Question

three objects are arranged as shown: a solid board of length $l_0$ and uniform density, a board that, except for a small hole in its upper half, is identical to the solid board, and a small cube whose center is at a height of $\frac{l_0}{2}$. which of the following correctly ranks the $y$-coordinates of the center of mass $y_{sb}$, $y_{bh}$, and $y_{c}$ of the solid board, the board with the hole, and the small cube, respectively?
(a) $y_{sb}>y_{bh}>y_{c}$
(b) $y_{bh}>y_{sb}>y_{c}$
(c) $y_{sb}=y_{c}>y_{bh}$
(d) $y_{bh}>y_{sb}=y_{c}$

Explanation:

Step1: Determine center - of - mass of solid board

For a uniform - density solid board of length $L_0$ along the $y$ - axis, the center of mass is at its geometric center. So, $y_{SB}=\frac{L_0}{2}$.

Step2: Determine center - of - mass of board with hole

The board with a hole in its upper half has less mass in the upper part. The center of mass of this board will be lower than that of the solid board. So, $y_{BH}<\frac{L_0}{2}$.

Step3: Determine center - of - mass of small cube

The center of the small cube is at a height of $\frac{L_0}{2}$, so $y_{C}=\frac{L_0}{2}$.

Answer:

C. $y_{SB}=y_{C}>y_{BH}$