Two metal rods P and Q of equal original length have linear expansivities in the ratio 2:3. If they are heated through the same temperature difference, find the ratio of the increase in their lengths.

Hey, good to see you again. a) The formula for linear expansion is given by: L = L_0 T where L is the increase in length, L_0 is the original length, is the linear expansivity, and T is the change in temperature. Step 1: Write the expressions for the increase in length for metal P and metal Q. For metal P: L_P = L_0P _P T For metal Q: L_Q = L_0Q _Q T Step 2: Form the ratio of the increase in lengths of P to Q. ( L_P)/( L_Q) = L_0P _P TL_0Q _Q T Step 3: Cancel out the common term T since the temperature difference is the same for both metals. ( L_P)/( L_Q) = L_0P _PL_0Q _Q Step 4: Rearrange the terms to use the given ratios. ( L_P)/( L_Q) = (L_0PL_0Q) × ((_P)/(_Q)) Step 5: Substitute the given ratios: The ratio of linear expansivities of P to Q is (_P)/(_Q) = (2)/(3). The ratio of their original lengths (P to Q) is L_0PL_0Q = (3)/(4). ( L_P)/( L_Q) = ((3)/(4)) × ((2)/(3)) Step 6: Calculate the product. ( L_P)/( L_Q) = (3 × 2)/(4 × 3) = (6)/(12) = (1)/(2) The ratio of the increase in lengths of P to Q is 1:2. Comparing this with the given options: (a) 1:2 (b) 2:1 (c) 8:9 (d) 9:8 The correct option is (a). The final answer is 1:2. Send me the next one 📸