¯�\-km}m��\-df{�l�dO¯|gAdf}mkic®\-�(\�d² # q ² ½ ¢
r/'¤ ¢ r('¤É¢�r ¢*�*�*¢ r/'¤É¢ &Ð� Ñ<¦wr�Og���d�{|}~jO��l�g�_%\Kc]`f}�dXd�\�g�l�j+�/jX}mg���\ ² ½ q � �
²$# q r/' #%( �o,.- ½
Ѥɢ0/(r
1
� ��®l�jf\]l ��c]}mkmk���}m¹�\®d�{|\Ok~l�`���\5jXd�¸�g�l�k�¹�\5k�e��}»dwº(��z�{|}~j}~j@�¯�\�dfeKd�{|\O¹�\5k�e��}»dwº e��*df{|\O^�\5g`�\�k~l«df}m¹�\°dfeZd�{|\���`�e�¯|g��i��ªyg���l(jX\Kl ��¬7df{�\�º�l�k�kE{�l#¹(\-lZ¹(\�kme7��}»dwº )X¯-+��� Oe«c®\�¹�\5`5¬Ac]{|\5gdf{�\�º��w¯|^���e�g|\�l«d�l�d�}�^�\(¬1d�{|\��5l�` }~j �Al�}mg|}mg|��¹�\5k�e��}�dwº�l�g���jfe�df{�l�d df{�\�¹�\5k�e��}»dwºe��°df{|\�^�\�g }~j��}m^Z}mg|}~jX{�\5�E� �7}mg���\�d�{|\�}m`�¹�\5k�e��}�dwº�}~j��}�^�}mg|}~jX{|\<�E¬¿df{|\5º¥��l�`f`�º�k�\<jfj^�e�^�\�gAd�¯|^��FHG�IKJ�L�MON ��QSR7TVU�P���Y���z1e�jfe�km¹�\°df{�}mj]��`fe(_|k�\5^�¬c¿\�¯�jf\ aO\5c�dfe�g�h j�k~l#cK�o pHqÎr�xªygVe(¯|`¿�5l�jf\�¬(df{|\-e�g|kmºZ��e�`���\bl(��d�}�g|��e�g�d�{|\°`�e��%\O}~j¿df{|\°��`�l#¹A}�d�l�df}me�g|g�l�k��e(`���\���ªËd�l(��d�je�g�k�º�e(g�l���\5`Xd�l�}mg���l�`fd®e���df{�\K`fe(��\(����\Kc]}mk�ki�5l�kmk*df{|\�^�l�j�j�e��1df{|}~j]jX�%\5��}»¸%�°��l�`fd�e��df{�\K`fe(��\ r�� �Åab\�c�dfe(gih j�kml#c>c]}�kmki_%\5��e(^�\r � ��qÎr�xaOe�df\�{|\5`f\�d�{�l«d r }mj�d�{|\�d�e�d�l�k^�l�j�jKe��®df{|\�`�e��%\���ªË��c®\Vl�j�jf¯|^�\�d�{�l«dKd�{|\Vkm}�g�\5l�`�\5g�jX}�dwºVe��1d�{|\K`fe(��\�}~j]��e(g�jwd�l�gAd5¬7d�{|\K^�l�j�j r�� }~j]�(}�¹(\�g�_Aºr � q
� Á µ Ã(Äw � ·1 �ÔX¦¿qr�·1 vÔX¦
�bjf}mg|� xZq��� ��t� t � ¬|c®\-�(\�d ³ ·� �ÔX¦�Ô ³ �
��·1 �ÔX¦¿q �
z�{|}~j@jf\5��e�g��Ze�`��\5`@�}��%\5`f\5g(d�}ml�k�\<±(¯�l«df}me�g��5l�g�_%\°jfe�km¹�\5�Z_7º�df{|\O^�\�df{|e��e��E��e�g�jXd�l�gAd��e7\�����}�\5gAd�j+�v��� �w�Ål�g7ºV^�l«df{�d�\�¾7df_%e7e�� ���Åz�{|\Kjfe�km¯df}me�g�}~j·1 �ÔX¦¿q�����t ¢��
� ( �t
c]{|\5`f\�� q�� � Á� �z1e�\5¹#l�k�¯�l«df\°df{�\-��e�g�jXd�l�g(d�j � l�g�� � ¬c¿\-{�l#¹�\bdfe�d�l���\-}�g���e(g�jX}~�\5`�l�df}me�g�df{�\-}mg|}�df}~l�k��e(g��}�df}me�g�j5��ªyg�e�¯|`O�5l�jf\�¬7c¿\K{�l#¹�\ ·1 � ¦Åq � ½�·� �ÔX¦�Ô!
