classical mechanics -douglas gregory_solution_manual

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1、CLASSICALMECHANICSSOLUTIONSMANUALR.DouglasGregoryNovember2006Pleasereportanyerrorsinthesesolutionsbyemailingcm:solutions@btinternet:com2Contents1Thealgebraandcalculusofvectors32Velocity,accelerationandscalarangularvelocity273Newton'slawsofmotionandthelawofgravitation624Problemsi

2、nparticledynamics765Linearoscillationsandnormalmodes1396Energyconservation1797Orbitsinacentralfield2218Non-linearoscillationsandphasespace2769Theenergyprinciple30610Thelinearmomentumprinciple33511Theangularmomentumprinciple38112Lagrange'sequationsandconservationprinciples42913The

3、calculusofvariationsandHamilton'sprinciple47314Hamilton'sequationsandphasespace50515Thegeneraltheoryofsmalloscillations53316Vectorangularvelocity57717Rotatingreferenceframes59018Tensoralgebraandtheinertiatensor61519Problemsinrigidbodydynamics646ChapterOneThealgebraandcalculusofv

4、ectorscCambridgeUniversityPress,2006Chapter1Thealgebraandcalculusofvectors4Problem1.1Intermsofthestandardbasissetfi;j;kg,aD2ij2k,bD3i4kandcDi5jC3k.(i)Find3aC2b4candjabj2.(ii)Findjaj,jbjanda
b.Deducetheanglebetweenaandb.(iii)Findthecomponentofcinthedirectionofaandinthedirec

5、tionofb.(iv)Findab,bcand.ab/.bc/.(v)Finda
.bc/and.ab/
candverifythattheyareequal.Isthesetfa;b;cgright-orleft-handed?(vi)Byevaluatingeachside,verifytheidentitya.bc/D.a
c/b.a
b/c.Solution(i)3aC2b4cD3.2ij2k/C2.3i4k/4.i5jC3k/D8iC17j26k:jabj2D.ab/
.ab/D.ijC2k/
.

6、ijC2k/D.1/2C.1/2C22D6:(ii)jaj2Da
aD.2ij2k/
.2ij2k/D22C.1/2C.2/2D9:HencejajD3.jbj2Db
bD.3i4k/
.3i4k/D32C.4/2D25:HencejbjD5.a
bD.2ij2k/
.3i4k/


D23C.1/0C.2/.4/D14:cCambridgeUniversityPress,2006Chapter1Thealgebraandcalculusofvectors5Theangle˛betweenaandbis

7、thengivenbya
bcos˛Djajjbj1414DD:3515Thus˛Dtan114.15(iii)Thecomponentofcinthedirectionofaisac
baDc
jaj2ij2kD.i5jC3k/
j2ij2kj


12C.5/.1/C3.2/D31D:3Thecomponentofcinthedirectionofbisbc
bbDc
jbj3i4kD.i5jC3k/
j3i4kj


13C.5/0C3.4/D59D:5(iv)abD.2i

8、j2k/.3i4k/ˇˇˇijkˇˇˇDˇˇ212ˇˇˇ304ˇ


D4