Question

Find the center of mass of a uniform L shaped lamina (a thin flat plate) with dimensions as shown. The mass of the lamina is 3 Kg.

(A) 5/6 m, 5/6 m
(A) 3/4 m, 3/4 m
(A) 5/8 m, 5/8 m
(A) 3/5 m, 3/5 m

Answer 1

Hint
The center of mass is a position defined relative to an object or system of objects. It is the average position of all the parts of the system, weighted according to their masses. ... Sometimes the center of mass doesn't fall anywhere on the object.

Complete step by step answer
Choosing the X and Y axes. We have the coordinates of the vertices of the L. shaped lamina as given in the figure. We can think of the L-shape to be constant of 3 squares each of length 1m. The mass of each square is 1 Kg. Since the lamina is uniform.
The center of mass of each square in X,Y coordinate $C_1(1/2,1/2)$,$C_2(3/2,1/2)$ and $C_3(1/2,3/2)$ of the square are by,
Formula for center of mass,
$\Rightarrow X={\displaystyle \frac{m_a{x}_a+m_b{x}_b+m_c{x}_c}{m_a+m_b+m_c}}$ ...(1)
And, $Y={\displaystyle \frac{m_a{y}_a+m_b{y}_b+m_c{y}_c}{m_a+m_b+m_c}}$ ....(2)
Now put the values in the equation 1 and 2 we get,
 $\Rightarrow X={\displaystyle \frac{[1(1/2)+1(3/2)+1(1/2)]}{(1+1+1)}}={\displaystyle \frac{5}{6}}$ m
Now continuing the equation 2
 $\Rightarrow Y={\displaystyle \frac{[1(1/2)+1(1/2)+1(3/2)]}{(1+1+1)}}={\displaystyle \frac{5}{6}}$ m
So, the center of mass for the L shape is $\left({\displaystyle \frac{5}{6}},{\displaystyle \frac{5}{6}}\right)$.
Option (A) is correct.

Note
The velocity of the system's center of mass does not change, as long as the system is closed. The system moves as if all the mass is concentrated at a single point. The final location will be at the weighted distance between the masses.