Three uniform thin aluminium rods each of length 2 ~m form an equilateral triangle P Q R as shown in the figure. The mid point of the rod P Q 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 centre of mass of the system of the rods is ............ mm . (Coefficient of volume expansion of aluminium =12 3 10⁻⁶ ~K ⁻¹ )

Step-by-step solution

Initially, aligned & r _ COM = m₁ r ₁+m₂ r ₂+m₃ r ₃ (m₁+m₂+m₃ ) & r _ COM = 1 3 ( 3 2 1 2 ) j r _ COM = 1 3 j aligned Final length, l_f=l_i(1+ T) or, l_f=l_i (1+ 3 T ) [ =3 ] or, l_f=2 (1+ 12 3 10⁻⁶ 3 50 ) or, l_f=2 (1+4 50 3 10⁻⁶ ) or, l_f=2 (1+2 3 10⁻⁴ ) aligned & So, ( r _ COM )_ final = 1 3 3 2 (1+2 3 10⁻⁴ ) 2 j & = ( 1 3 +2 10⁻⁴ ) j aligned So, y= ( r _ СOM )_ final - ( r _ СоM )_ initial =2 10⁻⁴ ~m =0.2 ~mm