Calculate the ratio of the moments of inertia I1/I2 of two beams of rectangular cross section of the same width, one with a height h and the other, made up of three thinner beams of height h/3.
I1/I2 = ?
To calculate the ratio of the moments of inertia I1/I2, we need to use the parallel axis theorem, which states that the moment of inertia of a body about an axis parallel to and at a distance 'd' from an axis through the center of mass of the body is given by:
I = I_cm + Md^2
Here, I is the moment of inertia about the new axis, I_cm is the moment of inertia about an axis through the center of mass, M is the mass of the body, and d is the perpendicular distance between the two parallel axes.
Let's denote the moments of inertia of the two beams as I1 for the beam with height h and I2 for the beam made up of three thinner beams of height h/3.
For the first beam with height h, the moment of inertia, I1, can be calculated using the formula for the moment of inertia of a rectangle about its centroid:
I1 = (1/12) * bh^3
Where b is the width of the rectangle. Since both beams have the same width, we can eliminate the 'b' term from the equation.
For the second beam, made up of three thinner beams each with height h/3, we'll need to calculate the moments of inertia of each individual beam, and then add them up.
The moment of inertia of a thinner beam, I_3, can be calculated using the same formula, but with height h/3 instead:
I_3 = (1/12) * b(h/3)^3
Since we have three of these thinner beams, we can calculate the total moment of inertia of the second beam, I2, by adding up the moments of inertia of each individual beam:
I2 = 3 * I_3 = 3 * (1/12) * b(h/3)^3
Now, we can calculate the ratio I1/I2 by substituting the values we have obtained:
I1/I2 = [(1/12) * bh^3] / [3 * (1/12) * b(h/3)^3]
Simplifying the equation, we can cancel out the common factors:
I1/I2 = (1/3) * h^3 / (1/3)^3 = (1/3) * h^3 * (3^3/1^3) = h^3 / 9
Therefore, the ratio of the moments of inertia I1/I2 is given by h^3 / 9.