Considering the diagram below, derive the standard formula for the second
moment of area and radius of gyration for a rectangle of length L and
breadth B about an axis through its centroid and parallel to the long edge.
To derive the standard formula for the second moment of area and radius of gyration for a rectangle about an axis through its centroid and parallel to the long edge, we can follow these steps:
Step 1: Understand the Problem
We are given a rectangle of length L and breadth B. We need to find the second moment of area (also known as the moment of inertia) and the radius of gyration about an axis passing through the centroid and parallel to the rectangle's long edge.
Step 2: Define Variables
Let's assume the centroid of the rectangle is located at its center. We can define the following variables:
- L: Length of the rectangle
- B: Breadth of the rectangle
- A: Area of the rectangle
- x: Distance from the centroid to an element within the rectangle
Step 3: Calculate the Second Moment of Area
The second moment of area, also known as the moment of inertia (I), can be calculated by summing the moments of inertia of all small elements that make up the rectangle. For a rectangular element with dimensions dx and dy, the moment of inertia about an axis passing through the centroid and parallel to the long edge is given by:
dI = (dy * dx^3) / 12
To find the total second moment of area, we need to integrate this equation over the entire area of the rectangle:
I = ∫ [(dy * dx^3) / 12]
To perform this integration, we need to define the limits of integration. Since we are integrating with respect to y (dy), the limits of integration are from -B/2 to B/2. The value of dx can be considered constant as it represents the thickness of the rectangle.
I = ∫ [(B * dx^3) / 12] from y = -B/2 to y = B/2
Simplifying the equation:
I = (B * dx^3 / 12) * [(B/2) - (-B/2)]
I = (B * dx^3 / 12) * (B)
I = (B^2 * dx^3 / 12)
Step 4: Evaluate the Integral
Integrating the equation, we can find the second moment of area:
I = (B^2 * dx^3 / 12) | from x = -L/2 to x = L/2
I = (B^2 * dx^3 / 12) * (L/2 + L/2)
I = (B^2 * dx^3 / 12) * (L)
I = (B^2 * dx^3 * L / 6)
I = (B^2 * L * dx^3 / 6)
Step 5: Simplify the Equation
Since dx represents the thickness of the rectangle, we can consider it as a small value that tends towards zero, which can be denoted as "d" (delta). The equation simplifies further:
I = (B^2 * L * d^3 / 6)
Step 6: Calculate the Radius of Gyration
The radius of gyration (k) is calculated as the square root of the ratio of the second moment of area to the total area:
k = √(I / A)
k = √((B^2 * L * d^3 / 6) / (L * B))
k = √((B * d^3) / 6)
Simplifying further:
k = √(B * d^3) / √6
Therefore, the standard formula for the second moment of area (moment of inertia) of a rectangle of length L and breadth B about an axis through its centroid and parallel to the long edge is I = B^2 * L * d^3 / 6, and the standard formula for the radius of gyration is k = √(B * d^3) / √6.