Question

In figure 9-37, three uniform thin rods, each of length L, form an inverted U. The vertical rods each have a mass m; the horizontal rod has a mass 3m. What are (a) thex coordinate and (b) the y coordinate of the system's center of mass? Give your answer in terms of the variables given.

Answer 1

for the left vertical rod,
centre of mass is at (0,-L/2)
for the right vertical rod
center of mass is at (L,-L/2)
for the horizontal rod
center of mass is at (L/2,0)
so x-coordinate of the system=[m*0+m*L+3m*L/2]/(m+m+3m)=L/2
so y-coordinate of the system=[m*(-L/2)+m*(-L/2)+3m*0]/(m+m+3m)=-L/5

Answer 2

x cm = 0.5*L

ycm = 0.2*L

Answer 3

First of all we have to mark the individual centre of mass
let us consider left vertical rod as 1 , horizontal rod as 2 and right vertical od as 3.
So
X₁ = 0 , Y₁ = -L/2
X₂ = L/2 . Y₂ = 0
X₃ = L , Y₃ = -L/2
Now we know that the centre of mass is
X_CM = (M₁X₁ + M₂X₂  + M₃X₃)/(M₁ +M₂ +M₃)
We know that M₁ = m , M₂ = 3m and M₃ = m
So on putting the value we get
X_CM = (m*0 + 3m*(L/2) + mL) /(m+3m+m) = (5mL/2)/(5m) = L/2
X_CM = L/2
Similarly for Y axis
Y_CM = (M₁Y₁ + M₂Y₂ + M₃Y₃)/(M₁+M₂+M₃)
Y_CM = (m*(-L/2) +3m*0 + m(-L/2)) /(m+3m+m) = (-mL)/5m = -L/5
Y_CM = -L/5