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

2024
Views:

266

Pages:

5
Author:

Jabin F.

Mathematical Introduction # Moment of Inertia (MI) For a system of particles, MI is defined about an axis of rotation as: For discrete system (discontinuous system): I = Σm_i * r_i² where r_i is the distance of m_i from the given axis. For continuous system: I = ∫dm * r² where r is the distance of dm from the given axis. # MI in terms of co-ordinates MI about z-axis: I_zz = Σm_i * r_i² where r_i² = x_i² + y_i² MI about x-axis: I_xx = Σm_i * (y_i² + z_i²) MI about y-axis: I_yy = Σm_i * (x_i² + z_i²) For continuous systems: I_zz = ∫dm * (y² + z²) I_yy = ∫dm * (x² + z²) I_xx = ∫dm * (x² + y²) D-25-1/10/19 Q: Calculate M.I. of given triangle uniform plate about axis shown. Solution: As plate is uniform: dm = M/A dA = M/a² dxdy As plate is continuous: I = ∫dm(x² + z²) = ∫(x² + z²) (9M/a³) dxdy I = ∫(0 to a) ∫(0 to x) (x² + z²) (9M/a³) dxdy = 9M/a³ ∫(0 to a) [x³(x-0)/3] dx = 9M/a³ ∫(0 to a) (x⁴/3) dx = 9M/a³ [(x⁵/15)](0 to a) = 9M/a³ (a⁵/15) = 1/5 Ma²