Problem 9.185 y y2 = mx Determine by direct integration y1 = kx2 y Determine by direct integration the moments of inertia of the shaded area with respect to the x and y axes.

Solving Problems on Your Own Determine by direct integration the moments of inertia of the shaded area with respect to the x and y axes. Problem 9.185 b y2 = mx a x y1 = kx2 y 1. Calculate the moments of inertia Ix and Iy. These moments of inertia are defined by: Ix = y2 dA and Iy = x2 dA Where dA is a differential element of area dx dy. 1a. To compute Ix choose dA to be a thin strip parallel to the x axis. All the the points of the strip are at the same distance y from the x axis. The moment of inertia dIx of the strip is given by y2 dA.

Solving Problems on Your Own Determine by direct integration the moments of inertia of the shaded area with respect to the x and y axes. Problem 9.185 b y2 = mx a x y1 = kx2 y 1b. To compute Iy choose dA to be a thin strip parallel to the y axis. All the the points of the strip are at the same distance x from the y axis. The moment of inertia dIy of the strip is given by x2 dA. 1c. Integrate dIx and dIy over the whole area.

m = y1 = k x2: b = k a2 k = y1 = k x2 then: x1 = y1 1/2 Problem 9.185 Solution b y2 = mx a x y1 = kx2 y Determine m and k: At x = a, y2 = b: b = m a m = y1 = k x2: b = k a2 k = b a b a2 Express x in terms of y1 and y2: y1 = k x2 then: x1 = y1 y2 = mx then x2 = y2 1 k1/2 m 1/2

dA = ( y2 - y1 ) dx = ( m x _ k x2 ) dx Problem 9.185 Solution y a To compute Iy choose dA to be a thin strip parallel to the y axis. b y2 dA = ( y2 - y1 ) dx = ( m x _ k x2 ) dx y1 x x dx a Iy = x2 dA = x2 ( m x _ k x2 ) dx = ( m x3 _ k x4 ) dx = [ m x4 _ k x5 ] = m a4 _ k a5 = b a3 a 1 4 1 5 a 1 4 1 5 1 20 Iy = b a3/ 20 Substituting k = , m = b a2 a

dA = ( x1 - x2 ) dy = ( y _ y ) dy 1/2 Ix = y2 dA = y2 ( y _ y ) dy Problem 9.185 Solution y a To compute Ix choose dA to be a thin strip parallel to the x axis. x2 dy b dA = ( x1 - x2 ) dy = ( y _ y ) dy 1 k1/2 m 1/2 y x x1 Ix = y2 dA = y2 ( y _ y ) dy = ( y _ y3 ) dy = [ y _ y4 ] = b _ b4 = a b3 1 k1/2 1/2 m 5/2 b 2 7 7/2 4 28 Ix = a b3/ 28 Substituting k = , m = b a2 a