Introduction:
To find the moment of inertia about the y-axis for three thin uniform rods, each of mass m and length l, we can divide the problem into three separate cases. We will consider each rod individually and then combine their moments of inertia to find the total moment of inertia about the y-axis.

Calculating the moment of inertia for each rod:
We can use the formula for the moment of inertia of a thin rod about an axis perpendicular to its length and passing through one end, which is given by I = (1/3) * m * l^2.

Case 1: Rod along the x-axis:
Let's consider the rod along the x-axis. The moment of inertia (I1) of this rod about the y-axis can be calculated using the above formula.

I1 = (1/3) * m * l^2

Case 2: Rod along the y-axis:
Now, let's consider the rod along the y-axis. Since the y-axis is the axis of rotation, the moment of inertia (I2) for this rod is zero. This is because all the mass of the rod lies along the axis of rotation, resulting in no contribution to the moment of inertia.

I2 = 0

Case 3: Rod along the z-axis:
Lastly, let's consider the rod along the z-axis. Similar to the rod along the x-axis, the moment of inertia (I3) of this rod about the y-axis can be calculated using the formula mentioned earlier.

I3 = (1/3) * m * l^2

Combining the moments of inertia:
To find the total moment of inertia (I_total) about the y-axis, we need to add the moments of inertia of each rod.

I_total = I1 + I2 + I3
= (1/3) * m * l^2 + 0 + (1/3) * m * l^2
= (2/3) * m * l^2

Therefore, the moment of inertia about the y-axis for the given system of three thin uniform rods is (2/3) * m * l^2.

Summary:
- The moment of inertia about the y-axis for each rod can be calculated using the formula I = (1/3) * m * l^2.
- The moment of inertia for the rod along the y-axis is zero.
- By adding the moments of inertia of each rod, we can find the total moment of inertia about the y-axis, which is (2/3) * m * l^2.

2ml²/3??