Equal mass of three liquids are kept in three identical cylindrical vessels A, B and C. the densities are `rho_A, rho_B, rho_C with rho_Altrho_Bltrho_C`. The force on the base will be

Options

A. maximum in vessel A
B. maximum in vessel B
C. maximum in vessel C
D. equal in al the vessels

Answer

To solve the problem of determining the force on the base of three identical cylindrical vessels containing equal masses of three different liquids with varying densities, we can follow these steps:

### Step 1: Understand the relationship between mass, density, and volume
The mass \( m \) of a liquid can be expressed in terms of its density \( \rho \) and volume \( V \) using the formula:
\[ m = \rho \cdot V \]
For a cylindrical vessel, the volume \( V \) can be expressed as:
\[ V = A \cdot h \]
where \( A \) is the base area and \( h \) is the height of the liquid column.

### Step 2: Express height in terms of mass and density
Since we have equal masses of the liquids in the vessels, we can rearrange the mass formula to express height \( h \):
\[ h = \frac{m}{\rho \cdot A} \]
This equation shows that the height of the liquid column depends on the mass of the liquid, its density, and the base area of the vessel.

### Step 3: Calculate the pressure at the base of the vessel
The pressure \( P \) at the base of the liquid column can be calculated using the hydrostatic pressure formula:
\[ P = \rho \cdot g \cdot h \]
Substituting the expression for \( h \) from Step 2 into the pressure equation gives:
\[ P = \rho \cdot g \cdot \left(\frac{m}{\rho \cdot A}\right) = \frac{m \cdot g}{A} \]

### Step 4: Calculate the force on the base of the vessel
The force \( F \) on the base of the vessel is given by the product of pressure and area:
\[ F = P \cdot A = \left(\frac{m \cdot g}{A}\right) \cdot A = m \cdot g \]
This shows that the force on the base depends only on the mass of the liquid and the acceleration due to gravity, and is independent of the density of the liquid.

### Step 5: Conclusion
Since all three vessels contain equal masses of liquid, the force on the base of each vessel will be the same:
\[ F_A = F_B = F_C = m \cdot g \]
Thus, the correct answer is that the force on the base will be equal in all three vessels.

### Final Answer
The force on the base will be equal in all the vessels.

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