Two point masses 3 kg and 5 kg are at 4 m and 8 m from the origin on X-axis. Locate the position of center of mass of the two point masses (from the origin and (1) from 3 kg mass.

Explanation

Let us take `m_(1)=3kg and m_(2)= kg`
(i) To find center of mass from the origin: The point masses are at positions, `x_(1)=4m,x_(2)=8m` from the origin along x-axis

The center of mass Xcy is located 6.5 m from the origin on X-axis, which is derived as below,
`X_(CM)=(m_(1)x_(1)+m_(2)x_(2))/(m_(1)+m_(2))`
`x_(CM)=((3xx4)+(5xx8))/(3+5)`
`x_(CM)=(12+40)/(8)=(52)/(8)-6.5m`
(ii) find the center of mass from 3 kg mass: The origin is shifted to 3 kg mass alone X-axis. The position of 3 kg point mass is zero `(x_(2)-4)` and the position of 5 kg point mass is 4 m from the shifted origin `(x_(2)=4m)`.
`X_(CM)=((3xx0)+(5xx4))/(3+5)`
`X_(CM)=(0+20)/(8)=(20)/(8)=2.5 m`

The center of mass is located 2.5 m from 3 kg point mass, (and 1.5 m from the 5 kg point mass) on x-axis.
This result shows that the center of mass is located closer to larger mass.
If the origin is shifted to the center of mass, then the principle of moments holds good `m_(1)x_(1)=m_(2)x_(2)3xx2.5=5xx1.5,7.5=7.5`
When we compare case (t) with case (ii), the X = 2.5 m from 3 kg mass could also be obtained by subtracting 4 m (the position of 3 kg mass) from 6,5 m, where the center of mass was located in case (i)