Question

current attempt in progress a ball is slightly too large to fit through a hole in a flat plate. the drawing shows two arrangements of this situation. in arrangement i the ball is made from metal a and the plate from metal b. when both the ball and the plate are cooled by the same number of celsius degrees, the ball passes through the hole. in arrangement ii the ball is also made from metal a, but the plate is made from metal c. here, the ball passes through the hole when both the ball and the plate are heated by the same number of celsius degrees. rank the coefficients of linear thermal expansion of metals a, b, and c in descending order (largest first). arrangement i (ball a, plate b) arrangement ii (ball a, plate c) options: 1. $\alpha_a, \alpha_b, \alpha_c$ 2. $\alpha_b, \alpha_c, \alpha_a$ 3. $\alpha_b, \alpha_a, \alpha_c$ 4. $\alpha_c, \alpha_b, \alpha_a$ 5. $\alpha_c, \alpha_a, \alpha_b$

Explanation:

Step1: Analyze Arrangement I

When cooled, the ball (metal A) and plate (metal B) change size. The ball passes through, so the plate (metal B) contracts more than the ball (metal A). Using the formula for linear thermal expansion \(\Delta L = L_0\alpha\Delta T\) (where \(\Delta L\) is the change in length, \(L_0\) is the original length, \(\alpha\) is the coefficient of linear thermal expansion, and \(\Delta T\) is the temperature change). Since \(\Delta T\) is negative (cooling) and \(L_0\) is positive (assuming initial non - zero lengths), \(\alpha_B>\alpha_A\) (because \(\vert\Delta L_B\vert>\vert\Delta L_A\vert\) for the hole to become larger relative to the ball).

Step2: Analyze Arrangement II

When heated, the ball (metal A) and plate (metal C) change size. The ball passes through, so the plate (metal C) expands more than the ball (metal A). Using \(\Delta L = L_0\alpha\Delta T\) (where \(\Delta T\) is positive (heating)), \(\alpha_C>\alpha_A\). Also, when comparing B and C:
In Arrangement I (cooling), B contracts more (for the hole to be large enough for A). In Arrangement II (heating), C expands more (for the hole to be large enough for A). Let's assume similar initial dimensions. The rate of change (related to \(\alpha\)): when cooled, B has a larger magnitude of contraction (for the hole to be useful), and when heated, C has a larger magnitude of expansion. If we consider the fact that for the hole (in the plate) to change size relative to the ball, and assuming similar initial geometries, \(\alpha_C>\alpha_B\) (because heating and cooling are just temperature changes of opposite sign, and the effect on the hole - ball fit implies a larger \(\alpha\) for C compared to B).

Answer:

\(\alpha_C,\alpha_B,\alpha_A\)