Question

Athin e plate (lamina) is of mass and dimensions a by We choose a coordinate system as in the figure such that the plate lies in the sy plane with originat a comer The inertia Tensor is then 4 -30 m 340 0 0 8 insanlar (a) The angular velocity of the plate is 10, what is the momentum Ly Is Laluned within the same direction? (b) Calculate the principal moments of inertia for the plate (c) Find the principal ases directions for this plate

Answer 1

RANK Ans. (@) I= ma? 4-3 -3 4 12 Anjulan momentum L %3D 4 -3 0 ma? %3D -3 4 -6w ma? %3D L is not alignecl with lav.

for prinipal (b) * Ineutia of moments det I rmatrix AI)=0 iarzing ma Let Io 12 -310 4Io-1 -3 I. 410-d 8 Io-1 - 316 4 Io-d 410-1 -3T0

Date Page KANNA 2. 2. OY (4T0-A)= 9,² %3D 8I, -A -0 Fither d,= 8ID 4T0-1= +9I0

A) = 8 ma? 13 = 7I0 Az= Io %3D 12 A3= 12 141=2ma² ma2 12 3. Paincipal (C) axes. for %3D Wx I Wt wy lWx amp Wy 4 -3 maz wy -3 4 12 Wx Wx 4-3 0 8. it Wy Wy -340 LOL Wx 3Wy %3D 3 Wy - 3wy

3Wy Date Page RANKA wy -3Wx +' 4 Wy -3uy=4Wy -3Wx Wy (i) usiny = 4 Wy 3Wy Wq is arbitary. for norrmatized Wi q Wy=0 Wy=D atso ['De=0]

: Wi= for d=dz I W2 I wi = d, w2 Wx 2. %3D ma? Wy Wy 12 WX 4-3 0 Wy wy -3 4 DO Wx =Wy = Wx HWx -3Wy Let Wz =0 is asrbit any so that it 18 Oerthe Orthojorat to w,

Date Page Now W2= normalized w For wx' = 1/½ for dg = d3. I Wz = d3 Wg 7ma?/ Wx Wy 2. Wx -3 %D wy 12 - 3 7Wx Wx 4-3 Twy wy -3 8. lWz=D0 8 WI =7 Wq => - TWx => 3Wx = -3Wy Jux=-wWg) 4Wx -3 Wy Wx Now for normalized Wa wx'twx?+o =/ Twx=1 - Wx juxI -1 u ولار Panopal axes are