A vessel in the form of a truncated cone has base area `a_(1)` and top area `a_(2)`. It is filled to the brim with a liquid. The force at the bottom of the vessel is

Options

A. a.Equal to weight of the liquid, if `a_(1) lt a_(2)`
B. b.Greater than weight of the liquid, if `a_(2) lt a_(1)`
C. c.Less than weight of the liquid, if `a_(1) lt a_(2)`
D. d.Equal to weight of the liquid, for any value of the areas

Answer

To find the force at the bottom of a truncated cone vessel filled with liquid, we can follow these steps:

### Step 1: Understand the System
The vessel is in the form of a truncated cone with a base area \( A_1 \) and a top area \( A_2 \). It is filled with a liquid, and we need to analyze the forces acting at the bottom of the vessel.

### Step 2: Identify the Weight of the Liquid
The weight of the liquid (\( W \)) can be expressed as:
\[ W = \rho V g \]
where \( \rho \) is the density of the liquid, \( V \) is the volume of the liquid, and \( g \) is the acceleration due to gravity.

### Step 3: Calculate the Volume of the Liquid
The volume \( V \) of the liquid in a truncated cone can be calculated using the formula:
\[ V = \frac{1}{3} h (A_1 + A_2 + \sqrt{A_1 A_2}) \]
where \( h \) is the height of the liquid column.

### Step 4: Determine the Force at the Bottom
The force \( F \) at the bottom of the vessel is the sum of the weight of the liquid and the normal force exerted by the liquid on the bottom surface. The normal force acts vertically upward and balances the weight of the liquid. Therefore, the total force at the bottom can be expressed as:
\[ F = W + N_v \]
where \( N_v \) is the vertical component of the normal force.

### Step 5: Analyze the Cases
1. **Case 1**: If \( A_1 < A_2 \) (the base area is smaller than the top area), the force at the bottom will be equal to the weight of the liquid since the normal force will not contribute additional force.
2. **Case 2**: If \( A_1 > A_2 \) (the base area is larger than the top area), the force at the bottom will be greater than the weight of the liquid due to the additional normal force acting downwards.

### Step 6: Conclusion
From the analysis, we can conclude:
- If \( A_1 < A_2 \), then \( F = W \).
- If \( A_1 > A_2 \), then \( F > W \).
- If \( A_1 = A_2 \), then \( F = W \).

Thus, the force at the bottom of the vessel is equal to the weight of the liquid when \( A_1 < A_2 \) and greater than the weight of the liquid when \( A_1 > A_2 \).

### Final Answer
The correct options based on the analysis are:
- Greater than the weight of the liquid if \( A_1 > A_2 \).
- Equal to the weight of the liquid if \( A_1 < A_2 \).