A vessel containing water is given a constant acceleration a towards the right,a straight horizontal path. Which of the following diagram represents the surface of the liquid.

Options

A.
B.
C.
D. none of these

Explanation

To solve the problem of how the surface of the liquid in a vessel behaves when the vessel is given a constant acceleration to the right, we can follow these steps:

### Step 1: Understand the Situation
When the vessel containing water is accelerated to the right with a constant acceleration \( a \), the water inside the vessel will not remain at rest. Instead, it will experience a pseudo force acting in the opposite direction of the acceleration (to the left).

### Step 2: Analyze the Forces
The liquid particles experience a pseudo force due to the acceleration of the vessel. This pseudo force can be represented as \( F_{\text{pseudo}} = m \cdot a \), where \( m \) is the mass of the liquid particle. This force acts horizontally to the left on the liquid particles.

### Step 3: Determine the Effect on the Liquid Surface
Due to the pseudo force, the liquid will not remain horizontal. Instead, the surface of the liquid will tilt. The liquid particles will accumulate more towards the left side of the vessel, creating an inclined surface.

### Step 4: Calculate the Angle of Inclination
The angle of inclination \( \theta \) of the liquid surface can be determined using the relationship between the gravitational force acting downwards and the pseudo force acting horizontally. The tangent of the angle \( \theta \) can be expressed as:
\[ \tan(\theta) = \frac{a}{g} \]
where \( g \) is the acceleration due to gravity.

### Step 5: Identify the Correct Diagram
Among the given options, the diagram that shows the surface of the liquid inclined to the horizontal at an angle \( \theta \) (where the left side is higher than the right side) represents the correct shape of the liquid surface when the vessel is accelerated to the right.

### Conclusion
The correct option that depicts the shape of the liquid inside the vessel when it is accelerated towards the right is option number three.

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