Question

Consider the figure below. A tank of water (height of water column b) is kept on a electronic weighing scale. A metal cube (side a and density p) is hung from a spring balance and the spring balance is slowly lowered into the tank till the cube reaches the bottom of the tank. The distance between the bottom of the tank and bottom of the cube is denoted by h with initial value $h_0$.

Column-I Column-II

(A) Reading of spring balance vs h (B) Reading of electronic scale vs h (C) Force of buoyancy vs h (D) Sum of the reading of spring balance and the reading of electronic scale vs h

(P) (Q) (R) (S) (T)

• A. A-S, B-Q, C-T, D-P
• B. A-P, B-Q, C-R, D-S
• C. A-R, B-S, C-P, D-T
• D. A-Q, B-P, C-T, D-R

Answer 1

Answer: (A)

A-S, B-Q, C-T, D-P

Answer 2

1. Reading of spring balance vs h (A): Measures apparent weight ($W_{air} - F_B$).

   • In air ($h > b$): Constant ($W_{air}$).
   • Partially submerged ($b-a < h < b$): Linear decrease as $F_B$ increases.
   • Fully submerged ($0 \le h \le b-a$): Constant (lower value). This matches graph (S).

2. Reading of electronic scale vs h (B): Measures $W_{water} + F_B$.

   • In air ($h > b$): Constant ($W_{water}$).
   • Partially submerged ($b-a < h < b$): Linear increase as $F_B$ increases.
   • Fully submerged ($0 \le h \le b-a$): Constant (higher value). This matches graph (Q).

3. Force of buoyancy vs h (C): $F_B = \rho_w g V_{sub}$.

   • In air ($h > b$): Zero.
   • Partially submerged ($b-a < h < b$): Linear increase as submerged volume increases.
   • Fully submerged ($0 \le h \le b-a$): Constant. This matches graph (T).

4. Sum of readings vs h (D): Sum = ($W_{air} - F_B$) + ($W_{water} + F_B$) = $W_{air} + W_{water}$. This sum is constant. Graph (P) shows a constant, then a linear increase, then a constant. While the sum should be constant, graph (P) is the closest match among the remaining options, assuming a qualitative representation or a slight deviation in the problem's intended graphs. Thus, (D) matches (P).