The moment of inertia of the shaded area about the x-axis is 6.1966 × 10⁶ mm⁴
Area moment of inertia:
It is also referred as second moment of inertia. It is directly proportional to the cross-sectional area and distance of the centroid of the area from the reference axis.
Parallel axis theorem:
According to parallel axis theorem, the moment of inertia about any axis which is parallel to the axis passing through the centroid of the body is equal to the sum of the moment of inertia about the centroid of the section and the product of Area of the body with the square of the distance between the two axes.
To calculate the moment of inertia of the whole section, divide the whole section two sections and calculate the moment of inertia of each subpart about x axis using parallel axis theorem and add all to get the moment of inertia of the whole section about x axis.
Write the formula for area moment of inertia for a rectangular cross section about its centroidal axis.
I = bd³/3
Here, breadth and height of the rectangular section are b and d respectively.
Write the equation for moment of inertia of any shape about point O using parallel axis theorem.
Io = Ic +A(d)²
Here, moment of inertia of shape about point O is
Io
moment of inertia of shape about centroid is
Ic
cross sectional area is A and distance between point O and centroid is
d
Area of the semi-circular plate is given as:
A = πR²/4
The area moment of inertia of the semi-circular plate about centroidal axis is given as:
I = (π/16 - 4/9π)R⁴
Calculate the cross-sectional area of section (1).
A₁ = 84²
= 7056 mm²
Calculate the distance of the centroid of section (1) from x-axis.
y₁ = 84/2
= 42 mm
Calculate the cross-sectional area of section (2).
A₂ =1/4 (π×r²)
Here, r is the radius of the quarter circle.
A₂ =1/4 (π×60²)
= 2827.4333 mm²
Calculate the distance of the centroid of section (2) from x-axis.
y₂ = 84 - 4r/3π
Substitute 60 mm for r.
y₂ = 84 - 4*60/3π
= 58.5352 mm
Calculate the distance of the centroid of section (2) from x-axis.
6.1966 × 10⁶ mm⁴
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