Two metre scales, one of steel and the other of aluminium, have exactly the same readings at 20^ C . Find the ratio aluminium-centimetre/steel-centimetre at 40^ C . Given that for steel = 1.1 10⁻⁵ ^ C⁻¹ and for aluminium = 2.3 10⁻⁵ ^ C⁻¹ .

Step-by-step solution

Let the true length of 1 cm on both scales at 20^ C be l₀ . At 40^ C , the physical length of 1 cm on the aluminium scale (aluminium-centimetre) becomes: l_ Al = l₀(1 + _ Al T) Similarly, the physical length of 1 cm on the steel scale (steel-centimetre) becomes: l_ St = l₀(1 + _ St T) The ratio of aluminium-centimetre to steel-centimetre at 40^ C is given by: l_ Al l_ St = 1 + _ Al T 1 + _ St T Since _ St T 1 , using the binomial approximation (1 + x)⁻¹ 1 - x : l_ Al l_ St (1 + _ Al T)(1 - _ St T) 1 + ( _ Al - _ St