t- ½q �
"
ªË��c¿\K��e(^��|¯df\�d�{|\5jf\K±A¯�l�gAdf}�dwºV¯�jf}mg|�Zdf{|\ jfe�km¯df}me�g�¬c¿\K�(\�d·1 � ¦@q���¢�� q � ½�·� �ÔX¦(Ô t- ½q �
� � � � q ���\K{�l#¹�\K��\5±A¯�l«d�}�e(g�j5¬|l�g�����¯|g|�7g|e«c]g�j5�Åz�{|\�¹«l�km¯|\Ke��1df{|\ ��e�g�jXd�l�g(d�j]l�`�\
�>q � q � ½¶z�{|\5`f\���e�`�\°e(¯|`OjXe(k�¯|df}me�g�_�\<��e�^�\<j ·� �ÔX¦®q
� ½ ������� � ÔX¦c]{|\5`f\�c®\-¯�jX\<�
����� v·%¦®q � ¢ � ( ¶
FHG�IKJ�L�MON ��QSR7TVU�P���Yz�{|\-��e�`���\�}~j]��}m¹�\5g�_7º p¡q ��Ô q (r�Ô ² ¢�r
²�Ôz�{|\-`�l�}�gV�vl�kmkmj®l«d]��e(g�jwd�l�gAd¿¹(\�kme��}�dwº�¬jXe�df{|\�jf\5��e(g���d�\�`�^£}mj®¼�\5`fe���z�{�\°^�l�j�j®��l�g�_�\c]`�}»dfdf\�g�l(j r q � ��c]{|\�`�\�)w_ +Z}~j�df{|\Vjf¯|`f�vl���\��|\�g�jf}»dwº�l�g�� )���+�}~j-df{�\Vl�`�\5l|��z�{|\l�`�\5l�}~j]��e�g�jwd�l�gAd5¬|jfe�c¿\���\�d p¡q ��Ô � ² ½��\K�Ag�e«c l�kmkEdf{�\5jf\�±A¯�l�gAdf}�df}m\5j5�Åz�{A¯�jp¡q � � � ¶�� &aOe«cK¬}���df{|\K¹(\�kme7��}»dwº�e���d�{|\K_%e«c]kE}mj ² � ¬|c¿\�c]}mk�k���\�dpHq � ��Ô �� ² ½ ¢ ² � ¦aOe�df\�{|\�`�\Kdf{�l«dbd�{|\�`�l«d�\ }~jbg|e«c¡�}��*\�`�\�gAd5�°ªËd-��l�g�_�\Z`�\�k~l«df\<��dfe�d�{|\Ze�df{�\�`b`�l«d�\ }mgdf{�\���e�kmk�e«c]}mg|�Zc®l#º(��z�{�\-`�l«d�\5j]�5l�g�_�\Kc]`�}»dfdf\5g�l(j �(Ô q �A² ½ � �(Ô q � ² ½ ¢ ² � ¦c]{|\5`f\ � }mj®d�{|\K¹�e(k�¯�^Z\��\5g�jX}�dwº��Åz�{�\��|\�g�jf}»dwº�}mj®d�{|\Kjfl�^Z\�}mg�d�{|\�dwc®eZ�5l�jf\��z�{|\�`�\����e�`�\�¬c®\�{�l#¹�\ � ��Ô q ² ½ ¢ ² �² ½
���
�7eZdf{�\�¸�g�l�kil�g�jfc¿\5`®}~j p¡q � �(Ô � ² ½ ¢ ² � ¦ ³² ½ªË��c¿\��|¯d]d�{|\Kg7¯|^�_%\�`�j5¬c¿\�{�l#¹(\ p¡q � � ����� &FHG�IKJ�L�MON WVQSR ��U���Y��\�c�l�gAdOd�e�¸�g���df{�\�¹�\5k�e��}»dwº�e��d�{|\�jX^�l�kmk�_|k�e��� ��^�l(jfj�) r + �]c]{�\�g }�d�k�\<l#¹�\5jOdf{|\_|}m��e�g|\(�]Ê1}m`�jwd<¬�c®\ ��e�^��|¯|df\Kd�{|\ d�e�d�l�ki\�g�\�`���º�_%\���e(`f\�d�{|\�_�k�e���jOjXd�l�`fdO^�e«¹7}�g|���¿ªËd}~j���}m¹�\�g�_7º � tvuXt q � ¢ � qsr�� �c]{|}~��{ }~j-e�g|kmº�df{|\Z�%e�df\5gAdf}~l�k�\�g|\5`f�(º�e��Åd�{|\�jX^�l�kmk�_|kme7��� �v_�\���e�`�\�}�d jwd�l�`fd�j°^Ze«¹7}mg|��¬}�db}mjOdf{|\�e�g|kmº���e�gAdf`�}m_|¯df}me�g�d�e�df{|\�\�g|\5`f�(º ���®abe�d�\ {|\�`�\Kd�{�l«dbc®\ �|¯|dOdf{|\�� q � l«ddf{�\Z��kml(��\ d�{|\�jX^�l�k�k�_|kme����k�\<l#¹�\5jbdf{�\Z_�}���e�g|\(��z�{|\5gi¬Ec¿\���l�g���e�^��|¯d�\�d�{|\�dfe�d�l�k\�g�\�`���ºVl«� d�\�`�d�{|\ jX^�l�k�kE_|kme���Vkm\5l#¹�\<j���ªËdO}~j
�  ´ t Ä � q � ¢ � � ¢ � ³q Ѷ ¤ ²7³�
¢ Ѷ r ²A³³�7}mg���\�df{�\�`�\�}~jZg|e��|}mj�jX}m��l�df}m¹�\V��e(`���\5j ��km}��(\���`�}~��df}me�g ��¬@c®\���l�g©¯�jf\���e�g�jf\�`�¹«l«df}me�g�e��\�g�\�`���º�dfeVj�l#º�df{�l«d�  ´ t Ä � q � tvuXt ���7e�c¿\�{�l#¹(\°e(¯|`�¸�`�jwd]`�\�k~l«d�}�e(ginѶ ¤ ²A³�¢ Ѷ r ²7³³
qÎr�� �z�{|\5`f\�}mj°g|e�\�¾7df\5`fg�l�k���e(`���\5j5�-z�{�\�`�\���e�`�\�¬%c®\���l�g ¯�jf\���e�g�jf\�`�¹«l«d�}�e(g�e��@^�e�^�\5g(d�¯|^dfe��(\�dbl�jX\<��e(g���`�\�k~l«d�}�e(gi� ¤ ² � qÎr ² ³��\-{�l#¹�\K� \<±(¯�l«df}me�g�j®l�g���� ¯|g��Ag�e«c]g�j�����\ ��l�g�jfe�km¹�\°df{|\��|`�e�_|km\�^��z�{|\�¹�\5k�e��}»dwºe��1df{|\ jf^�l�kmkE_|kme7����c]}�kmki_%\
