two rods of equal lenth l and equal mass m are kept along x and y axes respecdtively such that their centre of mass lies at origin. what will be the moment of inertia about an line y=x ?

To find the moment of inertia of two rods about the line \( y = x \), we first need to understand the setup and the properties of the rods. We have two rods of equal length \( l \) and mass \( m \). One rod is aligned along the x-axis, and the other along the y-axis, with their center of mass located at the origin. The moment of inertia is a measure of an object's resistance to rotational motion about a given axis.

Understanding the Moment of Inertia

The moment of inertia \( I \) about an axis can be calculated using the formula:

• I = ∫ r² dm

where \( r \) is the distance from the axis of rotation to the mass element \( dm \). In our case, we want to calculate the moment of inertia about the line \( y = x \).

Coordinate Transformation

To simplify our calculations, we can rotate our coordinate system. The line \( y = x \) can be treated as an axis at a 45-degree angle to the x and y axes. To find the moment of inertia about this line, we can use the parallel axis theorem and the perpendicular distance from each rod to the line \( y = x \).

Calculating the Moment of Inertia for Each Rod

Let's denote the moment of inertia of the rod along the x-axis as \( I_x \) and the rod along the y-axis as \( I_y \).

• For the rod along the x-axis: The moment of inertia about its own axis is given by:
• I_x = (1/3) m l²
• For the rod along the y-axis: Similarly, its moment of inertia about its own axis is:
• I_y = (1/3) m l²

Using the Perpendicular Distance

The distance from the center of each rod to the line \( y = x \) is crucial. The perpendicular distance \( d \) from the center of the x-axis rod to the line \( y = x \) is:

• d_x = (l/2) / √2 = l / (2√2)

For the y-axis rod, the distance is the same:

• d_y = (l/2) / √2 = l / (2√2)

Applying the Parallel Axis Theorem

Now we apply the parallel axis theorem, which states:

• I = I_cm + md²

For the x-axis rod:

• I_x' = I_x + m d_x² = (1/3) m l² + m (l / (2√2))²

For the y-axis rod:

• I_y' = I_y + m d_y² = (1/3) m l² + m (l / (2√2))²

Final Calculation

Now, we can sum the moments of inertia of both rods about the line \( y = x \):

• I_total = I_x' + I_y'

Substituting the values, we get:

• I_total = 2 * [(1/3) m l² + m (l² / 8)]

After simplifying, we find:

• I_total = (2/3) m l² + (1/4) m l² = (8/12) m l² + (3/12) m l² = (11/12) m l²

Conclusion

The moment of inertia of the two rods about the line \( y = x \) is \( \frac{11}{12} m l² \). This result illustrates how the distribution of mass and the choice of the axis of rotation significantly influence the moment of inertia. Understanding these principles is essential in fields like physics and engineering, where rotational dynamics play a critical role.