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

2024
Views:

305

Pages:

5
Author:

Jabin F.

# Fundamental PB -> same as CB single CB = (1/2) PB double CB = (1/4) PB {ai, bi} = Sij Kronecker delta = {1 if i = j, 0 if i ≠ j} {f1(x1), f2(x2)} = 0 {f1(x1), f3(x3)} = 0 {f2(x2), f3(x3)} = 0 {f1(x1), f4(x4)} = 0 {f2(x2), f4(x4)} = 0 {f3(x3), f4(x4)} = 0 {f1(x1), f5(x5)} = 0 {f2(x2), f5(x5)} = 0 {f3(x3), f5(x5)} = 0 {f4(x4), f5(x5)} = 0 Question: If y = -x + a, calculate the charge in the shaded region where charge density σ = σ₀x. Solution: Area = (1/2) * a * a = a^2/2 If we use Cartesian coordinates and boundaries are not parallel to coordinate axes, then y limits will be a function of x or x limits will be a function of y. Dependence of y on x or dependence of x on y is written using the equation of boundaries. y = -x + a dA = dxdy dQ = σ dA = σ₀x dxdy Q = ∫∫σ₀x dxdy = σ₀ ∫₀ᵃ ∫₀^(-x+a) x dxdy Q = σ₀ ∫₀ᵃ (x^2/2 + ax) dx = σ₀ (a^3/3 + a^2/2) Q = (σ₀a^3)/3 + (σ₀a^2)/2 # Some Points Related to M.I.: Ixx ≥ 0, Iyy ≥ 0, Izz ≥ 0 (i.e. MI is zero or +ve) Ixx + Iyy ≥ Izz, Ixx + Izz ≥ Iyy, Iyy + Izz ≥ Ixx Ixx + Iyy = Izz Σ(mi * zi^2) If z = 0, then Ixx + Iyy = Izz, which is called 1x axes theorem (1d & 2d) ⇒ Perpendicular axes theorem: (applicable for 1d & 2d) If an object lies on the xy plane then Ixx + Iyy = Izz ⇒ Parallel axes theorem: (for 1d, 2d, & 3d) For any axis parallel to another axis through the center of mass (com): I = Ic + Md^2 ⇒ M.I. of a big object formed by combining smaller objects has the same formula as M.I. about smaller objects.