² ³q� ¶#� � ¤rÆ¢©¤
�
Assignment 5. 30-Sept-2005
1. Kleppner and Kolenkow, chapt.4, problem 4.3
2. Kleppner and Kolenkow, chapt.4, problem 4.6
3. Kleppner and Kolenkow, chapt.4, problem 4.11
4. Kleppner and Kolenkow, chapt.4, problem 4.13
5. Kleppner and Kolenkow, chapt.4, problem 4.21 (part a only).
6. Kleppner and Kolenkow, chapt.4, problem 4.23
Assignment 6. 7-Oct-2005
1. Kleppner and Kolenkow, chapt.4, problem 4.27
2. Solve problem 2.34 in Kleppner and Kolenkow, chapt.2, using the ideas ofenergy and angular momentum.
3. Kleppner and Kolenkow, chapt.6, problem 6.9
����������� ������������������������
!#"%$�&('*),+� -),./.,),'%$10�2�'*3�!54�67"%38*9;:=<?> & : '*@A6B! 8%C%CD9
EGF�HJI�K�LNM O PRQBSUTWV�X�YZ\[�]_^;`baAced_f(gehbi�jlknmoiNm�pq[B]]_fJrs[;t�uvkn]W^eiwpn[exnhbiyfqzDc;pnd{]W`b|�h}iN~�`}[�]W^eiNh�pqg�z�d�pqj�in����imy`bh}hD�;a_i�|�fn[sa{i5d_��p�]W`}f([�fqz�i�[ei5d_x(�vpq[;tUj�f(j�i�[B]_�ejl���oi�z�f(d_i�]_^;i�|�fnhbh}`�a_`}f([�k7c;pqd_]_`�|�hbi-~`�a�j�f��7`b[exvp�]y�ni5h}f�|�`�]������v����^7�;a5k��;a_`}[;x�|�fn[;a_i�dW��p�]W`}f([Ufqz�i�[;i�dWxn�nk7moi |�pn[�aWp#��]_^;pq]
��*� �B�� �
��D� �B��l� � �B��
fnd� �� � � �� � � � ��my^ei5d_i�� � pn[;t�� � pqdWi-]_^eiJ�(i�hbf7|�`�]���pqz�]_i5dy]_^ei |�fnhbh}`�a_`}f([�fnz1c;pnd{]W`b|�h}i ~ pn[;t��-�;��^;i�[�k
moi�|�pn[��;a_i�|�fn[;a_i�dW��p�]_`bfn[vfnz�j�fnj�i�[B]_�;j¡`}[U]_^eiw¢Upq[;tv�vt�`bdWi5|£]W`}f([�����i-^;p#�(iN`}[U]_^ei�Ut�`bdWi5|£]W`}f([�¤
¥%¦¨§�©#ª_«W§¬ �® � ¥ ¬ � � ¥ ¬ � � � � ��¯{°b± uJ² � � � � ¯_°b±´³�µ7¶n·=¸fnd� ��¯{°b± u �?¹ � � �
Z\[�]_^ei�¢7º\t�`}dWi5|�]_`bfn[�¤� ��� � � � ��»�¼ ¯ u � � � � � »�¼ ¯�³½µB¶ · ¸fnd
��� � � ��»�¼ ¯ u � ¹ � � ���^ei5[�kDm¾i�^;p#�(i�¿�i5ÀB�;p�]W`}f([;a�pq[;t�¿U�;[eÁ7[ef�my[;a5�J��iv|�pn[�a_fnhb�ni�]_^ei�a{��a{]_i5j���Âwa_`}[ex]_^;iwÃ-i5ÀB�;p�]W`}f([;a�fqzD|�f([;a_i�dW�#pq]_`bfn[�fqz*j�fnj�i�[B]_�;jÅĽ`}[�]_^;iy¢�pn[;t���t�`bd_i=|£]W`}f([sÆ£k(m¾i�r;[st
��� � � � ³ »�¼ ¯ u � ¯{°b± u ¸��^ei5[�kemoi�|5pq[�ce��]y]W^e`baNi5ÀB�;p�]W`}f([�`b[�]_^eiJf([ei-z�fnd�]W^ei |�f([;a{i5d_��pq]_`bfn[Ufnz1i5[ei�dWxn�(�
� �� ³ »�¼ ¯ u � ¯_°}± u ¸ � � � �� � � �� ¯{°b± � uÇ ¯{°b± � u � »�¼ ¯ � u � � »�¼ ¯ u ¯_°}± u � � � ¯_°}± � u
Ç�È�É ± u � �Ê f�]_^;i pq[ex(h}i�my`bh}hDg%i
u � ÉqË » È�É ±�³ � ¸
Ì
EGF�HJI�K�LNM V�PRQeV�T��JSvY���������
Z\[e`}]_`�pqhbh}���g%i�z�fndWi�]_^;i a�]Wd_`b[ex�`ba�c;�eh}hbi5tsƣk�]W^eiJjvpnaWa�^;pnayp�|�i5d{]�pq`b[�j�f(j�i5[B]_�ejl�� � � �� ������
~wa]_^eiNa{]_dW`}[;x-`�ace�;h}hbi5tDkn]_^eiyd�pnte`}�;apn[;t�]_^eiNpn[exn�;hbpnd�ni�hbf�|�`}]�� my`bh}heg%iN|�^;pn[exni=tD�1��^eij�fnj�i�[B]W�ej my`bh}h�g%i5|�fnj�i
� ³��{¸ � �� ³��{¸ �#� ³��{¸�o���7`}d_]_�;i-fnz1]W^ei |�f([;a_i�dW�#pq]_`bfn[�fqz�pn[exn�;hbpndoj�f(j�i5[B]_�ejlkemoi�^;p#�ni��� � � ³��{¸
�� ��� ��J� �� ³��{¸ � � ³��{¸��^ei�dWp(t�`}�sa � ³��{¸ `bao|�^;pn[exni=tvp�]op�|�f([;a�]�pq[B]A�(i�hbf7|�`�]���� Ä�� «������� �
� � Æ�my`�]W^ � ³ ¸ � � �(���^7�;a
� ³��{¸ � � �N²�� ���^ei5[�kemoi�|5pq[la{f(h}�(iwz�f(d�]_^ei pn[exn�;hbpndo�(i�hbf�|�`}]��n�
³��{¸ � � � ��³ �5� ²�� �{¸ ����������
��^ei-z�fnd�|�i�[ei5i5t�i=t�z�f�c;�eh}h�]_^ei a{]_dW`}[;x�my`bhbh�g;p�]_^ei |�i�[B]_dW`bce��]Wpnh*z�fnd�|�i(��Z¨]N`�ayx(`}�(i�[�gB� ³��{¸ � � � ³��{¸ � ³��{¸
��^7�;a ³��{¸ � �� �� �"!�³ � �N²�� �{¸$#
