Which expression represents the parallel axis theorem for moment of inertia about a parallel axis at distance d from the centroidal axis?

Which expression represents the parallel axis theorem for moment of inertia about a parallel axis at distance d from the centroidal axis?

• A. I = I_c + A d^2
• B. I = I_c
• C. I = I_c - A d^2
• D. I = I_c + 2A d^2

Answer: (A)

The main idea is the parallel axis theorem for the second moment of area: when you shift the axis to be parallel to the centroidal axis by a distance d, the moment of inertia about the new axis equals the centroidal moment plus Ad^2, where A is the cross-sectional area. So I = I_c + A d^2. This accounts for the extra spread of area relative to the new axis due to the offset. The other forms would either ignore the offset (I = I_c), subtract the offset (I = I_c − A d^2), or use an incorrect coefficient (I = I_c + 2A d^2), which don’t reflect how the area distribution contributes to the inertia about a parallel axis.

The main idea is the parallel axis theorem for the second moment of area: when you shift the axis to be parallel to the centroidal axis by a distance d, the moment of inertia about the new axis equals the centroidal moment plus Ad^2, where A is the cross-sectional area. So I = I_c + A d^2. This accounts for the extra spread of area relative to the new axis due to the offset.

The other forms would either ignore the offset (I = I_c), subtract the offset (I = I_c − A d^2), or use an incorrect coefficient (I = I_c + 2A d^2), which don’t reflect how the area distribution contributes to the inertia about a parallel axis.