Question

1 at 13:30. 2 Phys. 335 classical Mechanics Due: Jan. 10, 2020 (Homework II) Three equal masses are located at coordinate points (1,0,0), (0, 1, 2) and (0,2,1). a) find the inertia tensor about the origin b) Find the principal moments of inert c) Find the principal axes.. 2. Calculate the principal moments of inertia and principal axes of a thin uniform circular disk of mass m and fadius a. 3. A homogeneous cylinder of mass in and radius r rolls without slipping on a horizontal plane.. The cylinder is connected to the x as wall through a spring of spring constant k shown in the figure. a) Write down the rolling constraint. 0 Pr:3 6 Write down the total energy of the system. c) Using part (b), find the equation (s) of motion and find the frequency of oscillations. 79 Pr:2, ed to the 4. A solid sphere of mass M and radius a centered on the Z-axis a distance e hon the origin. (Tom) spekee = 2402) - 12 Find the moments of inertia of the system relastive to (oxyt) coordinate system. प

Answer 1

1. 11,0,00 - (0,1,2) - (0,2,1) By definition m, = m Ixx - {mith meam mg = m = mx2²t mxoht mxo². - m Iyy =ę mig = mxoht m x 1²7 mx 2² - 5 m Izz. Ę miz? = mx 6 mx 24 mx12 = sm - isimine so s*kscht I (2xoxwt xoxe tax toxu zitive Š - 362I=Zhb (vxzx uetzxxxutoxotive ) - =-4m Izx = Iyz =-I m;a;Z; Moment of Inertia tensor l Ixx Ixy Ixz ) Iyx Iyy IyZ т., 31 322 m 0 0 o 5m-am IO - Am 5m Scanned with CamScanner

(6) To find principal moment of inertian we use the characteristics equation: I H Ixx I Iyx Izx Ixy Iyy-I Izy и ни Ixz .. I47 -0 Izz-I P a . taria J-1,,22,13 Three roods 1 05m-I - 41 = (-2) | 15m-1) 2- ism] = cm-ig 7-35m-lometitcm? Ico = (masn s?lomitom? Izo (-a) i I? OmI-mIt gm2),20 = (mat) | I(3-956). m(3-9m2. t=0 = (m-1(I-am (I-m.co 1. Im m= Iz 1. These are the principal moment of inertia. . es we have ,1 ܕ14 ܟܟ ܕ ܓܟ ܪ Iyy-I SYZ Izx; -; Iqx 542-3 . for 3 ] 2 0 cs scanned with Can Scanner 4m 4m - 4m um

: ] ܘ ܢ ܨܒܝ -ܟܝܗܢ; ,; ܟܡܗܕܪ ܪܬܡܗ ܀ ܒܝ ܨ ܒܝ . ܐܟܝ - ܕܡܘܳܢ ܘܐܗܕܼ ܐ . ܟܰܢ݈ܩܬ݂ ܪܶܕ݂ܟ .- ܙ (& ܟܪܢܝܚ ,* , c( corresponding. anis principal A eܟܬ ܣܪ sܢ ܗ : ,1 - 1 1 1 ܘܪ an m direction Ger For I am ܕܡܰܟ݂ - ܫܟ -) _ ܘ ܢ ܘ ܀ ܢܳ ܢܝ ܟܟ ,- ܂ - ܘ - , ܚ ܗ ܝ -,- ܘ : ܢ ܢܝ ." 3 ܟܝ - ܢ ,. ܢܩ ܥ ܢ ܝ ܒܝ ܀ ܢ ܟܝ .. ܢܝ - - ܪܰܢ , . . A principal axis corresponding in the is 9m I-I to ( ܪܶ- ܟܢ ܖ ܖ ܀ - ܪܶ ܢ ܫܘܐܠܐ-ܡܝܵܐ cs Scanned with CamScanner