A horizontal aluminium plate has a hole with a diameter of 2.000 cm . A steel sphere having a diameter of 2.005 cm rests on this hole. These lengths are measured at an initial temperature of 10^ C . The temperature of the entire system is then gradually increased. Determine the temperature at which the ball will fall through the hole. The coefficient of linear expansion for aluminium is 23 10⁻⁶ ^ C ⁻¹ and for steel i

Step-by-step solution

Let the initial temperature be T₁ = 10^ C and the final temperature be T₂ . The ball will fall through the hole when the diameter of the hole becomes equal to the diameter of the sphere. D_ Al (1 + _ Al T) = D_ Fe (1 + _ Fe T) T = D_ Fe - D_ Al D_ Al _ Al - D_ Fe _ Fe T = 2.005 - 2.000 2.000 23 10⁻⁶ - 2.005 11 10⁻⁶ T = 0.005 (46 - 22.055) 10⁻⁶ T = 0.005 23.945 10⁻⁶ 208.8^ C T₂ = T₁ + T = 10^ C + 208.8^ C = 218.8^ C 219^ C