The pressure `P_(1)` at the top of a dam and `P_(2)` at a depth h from the top inside water (density `rho`) are related as :

Options

A. `P_(1)gtP_(2)`
B. `P_(1)=P_(2)`
C. `P_(1)-P_(2)=hrhog`
D. `P_(2)-P_(1)=hrhog`

Answer

To find the relationship between the pressure \( P_1 \) at the top of a dam and the pressure \( P_2 \) at a depth \( h \) from the top inside the water with density \( \rho \), we can follow these steps:

### Step 1: Understand the Concept of Pressure in Fluids
Pressure in a fluid increases with depth due to the weight of the fluid above. The pressure at a certain depth can be calculated using the formula:
\[ P = P_0 + \rho g h \]
where:
- \( P \) is the pressure at depth,
- \( P_0 \) is the pressure at the surface (or at the top),
- \( \rho \) is the density of the fluid,
- \( g \) is the acceleration due to gravity,
- \( h \) is the depth from the surface.

### Step 2: Identify the Pressures
In this case:
- \( P_1 \) is the pressure at the top of the dam (which is \( P_0 \)).
- \( P_2 \) is the pressure at depth \( h \).

### Step 3: Write the Pressure Relation
According to the formula, the pressure at depth \( h \) can be expressed as:
\[ P_2 = P_1 + \rho g h \]

### Step 4: Rearranging the Equation
To find the relationship between \( P_2 \) and \( P_1 \), we can rearrange the equation:
\[ P_2 - P_1 = \rho g h \]

### Step 5: Conclusion
Thus, the relationship between the pressures is:
\[ P_2 - P_1 = \rho g h \]
This indicates that the pressure at depth \( h \) is greater than the pressure at the top of the dam by an amount equal to the weight of the water column above that depth.

### Final Answer
The correct relationship is:
\[ P_2 - P_1 = \rho g h \]

---