An open U-tube contains mercury. When 11.2 cm of water is poured into one of the arms of the tube, how high does the mercury rise in the other arm from its initial level ?

Options

A. 0.82 cm
B. 1.35 cm
C. 0.41 cm
D. 2.32 cm

Explanation

To solve the problem of how high the mercury rises in the other arm of the U-tube when 11.2 cm of water is poured into one arm, we can follow these steps:

### Step 1: Understand the Pressure Balance
In a U-tube, the pressure at the same horizontal level in both arms must be equal. When water is poured into one arm, it exerts pressure that causes the mercury to rise in the other arm.

### Step 2: Identify the Variables
- Let \( h \) be the height to which mercury rises in the other arm.
- The height of water poured is \( H_w = 11.2 \, \text{cm} \).
- The density of water \( \rho_w = 1000 \, \text{kg/m}^3 \).
- The density of mercury \( \rho_m = 13600 \, \text{kg/m}^3 \).

### Step 3: Write the Pressure Equations
The pressure exerted by the column of water in one arm is equal to the pressure exerted by the mercury column in the other arm:

\[ P_{\text{water}} = P_{\text{mercury}} \]

This can be expressed as:

\[ \rho_w g H_w = \rho_m g h \]

Where \( g \) is the acceleration due to gravity, which cancels out from both sides.

### Step 4: Cancel Out the Gravitational Acceleration
Since \( g \) is present in both terms, we can simplify the equation to:

\[ \rho_w H_w = \rho_m h \]

### Step 5: Solve for \( h \)
Rearranging the equation to solve for \( h \):

\[ h = \frac{\rho_w H_w}{\rho_m} \]

### Step 6: Substitute the Known Values
Substituting the known values into the equation:

\[ h = \frac{1000 \, \text{kg/m}^3 \times 11.2 \, \text{cm}}{13600 \, \text{kg/m}^3} \]

### Step 7: Convert Units
Convert \( 11.2 \, \text{cm} \) to meters for consistency:

\[ 11.2 \, \text{cm} = 0.112 \, \text{m} \]

Now substitute:

\[ h = \frac{1000 \times 0.112}{13600} \]

### Step 8: Calculate \( h \)
Calculating the value:

\[ h = \frac{112}{13600} = 0.00824 \, \text{m} \]

Converting back to centimeters:

\[ h = 0.00824 \times 100 = 0.824 \, \text{cm} \]

### Final Answer
The mercury rises approximately **0.824 cm** in the other arm of the U-tube.

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