Question

Find the Moment of inertia of:

a) The rectangular solid formed by 0≤x≤a,0≤y≤b, and 0≤z≤c by calculating I_x, I_y, I_z. [Hint: Compute one of the moments directly and then reason about the other cases via symmetry].

b) The x, y and z axes of a thin plate bounded by the parabola x=−y² and the line x=−y with the density function defined as δ(x,y) = 1/y.

Find the Moment of inertia of: (a) (15 points) The rectangular solid formed by 0 < x <a, 0 <y <b, and 0 <z 5c by calculating Ic, Iy, Ix. (Hint: Compute one of the moments directly and then reason about the other cases via symmetry]. -y? (b) (20 points) The x, y and z axes of a thin plate bounded by the parabola x = and the line x = -y with the density function defined as 8(x, y) = y

Answer 1

Solution is formed by The given rectangular solid je side along x-axis 0<xsa os y sb de side b along y-axis je side с along z-axis. OSZEC the axes about which we let's choos - ахи МОТ need to calculate now then we can consider the rectangular yz plate as combination of two rods with length balong y axu length c along z-axis, t 2 then Ix 12 I melength)² + 1 m (lengtha) 2 12 Id = m ( b? +c) Similarly Iy= 12 m ( a + (²) I, = tem Cat+62) (b) The thin plate bounded by Parabola x 2-y line x = -4 ucy density function 8(x,y) i y x=-42 will use it in calculating mass element we

Ma час,9) qА h-= curv where x goes from x=-4² to vrye toi da dy at boundries. ay²=-y 0-ye 2 Y=oto 1 Mx 1 x dy -Y -yhty) dy My I 6 T.X dady Similarly My m-CENY -7w-vza jydy + 2 fly day My - му يا +1 2 [], 2 txt My = -1 + 2 + 2 x 2 My j 8