Moment of Inertia Parallel axis theorem

Question

Two rods of length a and mass m are joined together at right angle to form a T-squre. Show that MI about center of the T pendulum is (17m*a^2)/12 using parallel axis theorem

Answer 1

I got an answer which differs from that listed in the question.

My method is to compute the moment of inertia about the junction of the rods in two ways.

The first way is to use the well known results that the moment of inertia about the end of a uniform rod is

(1/3) m a² and the moment of inertia of a uniform rod about its center is (1/12) m a² .

Adding these gives ( 1/3 + 1/12 )m a² = (5/12) m a²

The second way is to note that the center of mass of the T square is a distance a/4 away from the junction (along the upright rod). The parallel axis theorem then requires

(5/12) m a² = I_CM + 2m (a/4)²

Doing the algebra gives I_CM = (7/24) m a²