A U-tube is partially filled with water. Oil which does not mix with water is next poured into one side untill water rises by 25 cm on the other side. If the density of oil be 0.8, the oil level will stand higher then the water level by

Options

A. 6.25cm
B. 12.50 cm
C. 31.75 cm
D. 62.50 cm

Answer

To solve the problem, we need to analyze the situation in the U-tube with water and oil. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have a U-tube partially filled with water. When oil is poured into one side, it causes the water level to rise by 25 cm on the other side. We need to find out how much higher the oil level is compared to the water level.

### Step 2: Define the Variables
- Let \( h_w = 25 \, \text{cm} \) (the rise in water level).
- Let \( \rho_w = 1 \, \text{g/cm}^3 \) (density of water).
- Let \( \rho_o = 0.8 \, \text{g/cm}^3 \) (density of oil).

### Step 3: Set Up the Pressure Equation
At equilibrium, the pressure exerted by the water column on one side must equal the pressure exerted by the oil column plus the water column on the other side.

The pressure due to the water column on the left side is:
\[ P_w = \rho_w g h_w \]
The pressure due to the oil column on the right side is:
\[ P_o = \rho_o g h_o \]
where \( h_o \) is the height of the oil column above the original water level.

### Step 4: Apply the Equilibrium Condition
At equilibrium, the pressures are equal:
\[ \rho_w g h_w = \rho_o g h_o \]
Since \( g \) (acceleration due to gravity) is constant and can be canceled out, we have:
\[ \rho_w h_w = \rho_o h_o \]

### Step 5: Substitute the Known Values
Substituting the known values into the equation:
\[ 1 \times 25 = 0.8 \times h_o \]
This simplifies to:
\[ 25 = 0.8 h_o \]

### Step 6: Solve for \( h_o \)
Now, solve for \( h_o \):
\[ h_o = \frac{25}{0.8} = 31.25 \, \text{cm} \]

### Step 7: Find the Height Difference
Now, we need to find how much higher the oil level is compared to the water level. The height difference \( \Delta h \) is given by:
\[ \Delta h = h_o - h_w \]
Substituting the values we found:
\[ \Delta h = 31.25 - 25 = 6.25 \, \text{cm} \]

### Final Answer
The oil level will stand higher than the water level by **6.25 cm**.

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