To what height h should a cylindrical vessel of diameter d be filled with a liquid so that the total force on the vertical surface of the vessel be equal to the force on the bottom-

Options

A. h=d
B. h=2d
C. h=3d
D. h=d/2

Answer

To solve the problem of determining the height \( h \) to which a cylindrical vessel of diameter \( d \) should be filled with liquid so that the total force on the vertical surface of the vessel equals the force on the bottom, we can follow these steps:

### Step 1: Calculate the Force on the Bottom of the Cylinder
The force exerted at the bottom of the cylinder can be calculated using the formula:
\[ F_{\text{bottom}} = P \cdot A \]
where \( P \) is the pressure at the bottom and \( A \) is the area of the bottom.

The pressure at the bottom of the cylinder is given by:
\[ P = \rho g h \]
where \( \rho \) is the density of the liquid, \( g \) is the acceleration due to gravity, and \( h \) is the height of the liquid.

The area \( A \) of the bottom of the cylindrical vessel is:
\[ A = \pi r^2 = \pi \left( \frac{d}{2} \right)^2 = \frac{\pi d^2}{4} \]

Thus, the force on the bottom is:
\[ F_{\text{bottom}} = \rho g h \cdot \frac{\pi d^2}{4} \]

### Step 2: Calculate the Force on the Vertical Surface of the Cylinder
The force exerted by the liquid on the vertical surface of the cylinder can be found by integrating the pressure over the height of the liquid. The pressure at a height \( y \) is:
\[ P(y) = \rho g y \]
The differential force \( dF \) on a differential area \( dA \) at height \( y \) is:
\[ dF = P(y) \cdot dA = \rho g y \cdot dA \]
The differential area \( dA \) of the vertical surface at height \( y \) is:
\[ dA = 2\pi r \, dy = 2\pi \left( \frac{d}{2} \right) dy = \pi d \, dy \]

Thus, the total force on the vertical surface from height \( 0 \) to \( h \) is:
\[ F_{\text{vertical}} = \int_0^h \rho g y \cdot \pi d \, dy \]
Calculating this integral:
\[ F_{\text{vertical}} = \rho g \pi d \int_0^h y \, dy = \rho g \pi d \left[ \frac{y^2}{2} \right]_0^h = \rho g \pi d \cdot \frac{h^2}{2} \]

### Step 3: Set the Forces Equal
To find the height \( h \) such that the total force on the vertical surface equals the force on the bottom, we set:
\[ F_{\text{bottom}} = F_{\text{vertical}} \]
Substituting the expressions we derived:
\[ \rho g h \cdot \frac{\pi d^2}{4} = \rho g \pi d \cdot \frac{h^2}{2} \]

### Step 4: Simplify and Solve for \( h \)
Canceling \( \rho g \) and \( \pi \) from both sides:
\[ \frac{h d^2}{4} = \frac{h^2 d}{2} \]
Rearranging gives:
\[ h d^2 = 2h^2 \]
Dividing both sides by \( h \) (assuming \( h \neq 0 \)):
\[ d^2 = 2h \]
Thus, we find:
\[ h = \frac{d^2}{2} \]

### Conclusion
The height \( h \) to which the cylindrical vessel should be filled is:
\[ h = \frac{d}{2} \]