A solid spher of mass `m_(1)` density `rho_(1)` radius `R_(1)` and a solid cone of mass `m_(2)` density `rho_(2)`. Radius `R_(2)`, are arranged as shown in the figure. The centre of mass is located at

Options

A. (0, 3R) if `m_(1) = m_(2) = m` and `R_(1) = R_(2) = R`
B. (0, 3R) if `rho_(1) = rho_(2)` and `R_(1) = R_(2) = R`
C. `(0, (11R)/(3))` if `rho_(1) = 2rho_(2)` and `R_(1) = R_(2) = R`
D. always at `(0, 4R_(2))`

Answer

A, B, C

Explanation

(A) `y_(COM) = (m xx (h)/(4) + m (h + R_(1)))/(2m)`
`y_(COM) = (m xx (R ) + m (5 R))/(2m)`
`y_(COM = 3R`
`x_(COM) = 0`
(B) `rho_(1) = rho_(2) = rho`
`m_(1) = rho (4)/(3) pi R^(3)`
`m_(2) = rho xx (pi (R )^(2) (4R))/(3) = (4)/(3) pi R^(2) rho`
again `m_(1) = m_(2)` & `R_(1) = R_(2)`
so same as option (A)
(C ) Let `rho_(2) = rho`, then `rho_(1) = 2rho`
`m_(1) = (2rho) (4)/(3) pi R^(3) = (8 rho pi R^(3))/(3)`
`m_(2) = (rho) ((pi R^(2)) (4R))/(3) = (4)/(3) pi rho R^(3)`
`y_(COM) = (m_(1) (4 R + R) + m_(2) ((4R)/(4)))/(m_(1) + m_(2))`
`y_(COM) = (((8)/(3) rho pi R^(3)) (5R) + (4)/(3) pi rho R^(3) (R ))/((8)/(3) rho pi R^(3) + (4)/(3) pi rho R^(3))`
`y_(COM) = ((40)/(3) rho pi R^(4) + (4)/(3) rho pi R^(4))/((12 rho pi R^(3))/(3))`
`y_(COM) = ((44 rho pi R^(4))/(3))/((12 rho pi R^(3))/(3)) = (11R)/(3)`