EGF�HJI�K�LNM ��PRQ&%�T�'�Y� iNmopn[B]]_f�r;[;t�]W^eiNj�fq]W`}f([�fqz*]_^eiwg;pqd=�)(�`bdWa{]5kn]_i5h;�;a�|�fn[;a_`�t�i�d]W^eiy]Wfnd�À(�;i5apqdWfn�;[;ti5p(|�^ ce`b�nfn]�c%fn`b[B]5��*yi�j�i�j�gsi5d�]_^;pq]�]_^eioxnd�p#�7`�]�p�]_`bfn[;[;pqh(z�fnd�|�i¾`bap(|£]_`b[ex-fn[�]_^eiy|�i5[B]_i�dfqz�jvpnaWa�fqz�]_^eiJdWf�tD���^ei�]_f(dWÀB�ei�pqdWfn�e[st�]W^ei�r;d�a�]yce`b�nfn]�c%fn`b[B]N`ba
+�, � ² ³�- �. ³��{¸{¸ �0/ � � -�1 �my^ei5d_i 1 � `ba¾]W^ei�[;fndWj�pnh%z�fnd�|�i�te�eiw]Wf�]_^ei�a{i=|�f([;t�ce`b�nfn]oc%fn`b[B]5���^ei�a{i=|�f([;tv]Wfnd�À(�;imy`bh}h�g%i
+ � � ³2- ² . ³��{¸_¸ ��/ ² � -21 ,
Ã
my^ei5d_i 1 � `�aN]W^ei�[;fndWj�pnh�z�f(dW|�i�t��ei�]_f�]W^ei rsdWa{]�ce`b�nfn]wc%fn`b[(]=����^ei5d_i�`ba�[ef�pn[exn�;hbpndj�fq]W`}f([����^B�sa+ , � + � �®
Ç 1 , � 1 � ² . ³��{¸- �0/
��^ei�]_f(dWÀB�ei�i5ÀB�;pq]_`bfn[ x(`}�(i5a��;a�pUdWi�h�p�]W`}f([ g%i�]�moi�i�[ ]_^eivÃU[efndWjvpqh�z�fnd�|�i(���Nf�mJk*m¾i|�f([;a_`bt�i5dN]W^ei�hb`b[ei5pndwj�fq]W`}f([ fqz]W^ei�g;h}f�|�Á*����^ei5d_i�pqdWi�Ã�z�f(dW|�i5a�pn|£]W`}[;xU^ef(d_`���f([(]�pqhbh}�fn[�]_^;iJgeh}f�|�Á*¤�]_^ei�Ã�z�dW`b|�]_`bfn[�z�fnd�|�i=a��
, ��� 1 , � ��� 1 �
��^ei5[�ke�;a_`}[;x��wi�m�]Wfn[�� a�hbp#mJ¤ , ² � � �. ³��{¸
Ç � ³�1 , ² 1 � ¸ � �. ³��{¸�o��]Nmoi�z�f(�e[;tlp�d_i5hbpq]_`bfn[�gsi�]�m¾i5i�[�]_^efBa{i�[ef(d_jvpnh*z�fnd�|�i5a5�² � /
- . ³��{¸ � . ³��{¸��^e`�a�`bay]_^eiJi=À(�sp�]_`bfn[�fqz�pq[�^;pqdWj�fn[e`�|-f(aW|�`bh}h�p�]Wfnd=����^;i a{f(h}��]W`}f([�`�a
. ³��{¸ ��� ¯_°b± �� � /- ��� ��� »�¼ ¯ �� � /- ����1f�rs[;t�]_^ei |�fn[;a{]Wpn[(]yfnz1j�fq]W`}f([�~ pn[;tl��k�moi��;a_i�]_^eiJ`b[e`}]_`�pqh´|�fn[;t�`}]_`bfn[´�
. ³ ¸ � . �� . ³��{¸� ������ ���� �®
� iJxni�]. ³��{¸ � . � »�¼ ¯
� � /- � �
¿
����������� ����������������������! �"#"%$
��&('*)#+,�.-/+0)��21��43/16587%581�9:�;7#<�=,9>1.?@�4=,)A7#<()B?@=DC%3:�;�.=,?@�.-/+,)�?8��7%9>16)�EF7%->�8=D�4GH)JI>=,GH7#<G2K�5L581.�2?4K#&
H
R
M&ONP+D1.Q/Q�9>1.?R)#9/S�N�7%+D1�9/T#7VUPWX�Y3/)�Q/�.&[ZMW�Q/?47#EX+D1�5\ZM&,�#�
] &ONP+D1.Q/Q�9>1.?R)#9/S�N�7%+D1�9/T#7VUPWX�Y3/)�Q/�.&[ZMW�Q/?47#EX+D1�5\ZM&,��^
�>&ONP+D1.Q/Q�9>1.?R)#9/S�N�7%+D1�9/T#7VUPWX�Y3/)�Q/�.&[ZMW�Q/?47#EX+D1�5\ZM&[ �"
$M&ONP+D1.Q/Q�9>1.?R)#9/S�N�7%+D1�9/T#7VUPWX�Y3/)�Q/�.&[ZMW�Q/?47#EX+D1�5\ZM&[ ��
Assignment 8. 21-Oct-2005
1. Kleppner and Kolenkow, chapt.6, problem 6.24
2. Kleppner and Kolenkow, chapt.6, problem 6.26
3. Kleppner and Kolenkow, chapt.6, problem 6.32
4. Kleppner and Kolenkow, chapt.6, problem 6.36
5. Kleppner and Kolenkow, chapt.7, problem 7.1
����������� ������������������������
!#"%$�&('*),+� -),./.,),'%$10�2�'*3�!54�67"%38:9<;#=?> ':@%6BADCE6B!�F%G*GIH
JLK�MON�P�QSR T UWV7XZY\[-]_^`badcfehgjilkhm\npohqsr�tvuxwyr{z<m\i|c }�n(a<i|gj~:r5m�c\ehr{cfn(m\�B�hr�i-�(}�c\g�ah�vn(adr��p}�ed~hmf�htv����ehr_npa<q��� npm�}�r-c\e<��cS��k<khq�gsr5i���c\npm��B�hr�gji�c\ehrlcfr5a<i|gsnpa � npm�}�r(���ehr�m\r � n(mfr(u:w�rle<�#�(r
�����?�����1�D������L�-� �1������������1� � ����L� � ���
¡y�:cSm\r�t�r�t�o¢r5m�c\ehrOn�c\ehr�mS~hr�z<ahg�cfgsnpavn � c\ehrlcfn(m\�B�hr�£�¤� £¥{¦ £§��¨��e7�<i�5�D� ��© �����
ªhm\npt«c\ehgsi�r5�B�<��cfgsnpaxuBwyr��pr�cy��}�npa¢~:g¬c\g�n(aZo%r�cwyr�r�aZcfehrO®O��a<�p�hqj��my�p}�}�r�qsr�m���c\g�n(a<i¨��a¢~wyr-�(r�cS��avr�¯7k<mfr�ififg�n(a¤n � cfe<rlcfr�a¢i|gsnpax�� � �?� � �?�© � �� �L� �
��ehr5axuhwyrl}5��a�q�n7n(°¤��cS±Sr5w�cfn(ax² iSqs�#w�n(a�~:mf�<t´³ µ¶ § �¸·�¹
º �¼»�½�© � � ¹º »�½ �� � �d�?¹
