Three uniform thin aluminum rods each of length 2 m form an equilateral triangle PQR as shown in the figure. The mid point of the rod PQ is at the origin of the coordinate system . If the temperature of the system of rods increases by `50^(@)C` the increase in y-coordinate of the ceentre of mass of hte system of the rods is ....... mm.
(Coefficient of volume expansion of aluminium = `12 sqrt(3) xx 10^(-6) K^(-1)` )

Options

A. `0 . 0 5`
B. `0 . 8`
C. `0 . 1`
D. ` 0 . 2`

Answer

D

Explanation

Initially.
`r_("COM") = (m_(1) r_(1) + m_(2) r_(2) + m_(3) r_(3))/((m_(1) +m_(2) + m_(3)))`
`r_("COM") = (1)/(3) ((sqrt(3))/(2) xx1xx2) hat(j)`
`rArr r_("COM") = (1)/(sqrt(3)) hat(j)`
Due to expansion each rod length becomes
`l_(f) = l_(i) (1 + alpga Delta T)`
`l_(t) = l_(i) (1 + (gamma)/(3) Delta T)`
`l_(t) = 2 (1 + (12 sqrt(3) xx 10 ^(-6))/(3)xx 50)`
`l_(t) = 2 (1 + 4 xx 50 xx sqrt(3) xx 10 ^(-6))`
`= 2 (1 + 2 sqrt(3) xx 10 ^(-4))`
So, ` (r_("COM"))_("final")=(1)/(3)xx (sqrt(3))/(2) xx (1 + 2 sqrt(3) xx 10^(-4) ) xx 2 hat(j)`
`= ((1)/(sqrt(3)) + 2 xx 10 ^(-4)) hat (j)`
Hence, `Delta y - (r_("COM"))_("final") - (r_("COM"))_("final") - (r_("COM")) _("initial")`
`= 2 xx 10^(-4) m = 0 2 m m `