University:
Santa Fe College
Course:
PHY 2004 | Applied Physics 1
Academic year:

2024
Views:

66

Pages:

2
Author:

Jabin F.

Problems related to M.I. Uniform objects → standard formulae of M.I. (no integration used) Non-uniform objects → integration is used! @ Uniform plate shaped as equilateral triangle as shown in figure, Calculate: Solution: For uniform motion, dm = M/A * dA dm = (4M / √3a²) dxdy Iy = ∫ dm(x² + z²) Iy = ∫ (4M/√3a²) dxdy (x² + y²) = (4M/√3a²) ∫ [∫(x² + y²) dx] dy = (4M/√3a²) ∫ [(x³/3 + xy²) |_(0, a/2) + ∫(x² + y²) dx] dy = (4M/√3a²) ∫ [(a³y/24 + a³y/3) + ∫(x² + y²) dx] dy = (4M/√3a²) ∫ [a³y/8 + ∫(x² + y²) dx] dy = (4M/√3a²) ∫ [a³y/8 + (x³/3 + xy²) |_(0, a/2)] dy = (4M/√3a²) ∫ [a³y/8 + (a³y/24 + a³y/3)] dy = (4M/√3a²) ∫ (9a³y/24) dy = (9M/a²) ∫ (a³y/6) dy = (9M/a²) [a⁴y²/12] |_(0, a/2) = (9M/a²) (a⁴/48) Iy = Ma²/24 for equilateral triangle, Ixx = Iyy = Ma²/24