A lamina with constant density rhox y rho occupies the regi

A lamina with constant density rho(x, y) = rho occupies the region under the curve y = sinx from x = 0 to x =pi. Find the moments of inertia I_x and I_y and the radii of the gyration x and y

Solution

Solution :

The definition of inertia is a volume integral

I(i,j) := dV R(V) (r² (i,j) - x_i x_j )

where R(V) is the density and delta(i,j) = 1 if i=j and 0 otherwise. From this definition, we specify to your problem. I\'ll take R(V) to be R on the plane at z=0 and 0 everywhere else (i.e. the lamina). In this case, the inertial components are :

I(1,1) := I_x = R* dx{0 ... }dy{0 ... sin(x)}y²
I(2,2) := I_y = R* dx{0 ... }dy{0 ... sin(x)}x²

Now solve each:

I_x= R* dx{0 ... } sin³(x)
= R*[-1/3*(cos(x))*(sin²*(x)+2)]{0...}
= R*[1-2/3] = R/3

I_y = R* dx{0 ... } x²*sin(x)
= R*[2x*sin(x) + (2-x²)*cos(x)]{0 ... }
= R*[-(2-²) - 2] = (²-4)*R