Question

A vesel whose bottom has round holes with diameter of \(0.1 \mathrm{~mm}\) is filled with water. The maximum height to which the water can be filled without leakage is (S.T. of water \(=[(75\) dyne \(\left.\\} / \mathrm{cm}], \mathrm{g}=1000 \mathrm{~m} / \mathrm{s}^{2}\right)\) (A) \(100 \mathrm{~cm}\) (B) \(75 \mathrm{~cm}\) (C) \(50 \mathrm{~cm}\) (D) \(30 \mathrm{~cm}\)

Short answer

The maximum height to which the water can be filled without leakage is \(75 \mathrm{~cm}\).

Step-by-step solution

Write the expression for surface tension force

The force due to surface tension (F) is given by the product of the surface tension (S) and the circumference of the hole: F = S × 2πr Where, r is the radius of the hole.

Calculate the radius

Calculate the radius (r) from the given diameter of the hole (0.1 mm): r = diameter/2 = 0.1/2

Convert units

Convert the units of r from millimeters to centimeters: r = 0.1/2 × (1/10) = 0.005 cm

Write the expression for gravitational force

Gravitational force (F') is given by the product of volume, density (ρ), and acceleration due to gravity (g): F' = Vρg Where, V is the volume of the water cylinder inside the hole.

Write the expression for volume

Write the volume (V) of the water cylinder as the product of the cross-sectional area of the hole and the maximum height (H) of the water: V = πr^2H

Write the conditions for balance between forces

At equilibrium, the gravitational force (F') must be equal to the surface tension force (F): S × 2πr = πr^2Hρg

Solve for H

Divide both sides by πr and the given values for S, ρ and g to solve for H: H = \(\frac{(2S)}{rρg}\) = \(\frac{150~\text{dyne/cm} × 2}{0.005~\text{cm} × 1~\text{g/cm}^{3} × 1000~\text{cm/s}^2}\) H = 75 cm The correct answer is (B) \(75 \mathrm{~cm}\). The maximum height to which the water can be filled without leakage is 75 cm.

Key concepts

Gravitational Force

Gravitational force is a vital concept in understanding how objects interact under the influence of gravity. When dealing with liquids, this force plays a crucial role in determining the behavior of the liquid in various vessels. Gravity acts on every particle of the liquid, pulling it downward. This force can be mathematically expressed as \[ F' = V \rho g \]where:

• \( V \) is the volume of the liquid.
• \( \rho \) is the density of the liquid.
• \( g \) is the acceleration due to gravity.

In many problems, including this exercise, the gravitational force must balance with other forces, such as surface tension, to establish equilibrium. Recognizing this force helps us understand if and how the liquid will stay in place or flow out of a container.

Unit Conversion

Unit conversion is a fundamental skill in solving physics problems. It ensures that all measurements are in consistent units for accurate calculations. In this exercise, we converted the diameter of holes from millimeters to centimeters to match other units in the equation. Here's a simple way to carry out such conversions:

• Identify the given units and required units.
• Use conversion factors to change the units. For example, 1 mm = 0.1 cm.
• Perform the conversion calculation. For instance, the radius was calculated as \( r = 0.1/2 \times (1/10) = 0.005 \) cm.

This step is crucial for ensuring accuracy, especially when dealing with equations where precision is key. Consistent units help avoid calculation errors and make it easier to compare and analyze results.

Forces Balance

Understanding forces balance is essential to solve problems involving different forces acting on a system. In this context, forces balance refers to the equality of surface tension and gravitational forces, which ensures the liquid stays inside the vessel.Here's how you can understand this balance:

• Surface tension force is calculated as \( F = S \times 2\pi r \), where \( S \) is the surface tension and \( r \) is the radius of the hole.
• Gravitational force is \( F' = V \rho g \).
• At equilibrium, these forces must be equal: \( S \times 2\pi r = \pi r^2 H \rho g \).

This equation helps us solve for unknowns such as height \( H \) by rearranging the equation and substituting known values. Understanding forces balance is crucial in ensuring accurate conclusions about how far water can be filled without leaking. It bridges the gap between theoretical calculations and real-world phenomena.