±Sn�wOu:wyr-ahr�r�~��pahn�c\ehr�m�r��(�¢��cfgsnpa¤cfn_ifnpqs�pr � npm�cfe<rl�(}�}�r�qsr�m���cfgsnpa�����ehrlqsr�ah�pcfevn � cfehrc\�pk¢r(uho%r�cwyr�r�a�cfe<r�®�~hmf�ht_i5uhgji��pgs�pr�a�o7�*µ¾�¿ÁÀ| � ¾¨ÃyÄ ��Å � ÀxÄ ��Å � À
wÆehr5mfr¤¾EÃZgjiOcfe<r¤gsahg¬c\gs�pq�qsr�a<��cfe���` � w�r_c���°(r�®�cfgst�r5i cfehr�~:r�m\g�����cfgs�prZn � ¾�¿ÁÀ| wÆg¬c\em\r5ifk¢r�}�c�cfn{cfgst�rpu¢w�rl�pr�c�cfehr �(}�}�r5q�r5m\��cfgsnpa�n � ~hmf�htdz µ¹��ÉÈ � ¾�¿ÁÀ|Â
È À �� � � � Ä � � � � � � �
Ê
º ��� ¹� ���ehr5axuhwyrl}5��avi|n(q��(r � n(m�cfehr �(}�}�r�qsr�m���cfgsnpa��¹{���� »
JLK�MON�P�QSR [�UWV7XZY\[lX�^ª�g�m�i|c5u7cfe<rOt{n(t�r�aBcSn � g�ahr5m|c\gs� � n(mS�{}��7q�gsa<~:r5m���a<~���i|k<ehr�m\rO��m\r
�������I� ��x�L�l����� �� �� �L�l�
�¸ehr5a c\ehr�i|khe<r�m\r��pa<~ c\ehr�}��7q�gsa<~:r5m_��m\r�n(a cfe<r�gsa<}�q�gsahr5~ khqj��ahr(ucfe<rv}�npt�k%npahr5a(cn � c\ehr¤�(m\�#�7g�c\��cfgsnpa<�pq � npm�}�rZk%r�m\k%r�a<~:gj}��<qs�pm�c\n�cfehr�khqj��ahr�wÆg�qsqyo¢r�}���a¢}�r�qsqsr5~ o7��cfehrahn(mft_��q � npm�}�r(����a¤cfehr-n�c\ehr�mye<�pa<~IuBcfehrl}�n(t�k¢n(ahr�aBc�k<��m���qsq�r5q<c\n c\ehrlkhqs�pahr�wynpa�² c�o¢r}��pa<}�r5q�qsr5~d��a<~dwÆgsq�q�r�¯:r5m|c �¤cfnpm��B�hr � npadcfe<r{n(o��r5}�c5�{��ehr{cfnpm��B�hr{gsi�cfe<r�i\��t�r � npmo%n�cfe�c\ehr i|k<ehr�m\rO��a<~¤cfehr }��7q�gsa<~:r5m5�� gsa<}�r-cfe<r-n(o��r5}�c\iÆ~:n{ahnpcÆifqsg�kxu7c\ehrO��ah�(�hqj��m��p}�}�r�qsr�m���c\g�n(aZwÆg�qsq*cfr5q�qI�¢i�wÆehgs}�e�npahr�pn7r�i�~:n�wÆa � �pi|cfr�m�µ�cfe<r{n(ahr�wÆg�cfe cfehr{ohgs�p�(r5i|c-��a<�p�hqj��ml�p}5}�r�qsr�m���c\g�n(avwÆg�qsq�o¢r � �pi|cfr5m5���ehrO��a<�p�hqj��mÆ�p}5}�r5q�r5m\��cfgsnpa¤gji
� ����� � �������
����� {� �� ��
��ehr5axuhwyrl}�npt�kh�:c\r-cfehrOm���c\g�n�������� ��� �
���� � ����� ����
��ehr5mfr � npm\rpuBcfehr ifkhehr5mfrlwÆgsq�q�mfr��p}�eZc\ehrOo%n�c|c\npt´z<m\i|c5�JLK�MON�P�QSR �ZUWV7XZY���[�^`bahg�cfgj��qsq��(u7cfehrlz¢m\i|c�~:gjif}-e<�(iÆ��av��ah�(�hqs�pm�t{n(t�r�aBcf�htWn �
¾¨Ã �?��� Å Ã��a<}�rZcfe<r�® ~:gjif}5i{�pmfr�kh�:c�g�aBc\n�}�npaBc\�(}�c5u�c\ehr�� wÆgsqsqÆr�¯:r�mfc{� � npm�}�r�npa r��p}�e npcfehr5m5uwÆehgj}�evwÆg�qsqxgsa<~:�<}�r ��c\npm��B�hrp�¨³Æ}5}�npm�~:gsah�{cfnZ±�r�w�cfn(ax² iycfehgsm\~�qj�#w��/�p}�cfgsnpa�� mfr��p}�cfgsnpa���ucfe<r � n(m\}�r�wÆg�qsq:o¢r�c\ehrÆif�pt�rp�! �n�w�r5�pr5m5u�o%r5}5���<ifrEc\ehrÆm��p~:gsg:��m\r�~hg#"*r�m\r�aBc\i5u�c\ehrycfnpm��B�hr5iwÆgsq�qxo%rO~:g$"%r5mfr5a(c��¨��ehrlcfn(m\�B�hrln(a�~:gji\}lnpahrlgji
ÈÈ À¿ � � Å � ¿ÁÀ|Âf ��� �
®
¡y�:cÆcfe<rlcfnpm��B�hr-gsi���¢ic� �S� ½S� § �� n{cfe<r � npm�}�rlnpav~:gjif}ln(ahrlgsi
§ �Æ� ½ � �� ÈÅ � ¿ÁÀ|ÂÈ À��ifgsah�_�{ifgst{gsqj��mÆm\r5�pifnpa<g�ah�¢u7w�rO}��pa�}�npt�kh�:c\rlcfehr � n(m\}�r�n(a�cfehr ifr5}�npa<~�~:gjif}p�
§� � ½ � �¥ È Å �
¿ÁÀ|ÂÈ À¡yr5}5���<ifr�c\ehr � npm�}�r�iÆ��m\r�r��B�<��q � oh�:c�gsavnpkhk%n(ifg¬c\rO~:g�m\r5}�cfgsnpa���u:w�rle<�#�(r
ÈÈ À
� ���� Š�
¿ÁÀ|Â�Ä � �¥ Å �¿ÁÀ|Â�� ���
��ehr-�(�¢��aBcfg�c�_g�a¢i|gj~:rSc\ehr-k<��m\r�aBcfe<r5ifgsi�gsiEcfe<r-}�npa<ifr�m\�pr�~_�(�¢��aBcfg�c��g�a¤cfehrli|�:i|cfr5t�� � nwyr-e¢�#�pr � �� Å �
¿ÁÀ|Â�Ä � �¥ Å �¿ À| ���
�1n¤~:r�c\r�m\t�g�ahr cfehr{}�npa¢ic���aBc � u*w�r e¢�#�prlcfn¤�<i|r cfehr�g�ahg�cfgj��q�}�n(a<~:g�cfgsnpa<i Å � ¿ �  � Å Ã��a¢~ Å �¿ �  ��� �¨��e7�<i�wyrle<�#�pr �L���� Å ÃO��a<~
� �� Š�
¿ÁÀ|Â1Ä � �¥ Å �¿ÁÀ| � ���
� ÅÃ
