A solid is immersed completely in a liquid. The temperature coefficients of volume expansion of solid and the liquid are `gamma_(1)` and `gamma_(2) (lt gamma_(1))`. If temperatures of both are increased, then match the following.
`|{:(,"Table-1",,"Table-2"),((A),"upthrust on the solid will",(P),"increase"),((B),"apparent weight of the solid will",(Q),"decrease"),((C),"fraction of volume immersed in",(R),"remain same"),(,"the liquid if allowed to float will",,):}|`

Answer

To solve the problem, we need to analyze the effects of temperature changes on a solid immersed in a liquid, considering their coefficients of volume expansion. Let's break it down step by step.

### Step 1: Understand the Coefficients of Volume Expansion
The coefficients of volume expansion for the solid and the liquid are given as:
- \( \gamma_1 \) for the solid
- \( \gamma_2 \) for the liquid, where \( \gamma_2 < \gamma_1 \)

This means that the solid expands more than the liquid when the temperature increases.

### Step 2: Analyze the Upthrust (Buoyant Force)
When a solid is immersed in a liquid, it experiences an upthrust (buoyant force) equal to the weight of the liquid displaced by the solid. As the temperature increases:
- The volume of the solid increases more than the volume of the liquid due to the higher coefficient of expansion of the solid.
- Therefore, the volume of liquid displaced increases, leading to an increase in upthrust.

**Conclusion for A**: The upthrust on the solid will **increase**.

### Step 3: Analyze the Apparent Weight of the Solid
The apparent weight of the solid is given by the actual weight minus the upthrust. Since the upthrust increases (as established in Step 2), the apparent weight will decrease.

**Conclusion for B**: The apparent weight of the solid will **decrease**.

### Step 4: Analyze the Fraction of Volume Immersed
If the solid is allowed to float, the fraction of its volume that is immersed in the liquid is determined by the balance of weights (the weight of the solid and the upthrust). Since both the solid and the liquid expand, but the solid expands more, the fraction of the volume that is immersed will remain the same.

**Conclusion for C**: The fraction of volume immersed in the liquid will **remain the same**.

### Step 5: Match the Results with the Given Tables
Now we can match our conclusions with the options provided in the tables:

- A (upthrust on the solid will) matches with P (increase)
- B (apparent weight of the solid will) matches with Q (decrease)
- C (fraction of volume immersed in the liquid if allowed to float will) matches with R (remain same)

### Final Matching:
- A → P
- B → Q
- C → R

### Summary of the Solution:
- Upthrust on the solid will **increase** (A → P)
- Apparent weight of the solid will **decrease** (B → Q)
- Fraction of volume immersed in the liquid will **remain the same** (C → R)