four thin rod each of mass m and length l are joined to make a square find the momentem of interia af all. the four rods about any side a square

Question

Answer

yes the answer is 5/3ml^2 as it will be 0 for first rodby parallel axis theroem ml^2 for parallel rodand ml^2/3 for remaining rods making 0+ml^2+ml^2/3+ml^2/3=5/3ml^2

Answer

I think it is 2ml^2/3 as-1. there are 4 sides,when the system is rotated from one side(as axis) it becomes parallel to the other side so MI is 0 for each.2.and it(axis) is at the end of the two sides so the MI is ml^2/3 for each.3. so total MI is 0+0+ml^2/3+ml^2/3   = 2ml^2/3

Answer

Introduction:
To find the moment of inertia of the square formed by joining four thin rods, each of mass m and length l, we need to consider the individual moments of inertia of the rods and then combine them using the parallel axis theorem.

Individual Moment of Inertia:
The moment of inertia of a thin rod about an axis passing through its center and perpendicular to its length is given by the formula:

I = (1/12) * m * l^2

Since all four rods are identical, they will have the same moment of inertia when considered individually.

Combining the Moment of Inertia:
To find the total moment of inertia of the square, we can consider it as a combination of two perpendicular axes passing through its center.

Axis 1: Along the side of the square
Axis 2: Perpendicular to the side of the square

Using the parallel axis theorem, we can find the moment of inertia about each axis separately and then add them to obtain the total moment of inertia.

Moment of Inertia about Axis 1:
Since the axis passes through the center of mass of each rod, the moment of inertia of each rod about this axis will be the same as the individual moment of inertia calculated earlier.

Moment of Inertia about Axis 2:
Considering each rod as a point mass at its center, the moment of inertia of a point mass about an axis perpendicular to the plane of motion is given by the formula:

I = m * r^2

Since the length of each rod is l, the distance of each rod from axis 2 is l/2. Therefore, the moment of inertia of each rod about axis 2 is:

I = m * (l/2)^2 = (1/4) * m * l^2

Total Moment of Inertia:
To obtain the total moment of inertia about any side of the square, we need to add the moments of inertia about Axis 1 and Axis 2.

I_total = I_axis1 + I_axis2
I_total = (1/12) * m * l^2 + (1/4) * m * l^2
I_total = (1/12 + 1/4) * m * l^2
I_total = (1/3) * m * l^2

Therefore, the moment of inertia of the square formed by joining four thin rods about any side of the square is (1/3) * m * l^2.

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