Problem 9.185

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

2. Problem 9.185 y Solving Problems on Your Own y2 = mx Determine by direct integration the moments of inertia of the shaded area with respect to the x and y axes. b y1 = kx2 x a 1. Calculate the moments of inertia Ixand Iy. These moments of inertia are defined by: Ix = y2dA andIy = x2dA Where dA is a differential element of area dxdy. 1a. To compute Ixchoose dAto 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 y2dA.

3. Problem 9.185 y Solving Problems on Your Own y2 = mx Determine by direct integration the moments of inertia of the shaded area with respect to the x and y axes. b y1 = kx2 x a 1b. To compute Iychoose dAto 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 x2dA. 1c. IntegratedIxanddIyover the whole area.

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

5. Problem 9.185 Solution a 1 1 4 5 0 1 1 1 Iy = ba3/ 20 20 4 5 b b Substitutingk = , m = a a2 y a To compute Iy choose dA to be a thin strip parallel to the y axis. b y2 dA = ( y2 - y1 ) dx = ( mx_kx2 ) dx y1 x x dx a Iy = x2 dA = x2 ( mx_kx2 ) dx = ( mx3_kx4 ) dx = [ mx4_kx5] = ma4_ka5 = ba3 0 a 0

6. Problem 9.185 Solution dA = ( x1 - x2 ) dy = ( y_y ) dy 1 1 1/2 k1/2 m b 1 1 Ix = y2 dA = y2 ( y_y ) dy = ( y_y3 ) dy = [y_y4] = b_b4 = ab3 1/2 m k1/2 0 b 1 1 b 1 2 1 1 5/2 7/2 m 7 m k1/2 k1/2 4 0 0 1 1 1 1 2 Ix = ab3/ 28 7/2 m 28 4 7 k1/2 b b Substitutingk = , m = a a2 y a To compute Ix choose dA to be a thin strip parallel to the x axis. x2 dy b y x x1