Question

The parallel axis theorem states that the moment of inertia of a body about any axis is equal to:

• A. Its moment of inertia about a parallel axis through its center of mass plus the product of its mass and the distance between the axes squared
• B. The sum of the moment of inertia of individual components
• C. Its moment of inertia about the centroidal axis minus the square of the distance between the axes
• D. The product of its area and the square of the distance between the two axes

Hint:

Use the Parallel Axis Theorem: \( I = I_{\text{cm}} + Md^2 \) when shifting the axis of rotation parallel to the centroidal axis. It's a key tool in structural mechanics and dynamics.

Answer 1

The Correct Option is A

Step 1: The Parallel Axis Theorem is used to calculate the moment of inertia \( I \) of a body about any axis that is parallel to an axis through its center of mass (centroidal axis).
Step 2: The theorem states: \[ I = I_{\text{cm}} + Md^2 \] where \( I_{\text{cm}} \) is the moment of inertia about the centroidal axis, \( M \) is the mass of the body, and \( d \) is the perpendicular distance between the two axes.
Step 3: This theorem is especially useful when the centroidal moment of inertia is known, and we need to calculate it about another parallel axis (e.g., for beams and rotating objects).
Why the other options are incorrect:

• (B) Summing moments of inertia applies to composite bodies, not to a single body's shift in axis.
• (C) The correct relation is additive, not subtractive.
• (D) The area is not relevant in the standard form of the parallel axis theorem for mass moment of inertia.