��ehrOqj�pi|c�r5�B�<��cfgsnpa��pgs�prO�<ifrlcfe<r���ah�(�hqj��mÆ�pr5q�n:}�g¬c��n � npahr�~hgsi\}O�piS� � �<a<}�c\g�n(a�n � cfehr��a<�p�hqj��m��pr5q�n:}�g¬c�On � c\ehrÆn�c\ehr�m���` cE�(ifif�ht�r5i�c\e<��c�cfe<rS~:gsi\}�i¨wÆg�qsqhi|qsgsk���c�cfehrÆo%r��(g�aha<g�ah�n � cfehr-t�n�cfgsnpa��¨³ycym\r5�pq�qs��qs�pmf�(rScfgst�r �ÁÀ º� ��u7cfehrl~:gjif}5iywÆg�qsqIahn�cÆifq�gsk��paB�7t�npm\rpu7ifncfe<r�m\r-wÆgsqsqxo¢r �{m\r�qj��c\g�n(a<i|e<g�k�o¢r�cw�r5r�a�cfehr�®{�pah�p�hqj��m��(r�qsn7}�g¬c\g�r�i��
Å �¿  � � ¥ Å � ¿ Â
��ehr5axuhwyrl}5��avi|n(q��(r � n(m�cfehr �pah�p�<qs�pmy�(r�qsn:}�g�c�Zn � cfehrlz<m�i|c�~:gsi\}��
Å � ¿  � Å Ã �� Ä ·
JLK�MON�P�QSR ]¤UWV7XZY�� X�^�1nvifnpqs�pr{c\ehgsi khmfn(ohqsr�tvu�wyr�wÆg�qsq¨�<ifrZ}�n(a<i|r5mf����cfgsnpa n � r5ahr�m\�p� �pa<~D}�n(a<i|r5mf����cfgsnpa n �t�npt�r�aBc\�htv����npa<ifr�m\����cfgsnpa�n � r�ahr5mf�(�¤if�#�:i�cfe¢��c
������������� � ��������� �¡y�:c ������������� � �� · ��! �Ã
��������� � � �� ¿ · �ÆÄ · �\Â#" � Ä �� � � Å �� Ä �� � � Å ��$
wÆehr5mfr " gji¨c\ehr��(r�qsn:}�g�c��n � cfehr-}�r5a(c\r�myn � t_�pi\i � ��� ��u7�pa<~ � ����� �pmfrScfehr-t�npt�r�aBc�n �gsahr�mfcfgj�{n � t_�(ifi · ��� � ��m\np�<a<~Zc\ehr }�r5aBcfr�m�n � t_�pi\i5�� ��� � ��· ����� ¥ ������
Sr5mfr(u ¥ ��� � gjiycfehrO~:gji|c\��a¢}�rln � cfehrlt_�pi\i�cfn{cfehr ����n � cfehrOif�:ic\r�tv���ehr��¤�pmfr-�pgs�pr5ao7�¥ � � · �
· � Ä · ���¥ � � · �
· �ÆÄ · � �±Snpcfr �pqsifn{cfe<��cÆcfehr��pah�p�<qs�pmy�(r�qsn:}�g�cfgsr5iÆwÆgsq�q�o¢rOc\ehr i\��t�r Å � � Å � � Å �E��e<r�m\r � npm\rpu}�n(a<ifr�m\�#��cfgsnpa�n � r5ahr�m\�p��if�#�:i�cfe<��c
�� · � ! �à � �� ¿ · � Ä · � Â#" � Ä ��� · � � · �¿ · � Ä · �  � � � Ä
· � · � �¿ · � Ä · �  � � � � ÅE���ehgjiÆ}��pa�o%r i|gst�khqsg¬z<r�~I�· � ! �à � ¿ · � Ä · � Â#" � Ä · � · �
· � Ä · ��� � Å �±Sn�wOu:wyrlwÆg�qsqx�<ifrO}�n(a<i|r5mf����cfgsnpa¤n � t�n(t{r5aBcf�htÉc\nZi|n(q��(r � n(m�"Ç�
����������� � ������� �¡y�:c
�������#� � �¸· � !�à � � ����� � ¿ · �ÆÄ · �\Â#"
��ehr5mfr � npm\rpu7w�rle<�#�(rO��a�r�¯:khm\r5i\i|gsnpa�n � "Ç�" � · �
· � Ä · � !�Ã��ehr5axu¢wyr�}5��a�kh�hc�c\ehgsi�r�¯:khm\r5i\ifg�n(a�gsa�c\ehr�r��B�<��c\g�n(a�npo:c���gsahr5~ � m\npt�}�n7n(a<i|r5mf����cfgsnpan � r�ahr5mf�(�p�
· � ! �à � · � �· � Ä · � ! �Ã Ä · � · �· � Ä · ��� � Å �
º · � · � ! �à ��· � · � � � Å ���e7�<i5u:w�rO}��pa�ifnpqs�pr � n(m�cfehr �pah�p�hqj��m��(r�qsn7}�g¬c�(�Å �
!����a<}�r-wyrle<�#�pr�cfe<r ��ah�(�hqs�pm��pr�qsn:}�g�c�pu:wyrO}���av}�n(t�kh�:cfrlc\ehrlcfr5a<ifg�n(a�gsa�cfehr i|cfm\g�a<�<�
© � © � � © � �¸· � ¥ � Å � � · � ¥ � Å ���ehr5mfr � npm\rpuBcfehrlc\r�a<ifgsnpa�gsi© � · � · �
¿ · � Ä · �  ! �Ã�
JLK�MON�P�QSR ��UWV��{Y�T-^����������1n }�npt�kh�hcfr�c\ehrd�pah�p�hqj��m_�(r�qsn:}�g�c� n � c\ehr ehn7n(kxu�w�r�z¢m\i|c¤e<�#�(rvcfn }�npt�kh�:c\r�cfehr�pr5q�n:}�g¬c�_n � c\ehr ��� �h` cSgji��pgs�pr5a�o7�
! �� ���ehr5axu:cfe<r-�(r�qsn:}�g�c�Zn � cfehrOe<nBn(k�gjiycfe<r if�pt{rlo%r5}5���<ifr-cfehr5mfrOgji�ahn_i|qsgskhkhgsah�<�
! � ��Å ��� � �º Å ��� � � ��
