Two metre scales, one made of steel and the other of aluminium, agree at 20^ C . Determine the ratio aluminium-centimetre/steel-centimetre at 0^ C . The coefficient of linear expansion for steel is 1.1 10⁻⁵ ^ C⁻¹ and for aluminium is 2.3 10⁻⁵ ^ C⁻¹ .

Step-by-step solution

Let L₀ be the true length of 1 cm on both scales at 20^ C . At 0^ C , the temperature has decreased by T = 20^ C . The true length of 1 cm on the aluminium scale at 0^ C is: L_ Al = L₀(1 - _ Al T) The true length of 1 cm on the steel scale at 0^ C is: L_ steel = L₀(1 - _ steel T) The ratio of aluminium-centimetre to steel-centimetre at 0^ C is: Ratio = L_ Al L_ steel = 1 - _ Al T 1 - _ steel T Using binomial approximation since T 1 : Ratio (1 - _ Al T)(1 + _ steel T) 1 - ( _ Al - _ steel ) T Substituting the given