1. Divide the Square into Smaller Squares
Imagine dividing the square plate into smaller squares, each with side length "dx".
2. Consider a Single Small Square
Focus on one of these small squares located at a distance "x" from the corner where the axis of rotation passes.
* Mass of the Small Square: The mass of this small square is (dm) = (m/a²) * (dx)², where "a" is the side length of the large square.
* Distance from Axis: The distance of this small square from the axis of rotation is "x".
3. Moment of Inertia of the Small Square
The moment of inertia (dI) of this small square about the axis is:
dI = (dm) * x² = (m/a²) * (dx)² * x²
4. Integrate to Find Total Moment of Inertia
To find the total moment of inertia (I) of the entire square plate, integrate dI over the entire area of the square:
I = ∫dI = ∫(m/a²) * (dx)² * x²
The limits of integration will be from x = 0 to x = a (the side length of the square).
5. Calculation
Performing the integration, we get:
I = (m/a²) * ∫(x²) * (dx)² from x = 0 to x = a
I = (m/a²) * [ (x⁴)/4 ] from x = 0 to x = a
I = (m/a²) * [(a⁴)/4 - 0]
I = (m * a²) / 4
Therefore, the moment of inertia of a uniform square plate about an axis perpendicular to its plane and passing through one corner is (m * a²) / 4.
