Moment of Inertia of a Uniform Thin Rod about an Axis Passing Through a Point at a Distance of L/3 from one of its Ends and Perpendicular to the Rod

Introduction:
The moment of inertia of an object is a measure of its resistance to rotational motion about a particular axis. It depends on the mass distribution of the object and the axis of rotation. In this case, we are calculating the moment of inertia of a uniform thin rod about an axis passing through a point at a distance of L/3 from one of its ends and perpendicular to the rod.

Problem Statement:
Find the moment of inertia of a uniform thin rod of length L and mass M about an axis passing through a point at a distance of L/3 from one of its ends and perpendicular to the rod.

Approach:
To calculate the moment of inertia of the rod, we can use the parallel axis theorem, which states that the moment of inertia of an object about any axis parallel to an axis through its center of mass is equal to the sum of the moment of inertia about the center of mass and the product of its mass and the square of the perpendicular distance between the two axes.

Step 1: Identify the axis of rotation:
The axis of rotation in this case is passing through a point at a distance of L/3 from one of the rod's ends and perpendicular to the rod.

Step 2: Calculate the moment of inertia about the center of mass:
The moment of inertia of a uniform thin rod about an axis perpendicular to its length and passing through its center of mass is given by the formula: I_cm = (1/12) * M * L^2

Step 3: Calculate the perpendicular distance between the two axes:
The perpendicular distance between the axis passing through the center of mass and the given axis is L/3.

Step 4: Apply the parallel axis theorem:
According to the parallel axis theorem, the moment of inertia about the given axis can be calculated using the formula: I = I_cm + M * d^2, where I is the moment of inertia about the given axis, I_cm is the moment of inertia about the center of mass, M is the mass of the rod, and d is the perpendicular distance between the two axes.

Step 5: Substitute the values and calculate:
Using the given values and the formulas, we can substitute the values into the parallel axis theorem equation and calculate the moment of inertia about the given axis.

Final Answer:
The moment of inertia of a uniform thin rod of length L and mass M about an axis passing through a point at a distance of L/3 from one of its ends and perpendicular to the rod is given by the formula: I = (1/12) * M * L^2 + M * (L/3)^2. Simplifying this equation will give you the final answer.

Note: Remember to include appropriate units in your calculations and final answer.