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.

To calculate the ratio of moments of inertia (I1/I2) for two beams with rectangular cross sections, one of height h and the other made up of three thinner beams of height h/3, we need to use the parallel axis theorem.

The parallel axis theorem states that the moment of inertia of an object about an axis parallel to and a distance 'd' away from its centroid is equal to the sum of the moment of inertia about the centroid and the product of the object's mass and the square of the distance 'd'.

In this case, we have two beams with the same width but different heights. Let's denote the height of the first beam as h1 and the height of the second beam (made up of three thinner beams) as h2.

The moment of inertia for a rectangular beam (I) about its centroid can be calculated using the following formula:

I = (b * h^3) / 12,

where 'b' is the width of the beam and 'h' is the height of the beam.

For the first beam with height h1, the moment of inertia is given by I1 = (b * h1^3) / 12.

For the second beam made up of three thinner beams, the overall height is h2 = (h/3) + (h/3) + (h/3) = h. Each thinner beam has a height of h/3.

To calculate the moment of inertia for the second beam, we need to sum the moments of inertia for each thinner beam separately. Since the thinner beams are identical, we only need to calculate the moment of inertia for one of them. Let's denote the moment of inertia for one thinner beam as I2_thin.

Using the formula for the moment of inertia of a rectangular beam, we have:

I2_thin = (b * (h/3)^3) / 12.

Since we have three thinner beams, the total moment of inertia for the second beam is given by:

I2 = 3 * I2_thin = 3 * [(b * (h/3)^3) / 12].

Now, to find the ratio of I1/I2, we can divide I1 by I2:

I1/I2 = [(b * h1^3) / 12] / [3 * (b * (h/3)^3) / 12].

Simplifying the expression, we get:

I1/I2 = (h1/h)^3.

Therefore, the ratio of moments of inertia I1/I2 for the two beams is given by (h1/h)^3.

can Anyone answer please same question here