Final answer:
To calculate the moment of inertia for a rod with two masses at the midpoint, we consider not only the rod's mass but also the added masses. The total moment of inertia about the midpoint is the same as that of the rod alone since the masses at the midpoint contribute nothing additional due to their zero distance from the axis.
Step-by-step explanation:
The calculation of the moment of inertia for a rod with two masses at the midpoint includes not just the rod's mass itself but also the mass of the two attached objects. This is because the moment of inertia (I) measures how the mass is distributed with respect to the axis of rotation and any mass that is part of the rotating system must be accounted for.
Step-by-step explanation:
Identify the axis of rotation. In this case, it is specified as the midpoint of the rod.
Calculate the rod's moment of inertia about this axis. For a uniform rod of mass m and length L, rotating about its midpoint, the moment of inertia is Irod = m * L2 / 12.
Add the moments of inertia of the two masses. Assuming the masses are at the midpoint, their distance to the axis of rotation is zero, so if considered as point masses, their contribution would be simply mobject * r2, where r is the distance from the axis of rotation (which is zero in this case).
The total moment of inertia is then Itotal = Irod + 2 * mobject * r2. Since r = 0 for the midpoint, the additional moment due to the masses is zero.
Therefore, the moment of inertia for the rod with two masses at its midpoint, with respect to the axis through the midpoint, is Itotal = m * L2 / 12, the same as the moment of inertia for just the rod alone, unless the masses are positioned at some distance from the midpoint.