Moment of inertia of a triangular plate

see attachment

integration

The Attempt at a Solution

a. mH2/6 + mt2/12
b. mB2/2 + mt2/12
c. mH2/6 + mB2/2
d. -mBH/4
e. 0, 0
If the plate were thin t can be ignored.

Ok so e is because of symmetry so I get that. a-d on the other hand....seems like a bunch of complicated integrals. So double/triple integrals? I know integration needs to be done across the thickness, and then across the surface of the triangle, so 2 integrals? Any help would be appreciated.

Attachments

• The midplane of a uniform triangular plate is shown.pdf
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Find the frequency of small oscillations for a thin homogeneous plate if the motion takes
place in the plane of the plate and if the plate has the shape of an equilateral triangle and
is suspended (a) from the midpoint of one side and (b) from the apex.

A square sheet is constrained to rotate with an angular velocity ! about an axis passing
through its center and making an angle with the axis through the center of mass and
normal to the sheet (i.e. its axis of symmetry). At the instant the axis of rotation lies in
the plane determined by the axis of symmetry and a diagonal, the body is released. Find
the rate at which the axis of symmetry precesses about the constant direction of the angular
momentum.

Is anybody there can reply these?
I need help
they are due tomorrow......