Determine the mass moment of inertia of the uniform thin tri

Determine the mass moment of inertia of the uniform thin triangular plate of mass m about the x-axis. Also determine the radius of gyration about the x-axis. By analogy state I_yy and k_y. Then determine I_zz and k_z. Use the values m = 6.3 kg, h = 425 mm, and b = 635 mm.

Solution

For the given triangle, for any distance y along the y axis we can write that y = h - hx/b or, x = b(h-y)/h

So as to determine the moment of inertia about the x axis, we will assume a thin rod of thickness dy and at a distance y along the y axis and determine the moment of inertia of this thin rod and then integrate the same over the length along the y axis.

That is, dI_x = (2M/bh)xy²dy

hence, I_x = dI_x = (2M/bh)xy²dy

or, I_x = (2M/h²) (h-y)y²dy = (2M/h²) (h⁴/3 - h⁴/4) = Mh² / 6

Now comparing it with the form of MR² we get that radius of gyration is h/6

By analogy, we can say that I_y = Mb²/6 and radius of gyration is b/6

By perpendicular axis theorem we get: I_z = I_x + I_y

Hence I_z = M(h² + b²)/6 and radius of gyration would be sqrt[(h² + b²)/6]