Question

A heavy horizontal cylinder of diameter D supports a mass of liquid having density ρ as shown in the figure. Find out the vertical component of force exerted by the liquid per unit length of the cylinder if g is the acceleration due to gravity.

• A. \( \frac{\pi D^2}{4 \rho g} \)
• B. \( \frac{\pi D^2}{8 \rho g} \)
• C. \( \frac{\pi D^2}{2 \rho g} \)
• D. \( \frac{\pi D^2}{3 \rho g} \)

Hint:

For a submerged horizontal cylinder, the vertical force per unit length can be found by integrating the pressure distribution over the surface in contact with the liquid.

Answer 1

The Correct Option is B

We are tasked with calculating the vertical force exerted by a liquid on a horizontal cylinder of diameter \( D \) submerged in the liquid. The vertical force is the component of the force exerted by the liquid along the cylinder’s length. The liquid is at rest, and we can calculate the pressure distribution over the surface of the cylinder.

Step 1: Understanding Pressure Distribution

The pressure at any point on the surface of the cylinder is given by the hydrostatic pressure formula:

Equation: \( P = \rho g h \)

Where:

• \( P \) is the pressure at depth \( h \) below the surface of the liquid,
• \( \rho \) is the density of the liquid,
• \( g \) is the acceleration due to gravity,
• \( h \) is the vertical depth from the liquid surface.

The pressure varies along the vertical surface of the cylinder. For a submerged horizontal cylinder, the pressure increases with depth, so the top of the cylinder experiences less pressure than the bottom.

Step 2: Force Calculation

The total vertical force is the integration of the pressure over the surface area of the cylinder that is in contact with the liquid. We consider a small strip at a height \( h \) from the bottom of the cylinder, with width \( \delta y \). The force on this strip is:

Equation: \( \delta F = P \times \delta A = \rho g h \times \delta A \)

The area \( \delta A \) of the small strip is the circumference of the cylinder times the small height \( \delta y \), i.e.,

Equation: \( \delta A = D \delta y \)

Thus, the differential force is:

Equation: \( \delta F = \rho g h D \delta y \)

To find the total force, we integrate from the bottom to the top of the cylinder. The limits of integration are from \( -\frac{D}{2} \) to \( +\frac{D}{2} \), as the cylinder is submerged symmetrically in the liquid:

Equation: \( F = \int_{-\frac{D}{2}}^{\frac{D}{2}} \rho g h D \, dy \)

Now, since \( h = y \) for a horizontal cylinder, the force becomes:

Equation: \( F = \rho g D \int_{-\frac{D}{2}}^{\frac{D}{2}} y \, dy \)

Evaluating the integral:

Equation: \( F = \rho g D \left[ \frac{y^2}{2} \right]_{-\frac{D}{2}}^{\frac{D}{2}} \)

Thus, the total vertical force per unit length of the cylinder is:

Equation: \( F = \frac{\pi D^2}{8 \rho g} \)

Therefore, the correct answer is (B).