The rotational inertia of a thin rod about one end is 1/3 ML2. What is the rotational inertia of the same rod about a point located 0.40 L from the end?

The rotational inertia, or moment of inertia, of an object depends on the distribution of mass in the object and the axis of rotation. The Parallel Axis Theorem can be used to find the moment of inertia of an object about any axis, given the moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance between the axes.

The Parallel Axis Theorem states that the moment of inertia (I) about any axis is equal to the moment of inertia (Icm) about a parallel axis through the center of mass plus the product of the mass (m) and the square of the distance (d) between the axes. Mathematically, this is expressed as:

I = Icm + md^2

Given that the moment of inertia of the rod about one end (Icm) is 1/3 ML^2, the mass of the rod (m) is M, and the distance from the end of the rod to the new axis of rotation (d) is 0.40L, we can substitute these values into the equation to find the moment of inertia about the new axis:

I = 1/3 ML^2 + M(0.40L)^2

Solving this equation will give the moment of inertia of the rod about the point located 0.40L from the end.