Properties Of A Plane Area

**Strings (S _{i}P_{j}A_{jk}) = S_{7}P_{2}A_{21} Base Sequence = 12735 String Sequence = 12735 - 2 - 21**

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Properties Of A Plane Area

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Consider figure 124.1:

(a) Where is the centroid of the plane area?

(b) Write the general expression for the moment of inertia about the x axis

(c) Write the general expression for the moment of inertia about the y axis

(d) Write the general expression for the polar moment of inertia

(e) Write the general expression for the product of inertia

(f) Write the general expression for the radius of gyration with respect to the x axis

(g) Write the genaral expression for the radius of gyration with respect to the y axis

(h) Show that the polar moment of inertia is the sum of the moment of inertia about the x axis and the moment of inertia about the y axis.

**The strings**:
S_{7}P_{2}A_{21} (Identity - Physical Properties).
**The math**:

Pj Problem of Interest is of type *identity* (physical properties).

(a) Centroid is at G, the intersection of the lines *m* and *l*.

(b) Moment of inertia about x = I_{x} = ∫dA(x^{2}).

(c) Moment of inertia about y = I_{y} = ∫dA(y^{2}).

(d) Polar Moment of inertia = J_{z} = ∫dA(r^{2}).

Where r, is the distance from O to dA.

(e) Product of inertia = H_{xy} = ∫dA(xy).

(f) Radius of gyration = k_{x} = √(I_{x}/Area)

(g) Radius of gyration = k_{y} = √(I_{y}/Area)

(h) I_{x} + I_{y} = ∫dA(x^{2}) + ∫dA(y^{2}).

So, I_{x} + I_{y} = ∫dA[x^{2} + y^{2}] = ∫dA(r^{2}) = J_{z}.

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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