Some liquid is filled in a cylindrical vessel of radius R. Let `F_(1)` be the force applied by the liquid on the bottom of the cylinder. Now the same liquid is poured into a vessel of uniform square cross-section of side R. Let `F_(2)` be the force applied by the liquid on the bottom of this new vessel. (Neglect atmosphere pressure). Then

Options

A. `F_(1) = pi F_(2)`
B. `F_(1) = (F_(2))/(pi)`
C. `F_(1) = sqrt(pi)F_(2)`
D. `F_(1) = F_(2)`

Answer

To solve the problem, we need to analyze the forces exerted by the liquid in both vessels.

### Step-by-Step Solution:

1. **Identify the Force on the Bottom of the Cylinder (F1)**:
- The force exerted by the liquid on the bottom of the cylindrical vessel is equal to the weight of the liquid in the cylinder.
- The weight of the liquid can be calculated using the formula:
\[ F_1 = \text{Weight of liquid} = \text{Volume of liquid} \times \text{Density of liquid} \times g \]
- The volume of the liquid in a cylindrical vessel is given by:
\[ V = \pi R^2 h \]
where \( h \) is the height of the liquid.

2. **Identify the Force on the Bottom of the Square Vessel (F2)**:
- Similarly, the force exerted by the liquid on the bottom of the square vessel is also equal to the weight of the liquid.
- The volume of the liquid in the square vessel is given by:
\[ V = R^2 h \]
- Thus, the force on the bottom of the square vessel can be expressed as:
\[ F_2 = \text{Weight of liquid} = \text{Volume of liquid} \times \text{Density of liquid} \times g = R^2 h \times \rho \times g \]

3. **Equate the Forces (F1 and F2)**:
- Since the same amount of liquid is poured into both vessels, the volume of liquid remains constant. Therefore, the weight of the liquid in both cases is the same.
- This means:
\[ F_1 = F_2 \]

4. **Conclusion**:
- Since \( F_1 \) is equal to \( F_2 \), we conclude that:
\[ F_1 = F_2 \]

### Final Answer:
Thus, the force applied by the liquid on the bottom of the cylindrical vessel is equal to the force applied by the liquid on the bottom of the square vessel:
\[ F_1 = F_2 \]