Determine the mass moment of inertia Ixx of the uniform thin

Determine the mass moment of inertia I_xx of the uniform thin parabolic plate of mass m about the x-axis. State the corresponding radius of gyration k_x. Use the values m = 5.6 kg, b = 470 mm, and h = 310 mm. Answers: I_xx = kg m^2 K_x = mm

Solution

Here, we can write the parabolic equation as: y = Kx²

Now, we have y = h for x = b/2, hence we get: h = kb²/4

or, k = 4h/b²

Hence for any distance y along the y axis we have x = (b/2)(y/h)

Now, we will consider a thin rod of thickness dy along the y axis at a distance y.

Hence the mass of the thin rod we selected = dm = (3M/2bh)b(y/h)dy

Now as all the components of the rod are at a distance y from the x axis, we can say the inertia of the thin rod would be given as MR²

hence dI_x = (3M/2bh)b(y/h)y²dy

or, I_x =  dI_x = (3M/2h)(y/h)y²dy = (3M/2h^3/2)(2h^7/2/7) = 3Mh²/7

Also the radius of gyration would be given as Rg = (I/M) = h(2/7)