Mass and Length of the Rod

The uniform rod has a mass of 6m and a length of 6l.

Formation of an Equilateral Hexagon

The rod is bent to form an equilateral hexagon, which means each side of the hexagon has an equal length. Since the rod has a total length of 6l, each side of the hexagon will have a length of l.

Center of Mass

The center of mass (COM) of the rod is the point where the entire mass of the rod can be assumed to be concentrated. For a uniform rod, the COM is at its geometrical center, which is at a distance of 3l from each end of the rod.

Axis of Rotation

We need to calculate the moment of inertia (M.I.) of the rod about an axis passing through the COM and perpendicular to the plane of the hexagon. This axis of rotation will pass through the COM.

Moment of Inertia of a Rod

The moment of inertia of a rod about an axis perpendicular to the rod and passing through one of its ends is given by the formula: I = (1/3) * m * L^2, where m is the mass of the rod and L is its length. However, since the axis of rotation in this case passes through the COM, we need to use the parallel axis theorem.

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 through its COM is given by: I = I_COM + M * d^2, where I_COM is the moment of inertia about the axis through the COM and M is the total mass of the body.

Moment of Inertia Calculation

In this case, the moment of inertia about the axis passing through the COM is given by: I_COM = (1/12) * m * L^2. Since the axis of rotation passes through the COM, the distance 'd' is 0. Therefore, the moment of inertia about this axis is simply the moment of inertia about the axis through the COM, which is: I = (1/12) * m * L^2.

Final Answer

The moment of inertia of the bent rod about an axis passing through the COM and perpendicular to the plane of the hexagon is (1/12) * m * L^2.