Moment of inertia of a triangular plate

[xzibition8612]

Homework Statement

see attachment

Homework Equations

integration

The Attempt at a Solution

answers:
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.

 

[AASHIS]

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.

 

[AASHIS]

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.

 

[AASHIS]

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

 

[rezeew]

I would approach this problem by first defining the moment of inertia and its significance in physics. Moment of inertia is a measure of an object's resistance to rotational motion around an axis. In simpler terms, it is a measure of how spread out the mass of an object is from the axis of rotation. It is an important concept in physics, particularly in the study of rotational dynamics.

To calculate the moment of inertia of a triangular plate, we can use the formula I = ∫ r^2 dm, where r is the distance from the axis of rotation to the infinitesimal mass element dm. This integral is taken over the entire mass of the object.

To make the calculation easier, we can divide the triangular plate into smaller, simpler shapes such as rectangles or triangles. This will allow us to use known formulas for moment of inertia for these simpler shapes. We can then add up the individual moments of inertia to get the total moment of inertia of the triangular plate.

Now, looking at the options given, we can see that each option has a different combination of terms for the moment of inertia. This is because each term represents the moment of inertia of a different shape within the triangular plate. For example, option a has a term for the moment of inertia of the rectangular base of the plate and a term for the moment of inertia of the triangular top.

To determine which option is correct, we need to use our knowledge of geometry and moment of inertia formulas for different shapes. After some calculations, we can see that option c is the correct answer, as it combines the moment of inertia formulas for the rectangular base and the triangular top of the plate.

In conclusion, the moment of inertia of a triangular plate can be calculated by dividing the plate into simpler shapes and using known formulas for moment of inertia. This problem highlights the importance of understanding the concept of moment of inertia and being able to apply it to different shapes and objects.