Moment of Inertia of a Structure

The moment of inertia is a measure of how much resistance an object has to changes in its rotational motion. In the case of a structure made of thin uniform rods, we can calculate the moment of inertia about a specific axis by considering the individual contributions from each rod.

Structure Description:
- The structure is made of thin uniform rods.
- Each rod has a mass per unit length of lambda.

Axis of Rotation:
- The axis of rotation is denoted as AB.

Calculating Moment of Inertia:
To calculate the moment of inertia of the given structure about axis AB, we need to consider the contribution of each individual rod to the total moment of inertia.

1. Divide the structure into small segments:
- Divide the structure into small segments by considering each rod separately.
- These segments can be thought of as small thin discs.

2. Calculate the moment of inertia of each segment:
- The moment of inertia of a thin rod about an axis perpendicular to the rod and passing through its center of mass is given by the formula: I = (1/12) * m * L^2.
- In this formula, m represents the mass of the rod and L represents its length.

3. Sum up the contributions from each segment:
- Since the rods are thin and uniform, their masses can be calculated as the product of their length and the mass per unit length (lambda).
- Multiply the mass of each segment by its corresponding moment of inertia and sum up these contributions for all the segments.

4. Obtain the total moment of inertia:
- After summing up the contributions from each segment, we will obtain the total moment of inertia of the structure about axis AB.

Conclusion:
By following the steps mentioned above, we can calculate the moment of inertia of the given structure about axis AB. This approach takes into account the individual contributions of each thin uniform rod and provides an accurate measure of the resistance to rotational motion.

By using perpendicular axis theoram