Find the moment L about a perpendicular axis through the rod at a distance L/3 from the centre of mass.
To find the moment of inertia (L) about a perpendicular axis through the rod at a distance L/3 from the center of mass, we need to use the parallel axis theorem.
The parallel axis theorem states that the moment of inertia of a body about an axis parallel to and a distance "d" away from an axis passing through the center of mass is equal to the moment of inertia about the center of mass plus the mass of the body multiplied by the square of the distance "d".
In this case, let's assume the mass of the rod is "M" and the length of the rod is "2L" (half of the rod on each side of the center). The center of mass of the rod would be located at the middle, which is at a distance of L from either end.
Now, we want to find the moment of inertia (I) about an axis perpendicular to the rod and at a distance L/3 from the center of mass.
1. Find the moment of inertia (Icm) about the center of mass:
The moment of inertia of a rod about its center of mass is given by the formula:
Icm = (1/12) * M * L^2
2. Use the parallel axis theorem to find the moment of inertia (L) about the perpendicular axis:
L = Icm + M * (d^2)
Since the axis is L/3 away from the center of mass, the distance (d) would be L/3. Substituting this value, we have:
L = (1/12) * M * L^2 + M * (L/3)^2
Simplifying the equation, we get:
L = (1/12) * M * L^2 + M * (L^2/9)
Combining like terms, we have:
L = (1/12 + 1/9) * M * L^2
Adding the fractions, we have:
L = (3/36 + 4/36) * M * L^2
Simplifying further, we get:
L = (7/36) * M * L^2
Therefore, the moment of inertia (L) about a perpendicular axis through the rod at a distance L/3 from the center of mass is (7/36) * M * L^2.