Calculate the ratio of the moments of inertia I1I2 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 = ?
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To calculate the ratio of the moments of inertia, I1 and I2, for two beams of rectangular cross-section, one with a height h and the other made up of three thinner beams of height h/3, we need to use the parallel axis theorem.
The moment of inertia, I, is a measure of an object's resistance to rotational motion. For a rectangular cross-section beam with width b and height h, the moment of inertia about an axis through its centroid parallel to the height axis is given by the formula:
I = (1/12) * b * h^3
Using this formula, we can calculate the moments of inertia for the two beams.
First, let's calculate the moment of inertia, I1, for the beam with height h:
I1 = (1/12) * b * h^3
Now, let's calculate the moment of inertia, I2, for the beam made up of three thinner beams of height h/3:
Step 1: Calculate the moment of inertia of each thinner beam.
I_thinner_beam = (1/12) * b * (h/3)^3
Step 2: Since the thinner beams are parallel, we can use the parallel axis theorem to calculate the moment of inertia, I2.
By the parallel axis theorem, the total moment of inertia, I2, is equal to the sum of the individual moments of inertia of the thinner beams.
I2 = 3 * I_thinner_beam
Now, we can calculate the ratio of the moments of inertia.
I1/I2 = (1/12) * b * h^3 / (3 * ((1/12) * b * (h/3)^3))
= h^3 / 3 * (h/3)^3
= h^3 / (h/3)^3
= (h^3 / h^3) * (3^3 / 1)
= 27
Therefore, the ratio of the moments of inertia, I1/I2, for the two beams is 27.