Dynamics 17-1| Determine the moment of inertia Iy for the slender rod.

Determine the moment of inertia Iy for the slender rod. The rod's density p and cross-sectional area A are constant. Express the result in terms of the rod's total mass m.






Answer




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To determine the moment of inertia \( I_y \) for a slender rod about an axis perpendicular to the rod (denoted as the y-axis):

Given:
Density of the rod, \( \rho \)
Cross-sectional area, \( A \) (constant)
Length of the rod, \( L \) (assumed to be given)
Total mass of the rod, \( m \)
Axis perpendicular to the rod (assumed to pass through the center of mass of the rod)

Step 1: Expression for the Mass of the Rod
The total mass of the rod is given by:
m = \( \rho \cdot A \cdot L \)

Rearrange the equation to express the density in terms of the total mass:
\( \rho = \frac{m}{A \cdot L} \)

Step 2: Moment of Inertia of the Rod About the y-Axis
The moment of inertia of a slender rod about an axis perpendicular to the rod and passing through its center of mass is given by:
\( I_y = \frac{1}{12} m L^2 \)

Where:
\( m \) is the mass of the rod,
\( L \) is the length of the rod.

Final Result:
The moment of inertia of the slender rod about the y-axis, in terms of the rod's total mass \( m \), is:
\( I_y = \frac{1}{12} m L^2 \)

This is the moment of inertia for a slender rod rotating about an axis perpendicular to it and passing through its center of mass.
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