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 about a perpendicular axis through the rod at a distance L/3 from the center of mass, you need to use the parallel axis theorem. The parallel axis theorem states that the moment of inertia 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 object multiplied by the square of the distance 'd'.
In this case, the moment of inertia about the perpendicular axis through the rod at a distance L/3 from the center of mass can be calculated as follows:
1. Start by determining the moment of inertia about the center of mass of the rod. The moment of inertia of a straight rod about an axis perpendicular to the rod and passing through its center of mass is given by the formula:
I_cm = (1/12) * M * L²
Where:
- I_cm is the moment of inertia about the center of mass
- M is the mass of the rod
- L is the length of the rod
2. Calculate the distance 'd' between the center of mass and the perpendicular axis through the rod, in this case, L/3.
d = L/3
3. Apply the parallel axis theorem to find the moment of inertia about the perpendicular axis:
I = I_cm + M * d²
Substituting the values into the equation, we get:
I = (1/12) * M * L² + M * (L/3)²
Simplifying the equation further, we have:
I = (1/12) * M * L² + (1/9) * M * L²
Combining the fractions and simplifying, the final expression for the moment of inertia about the perpendicular axis is:
I = (7/36) * M * L²
So, the moment of inertia about the perpendicular axis through the rod at a distance L/3 from the center of mass is (7/36) times the mass of the rod multiplied by the square of the length.