Question

question if instead of heating the copper ring in part (a) the steel rod is cooled, would the magnitude of the required temperature change be larger, smaller, or the same? why? (dont calculate it!) larger, because steel has a smaller coefficient of area expansion larger, because the steel rod has a smaller radius smaller, because the steel rod has a smaller radius smaller, because steel has a larger coefficient of area expansion practice it use the worked example above to help you solve this problem. (a) a circular copper ring at 19.0°c has a hole with an area of 9.95 cm². what minimum temperature must it have so that it can be slipped onto a steel metal rod having a cross - sectional area of 10.0 cm²? 166.8°c (b) suppose the ring and the rod are heated simultaneously. what change in temperature of both will allow the ring to be slipped onto the end of the rod? (assume no significant change in the coefficients of linear expansion over this temperature range.) 422.7°c exercise hints: getting started | im stuck! a steel ring with a hole having area of 8.970 cm² is to be placed on an aluminum rod with cross - sectional area of 9.000 cm². both rod and ring are initially at a temperature of 25.0°c. at what common temperature can the steel ring be slipped onto one end of the aluminum rod? (the coefficient of linear expansion of steel is 11×10⁻⁶ (°c)⁻¹, and the coefficient of linear expansion of aluminum is 24×10⁻⁶ (°c)⁻¹.) °c resources

Explanation:

Step1: Area expansion formula

The formula for area expansion is \(A = A_0(1+\gamma\Delta T)\), where \(\gamma = 2\alpha\) ( \(\alpha\) is the coefficient of linear expansion). For the copper ring in part (a), when heated, \(A_{copper}=A_{0,copper}(1 + \gamma_{copper}\Delta T_{copper})\). If we cool the steel rod, \(A_{steel}=A_{0,steel}(1-\gamma_{steel}\Delta T_{steel})\). We want \(A_{copper,f}=A_{steel,f}\). Since \(\gamma_{copper}>\gamma_{steel}\) (copper has a larger coefficient of linear expansion, so larger \(\gamma\) for area expansion as \(\gamma = 2\alpha\)), from \(A = A_0(1\pm\gamma\Delta T)\), to get the same change in area (\(A_{copper,f}-A_{0,copper}=A_{0,steel}-A_{steel,f}\)), because \(\gamma_{steel}<\gamma_{copper}\), from \(\Delta A=A_0\gamma\Delta T\), \(\Delta T=\frac{\Delta A}{A_0\gamma}\). Given \(A_0\) (relevant to the problem - the initial areas are set for a fit) and \(\Delta A\) (the amount of area change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change 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Answer:

Step1: Area expansion formula

The formula for area expansion is \(A = A_0(1+\gamma\Delta T)\), where \(\gamma = 2\alpha\) ( \(\alpha\) is the coefficient of linear expansion). For the copper ring in part (a), when heated, \(A_{copper}=A_{0,copper}(1 + \gamma_{copper}\Delta T_{copper})\). If we cool the steel rod, \(A_{steel}=A_{0,steel}(1-\gamma_{steel}\Delta T_{steel})\). We want \(A_{copper,f}=A_{steel,f}\). Since \(\gamma_{copper}>\gamma_{steel}\) (copper has a larger coefficient of linear expansion, so larger \(\gamma\) for area expansion as \(\gamma = 2\alpha\)), from \(A = A_0(1\pm\gamma\Delta T)\), to get the same change in area (\(A_{copper,f}-A_{0,copper}=A_{0,steel}-A_{steel,f}\)), because \(\gamma_{steel}<\gamma_{copper}\), from \(\Delta A=A_0\gamma\Delta T\), \(\Delta T=\frac{\Delta A}{A_0\gamma}\). Given \(A_0\) (relevant to the problem - the initial areas are set for a fit) and \(\Delta A\) (the amount of area change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change change 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