An iron sphere having diameter D and mass M is immersed in hot water so that the temperature of the sphere increases by T . If is the coefficient of linear expansion of the iron then the change in the surface area of the sphere is

Step-by-step solution

Given, diameter of sphere =D Initial surface area, A=4 R^2=4 ( D 2 )^2= D^2 ...(i) Surface area after heating by temperature T , A^ =4 ( D^ 2 )^2= (D^ )^2 ...(ii) where D^ is the new diameter. From the equation of linear expansion, we have D^ =D(1+ T) ...(iii) putting the value of D^ from eq. (iii) in eq. (ii) we get A^ = D^2(1+ T)^2= D^2 (1+ ^2 T^2+2 T ) ...(iv) aligned & = D^2 (1+ ^2 T^2+2 T )- D^2 & = D^2 [1+ ^2 T^2+2 T-1 ] & = D^2 [ ^2 T^2+2 T ] aligned Change in surface area =A^ -A D^ 2 - D^2= D^2 T( . T+2)