A thin uniform rod of mass M and length L is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet

Verified step by step guidance

1. Identify the configuration of the rod after bending: The rod is bent at its center, forming two segments each of length L/2, perpendicular to each other.
2. Break down the problem into simpler parts: Consider each segment of the rod separately to calculate the moment of inertia for each part about the given axis.
3. Use the formula for the moment of inertia of a rod about an axis through one end, perpendicular to its length: $I = \frac{1}{3}mL^2$, where m is the mass of the rod segment and L is its length. Since each segment is L/2 long and has half the total mass (M/2), substitute these values into the formula.
4. Calculate the moment of inertia for each segment: $I_{\text{segment}} = \frac{1}{3}(M/2)(L/2)^2$. Simplify this expression to find the moment of inertia for one segment.
5. Add the moments of inertia of both segments to find the total moment of inertia of the bent rod about the axis: Since the rod is symmetric and the axis passes through the meeting point, the total moment of inertia is twice that of one segment.

Key Concepts

Moment of Inertia

Moment of inertia is a measure of an object's resistance to rotational motion about a specific axis. It depends on the mass distribution relative to that axis; the further the mass is from the axis, the greater the moment of inertia. For composite shapes, the total moment of inertia can be calculated by summing the moments of inertia of individual components, often using the parallel axis theorem.

Parallel Axis Theorem

The parallel axis theorem allows us to calculate the moment of inertia of a body about any axis parallel to an axis through its center of mass. It states that the moment of inertia about the new axis is equal to the moment of inertia about the center of mass axis plus the product of the mass and the square of the distance between the two axes. This theorem is particularly useful for complex shapes or systems of particles.

Composite Shapes

Composite shapes are formed by combining two or more simple geometric shapes. To find the moment of inertia of a composite shape, one can calculate the moment of inertia for each individual shape about the same axis and then sum them up. This approach simplifies the analysis of more complex objects, such as the bent rod in the question, by breaking them down into manageable parts.