A U-tube containing a liquid is accelerated horizontally with a constant acceleration `a_0`. If the separation between the vertical limbs is l find the difference in the heights of the liquid in the two arms.

Explanation

To solve the problem of finding the difference in heights of the liquid in the two arms of a U-tube that is being accelerated horizontally, we can follow these steps:

### Step 1: Understand the setup
We have a U-tube containing a liquid, and the entire U-tube is being accelerated horizontally with a constant acceleration \( a_0 \). This acceleration will affect the pressure distribution in the liquid.

### Step 2: Define the pressure difference
Let \( h \) be the difference in height of the liquid in the two arms of the U-tube. The pressure difference between the two arms due to this height difference can be expressed using the hydrostatic pressure formula:
\[ P_1 - P_2 = \rho g h \]
where:
- \( P_1 \) is the pressure at the bottom of the higher liquid column,
- \( P_2 \) is the pressure at the bottom of the lower liquid column,
- \( \rho \) is the density of the liquid,
- \( g \) is the acceleration due to gravity.

### Step 3: Consider the effect of acceleration
When the U-tube is accelerated with an acceleration \( a_0 \), there is an additional pressure difference created due to the horizontal acceleration. This pressure difference can be expressed as:
\[ P_1 - P_2 = \rho L a_0 \]
where \( L \) is the horizontal separation between the two vertical limbs of the U-tube.

### Step 4: Set the two pressure differences equal
Since both expressions represent the same pressure difference, we can set them equal to each other:
\[ \rho g h = \rho L a_0 \]

### Step 5: Simplify the equation
We can cancel \( \rho \) from both sides of the equation (assuming the density is constant and non-zero):
\[ g h = L a_0 \]

### Step 6: Solve for the height difference \( h \)
Now, we can solve for \( h \):
\[ h = \frac{L a_0}{g} \]

### Final Result
Thus, the difference in the heights of the liquid in the two arms of the U-tube is given by:
\[ h = \frac{L a_0}{g} \]

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