��ehgji�pgs�pr�i��<i�c\ehr���a<�p�hqj��m�(r�qsn7}�g¬c��n � cfehrSe<nBn(k_��m\np�ha<~�cfe<rS��¯:q�r(����e<r�m\rÆgsi¨�pahn�c\ehr�m}�n(aBcfm\g�oh�hcfgsnpa cfe<��cZgsi_}�npt�g�a<� � m\npt cfehr � �(}�c_c\e<��c_c\ehgsi¤��ah�(�hqs�pm��pr�qsn:}�g�c� }�e<�pah�pr�i~:gsmfr�}�c\g�n(ax�d��ehr�ehn7npk �pqsifn m\n�c���cfr�i��pq�n(ah� g�c\i{~:gj��t�r�c\r�m{wÆg¬c\e ��a �pah�p�<qs�pm��pr5q�n:}�g¬c�n � � ��ifg�ah��c\ehr m\g��(eBc�� e<��a<~�m\�hqsrpuhwyr }��pa ~:r�c\r�m\t{gsahrlc\ehr�~:gsmfr�}�c\g�n(a�n � cfehr��pah�p�<qs�pm�pr5q�n:}�g¬c�(� £Å �� ¿��� ½ �¥ Â�����������ehr5mfr���m\r $ }�n(a(c\mfgsoh�:c\g�n(a c\n¤c\ehr{cfn�c���q��a<�p�hqj��m-t�npt�r�aBcf�<t��{��ehr�z<m\i|clnpahr_}�n(t�r� m\npt cfehr-}�gsm\}��hqs�pmEt�n�c\g�n(a_n � c\ehr �����pmfn(�ha<~{c\ehr��� ��¯7gji5����ehr-i|r�}�n(a<~_}�n(a(c\mfgsoh�:c\g�n(a}�n(t�r5i � m\nptWcfe<r�m\n�c\��cfgsnpa�n � cfehr�ehn7npk ��m\np�<a<~�cfehr{��¯:qsrp����ehr�qs�(ic-}�npaBc\mfgsoh�:c\g�n(a�gsit�npm\rOi|�hohcfqsrp�¨` c�}�npt�r5i � m\npt cfehr m\n�c���c\g�n(a�n � c\ehrOehn7npk���m\np�<a<~Zg�c\i�~:gs�pt�r�cfr5m5�¨��e<gsigjiOcfe<rZmfnpc\��cfgsnpa�cfe<��c�}�e<�pah�pr�ilcfehr�~:g�m\r5}�cfgsnpa n � c\ehr¤�pah�p�hqj��m��pr�qsn:}�g�c� n � c\ehrZehn7n(k��m\np�<a<~_c\ehr ��¯:qsrp� £¾ ����� � £¾���� Ä £¾���� � �Ä £¾���� �� ����� ��1n z<a<~¤cfe<r-�pah�p�hqj��myt�npt�r�aBc\�ht ~:�hr�cfn�cfehr ��� u:wyr�}5��a�}�npa<ifgj~:r�m�cfehr-ehn7npk¤cfn{o¢r�{k¢n(g�aBcSk<�pm|c\gs}�q�r(� £¾!�"� � �L� � ����ehrO��a<�p�hqj��m�t�npt�r�aBc\�ht´n � c\ehrOehn7npkv��m\np�ha¢~_cfe<r ��¯:q�rlgji
£¾���� � � � ½ � ��� � � �¥ � ½��L� � �¥ª�g�a¢��qsq��(u7cfehrOqj�pi|c�}�n(a(c\mfgsoh�:c\g�n(a¤gji
£¾���� �� ����� � �?� ��� �� ����� � �� � �� �L�l� ��³Sa¢~¤c\ehrlcfnpc\�pqx��ah�(�hqs�pm�t{n(t�r�aBcf�htWgsi
£¾ ����� �$# �x�L�l� �� ½D�L�l� �¥wÆehgj}�e�gjiÆahnpcÆk<��m���qsq�r5q*cfn{cfe<r ��ah�(�hqs�pm��pr�qsn:}�g�c�Zn � cfehrOe<nBn(k¸£Å �
%
Assignment 9. 28-Oct-2005(Hints added, 2005-Nov-05)
1. Kleppner and Kolenkow, chapt.2, problem 2.20.
• Solve for the accelerations of all the elements of this pedagogical ma-chine, ~a1, ~a2 and ~a3.
• If mass 3 starts at t = 0 at a height h above the bottom of its well, findthe time when it hits the bottom.
Instructions: do this problem as an exercise in correct procedure - define thecoordinate system, write every vector in terms of its components and solvefor these components.
2. Kleppner and Kolenkow, chapt.7, problem 7.2
3. Kleppner and Kolenkow, chapt.7, problem 7.4
Hint: this is not hard to solve once you see the physics that governs thesituation. You should calculate the angular momentum ~L about a suitableorigin. Then d~L/dt tells you what the torque must be, and you can determinethe normal force from the torque. The choice of origin is critical becauseyou don’t want the unknown and irrelevant forces at the vertical axle tocontribute to the torque.
4. Kleppner and Kolenkow, chapt.7, problem 7.5
Hint: this problem is not hard to solve once you see the physics that governsthe situation. The first step is to understand why the car tends to tip. To dothis, calculate the angular momentum ~L(car) about a suitably chosen originand see that d~L(car)/dt requires that the normal forces on the wheels on theright and left side of the car be different. Obviously, the origin needs so bechosen in such a way that other unknown forces don’t enter the equations.Once you undestand what is happening, it is easy to add a flywheel in sucha way that d(~L(flywheel) + ~L(car))/dt doesn’t require a torque from thenormal forces. The orientation of the flywheel is a little surprising if you arethinking symmetry (I guessed wrong when asked on Friday) and you needto make sure that it works for both left turns and right turns.
