Question

A machinist starts with three identical square plates but cuts one corner from one of them, two corners from the second and three corners from the third. Rank the three according to the x - coordinates of their center of mass, from smallest to largest.

• A. $3,1,2$
• B. $1,3,2$
• C. $3,2,1$
• D. $1$ and $3$ tie, then $2$.

Answer 1

The correct option is B $1,3,2$


Consider a plate of mass $M$ equally distributed about the x-axis and y-axis as shown above.
Case 3
Add a small plate of mass $m$ at the bottom right corner of the above plate. Let the distance of COM of this plate from the origin be $x^{\prime }$
The x-coordinate of COM of the whole plate can be written as,
$x_3=\frac{M\left(0\right)+m\left(x^{\prime }\right)}{M+m}$
$x_3=\frac{m\left(x^{\prime }\right)}{M+m}$

Case 2
Now add a small plate of mass $m$ at the top right corner of the above plate.Then the distance of COM of this plate from the origin will also be $x^{\prime }$
The x-coordinate of COM of the whole plate can be written as,
$x_2=\frac{M\left(0\right)+m\left(x^{\prime }\right)+m\left(x^{\prime }\right)}{M+m+m}$
$x_2=\frac{2m\left(x^{\prime }\right)}{2m+M}$

Case 1
Now add a small plate of mass $m$ at the bottom left corner of the above plate.Then the distance of COM of this plate from the origin will be $(-x^{\prime })$
The COM for this plate will be given by
$x_1=\frac{M\left(0\right)+m\left(x^{\prime }\right)+m\left(x^{\prime }\right)+m(-x^{\prime })}{M+m+m+m}$
$x_1=\frac{m\left(x^{\prime }\right)}{3m+M}$
Comparing the COM for all three cases we get,
$x_2>x_3>x_1$
Hence the correct order is 1,3,2

For detailed solution, see next video.