Assignment 10. 11-Nov-2005(Reminder: this assignement will not be collected or graded.)
1. Kleppner and Kolenkow, chapt.8, problem 8.2
2. Kleppner and Kolenkow, chapt.8, problem 8.5
3. Kleppner and Kolenkow, chapt.8, problem 8.9
4. Kleppner and Kolenkow, chapt.8, problem 8.10
5. Kleppner and Kolenkow, chapt.7, problem 7.5.
Instructions: do the calculation in the rotating coordinate system in whichthe car is at rest.
Assignment 11. 18-Nov-2005(Reminder: this assignement will not be collected or graded.)
1. Kleppner and Kolenkow, chapt.9, problem 9.6 (part 1 only).
2. Kleppner and Kolenkow, chapt.9, problem 9.8
3. Kleppner and Kolenkow, chapt.8, problem 9.10
Assignment 12. 25-Nov-2005(Reminder: this assignement will not be collected or graded.)
1. Kleppner and Kolenkow, chapt.10, problem 10.3.
2. Kleppner and Kolenkow, chapt.10, problem 10.5
3. Kleppner and Kolenkow, chapt.5, problem 5.2
������������ ����������������� ��������! #"%$&�(')*,+.- ��/10�2
3 ����')'�������� � �!�546�������'
798�:<;>=@?BADC�EGFIH�JBA,K�:#L5:<MDN�OPQ:#RSA<EUT)VXWQMYRSZ<[\ADC^]#_<_�`Ka=@;bA<EcT)d!Ee_�`�f1T)d!Eg`<`
h �6'D� #�6�<i�� 1�6'<j Kck�=lNa=@NU:XMDm@Z�NSA)J�no[\Z(Z<LpAY8B:<;pHQKckBArq�NSArZ<sGMD:<m@MDq�m@:#RSZ<C�N^=@N^t\ADCu;v=wRSRSA)J�Hx9Z�qy:#C�AzC�A){(q�=wC�A)J|RSZ}MDmwA):#CbR�kBAp;bA);bZ<C�O|Z<s^O<Z�q~MD:<mlMDq�m@:#RSZ<Cb[\ADsZ<CuA�R�k�AzN�R�:#C�RbZ<saR�kBAAY8B:<;pHz�^mlm9{(q�A)NSR�=wZ�?�N,MD:#CuC�O�R�kBAXN�:<;vAv��A)=@��k�Rr=@?�;v:#C�L�N,AD�<A)?|R�kBZ�qB��k|NSZ�;bA>;v:�O�[\Am@Z�?B�<ADCcR�k�:<?pZ<R�k�ADCuNDH
KakBADC�A�:#C�A�R��&Z�{!qBA)NSR�=wZ�?�N�Z�?�R�k�=lN6RSA)N�RDH�KakBADC�A�:#C�A�NSZ�;bAcMYZ�?�N�R�:<?�R�N9:<?�JIs�Z<Cu;rq�m@:#A���=w�<A)?Z�?�R�kBA�NSA)MYZ�?�J�t�:#�<A
T#HU�&Z�?�N�=lJBADC�:�NSO�NSRSA);�MYZ�?�N�=lNSR�=@?B�zZ<s�:z[�Z�8�Z<s�;v:<N�N,���QZ�?�R�k�A>N�mwZ<t�=l?B��Nu=@JBA�Z<s&:��A)J��<A�Z<s�;v:<N�N����#H
m2
y
x
m1
x2x1
y1
�^mlm�R�kBA�N�qBC�s�:<MYA)NU:#C�A�s�Cu=@MYR�=wZ�?�mwA)N�Nc:<?�J�������_��g`5���<Ha��s�R�kBAI[\Z)8.:<?�J�R�kBA��&A)JB�<A:#C�A.=@?�=wR�=@:<m@mwO}:#R>C�A)NSRv�a=@R�k�R�kBAp[\Z)8�:#R>:���=w�<A)?�kBA)=w��k(Rb� � �¡ �¢&MD:<mlMDq�m@:#RSA�R�kBA�<A)mwZ�MD=wR�OzZ<s�R�kBA�[\Z)8r£�q�NSR�:<Na=wRc=lN�:#[�Z�q�R�RSZ>RSZ�q�M�k�R�kBA��<C�Z�q�?�J�H
]!Ha�^?�Z<[�£¤A)MYR�Z<s�;v:<N�Nc�¥=lN�t�m@:<MYA)Jp:#Ra:�Cu:<J�=@q�N&¦IZ�?.:�R�qBCu?(R�:#[�mwAQ�ak�=@M�k�C�Z<R�:#RSA)Na=@?R�kBA�kBZ<C�=w§DZ�?�R�:<m�t�m@:<?BA<H
r m
KckBA�R�qBCu?IR�:#[�mwA�N�R�:#C�R�N�sC�Z�;�C�A)NSR�:#R���� _U:<?�Jr=wR�N�:<?B��q�m@:#C6�<A)mwZ!MD=@R�O�=@?�MYC�A):<NSA)N�:#R�:MYZ�?�NSR�:<?(R�Cu:#RSA������� ��9H9Kck�AUMYZ!A�>MD=wA)?(R�Z<s�s�Cu=@MYR�=wZ�?X[\ADR��&ADA)?zR�kBAQZ<[�£¤A)MYR�:<?�J�R�kBAN�qBC�s�:<MYAUZ<s�R�kBAUR�qBCu?(R�:#[�mwAU=@N ��H�� s���� _��g]<`�N�� � ¢�¦�� ]I;�¢!:<?�J���� _��g`!¢���?�J>R�kBAR�=@;bAI�akBA)?�R�k�A�Z<[�£SA)MYR9£�q�NSRa[�AD��=l?�N�RSZzN�m@=wt�H���=@?�J.R�kBAI:<MDMYA)mwADC�:#R�=wZ�?���;v:#��?�=wR�q�J�A:<?�J�J�=@C�A)MYR�=wZ�?���£�q�NSR�:<NcR�k�A,Z<[�£SA)MYRa=@N�:#[\Z�qBR�RSZ>N�ml=wt�H
3 1�6'D������'�j� ���!�gdb; N�� �* ��1�����9 1���� ��� � 1 i�������Bi�('<j
�� ¦� � �"!¦$#¦&% ¦ !' #'� � ¦� � � �)(+*¦-,�¦ !' �/. #¦0%1(�¦ *' %~]2!¦ !' . #'
Recommended