Question

By sucking through a straw, a student can reduce the pressure in his lungs to \(750 \mathrm{~mm}\) of \(\mathrm{Hg}\) (density \(\left.=13.6\left(\mathrm{gm} / \mathrm{cm}^{2}\right)\right)\) using the straw, he can drink water from \(\mathrm{a}\) glass up to a maximum depth of (A) \(10 \mathrm{~cm}\) (B) \(75 \mathrm{~cm}\) (C) \(13.6 \mathrm{~cm}\) (D) \(1.36 \mathrm{~cm}\)

Short answer

In this problem, we need to find the maximum depth of water a student can drink using a straw after reducing the pressure in his lungs. The pressure reduction is equal to \(1 cm\) of Hg. Using the relation \(h'= h \times \frac{{\rho_{Hg}}}{{\rho_{H2O}}}\), where \(h'\) is the height of water column, \(h\) is the height of mercury column, \(\rho_{Hg}\) is the density of mercury, and \(\rho_{H2O}\) is the density of water, we find that the student can drink water up to a maximum depth of \(13.6 cm\). Thus, the correct answer is (C) \(13.6 cm\).

Step-by-step solution

Understand the Problem

In this problem, a student reduces the pressure in his lungs by sucking through a straw, and as a result, he can drink water from a glass up to a certain maximum depth. We are asked to find this depth. The problem involves the concept of atmospheric pressure and its relationship with a column of liquid. The atmospheric pressure is equal to the pressure exerted by a column of mercury of height 760 mm, and the student can reduce this to 750 mm of Hg.

Choose the Correct Concept

To solve this problem, we need to use the concept that the amount of pressure reduction corresponds to the height of water column that can be sucked up. In essence, we know the change in pressure in mercury units (760mm-750mm=10mm), our task is to convert this change in pressure to the height in water column units.

Convert Units

Before we make the conversion, we need to change the pressure change measure to cm (the same height units as water) since our options are also in cm units. Given \(1 mm = 0.1 cm\), hence, the pressure change is \(10 mm = 1 cm\) of Hg.

Use Pressure and Density to Find Height

We will use the relation \(h'= h \times \frac{{\rho_{Hg}}}{{\rho_{H2O}}}\), where \(h'\) is the height of water column, \(h\) is the height of mercury column, \(\rho_{Hg}\) is the density of mercury and \(\rho_{H2O}\) is the density of water. Using the given densities, we convert pressure reduction in Hg to equivalent height in water. Simply substitute the values we have into the equation: \(h' = 1 \times \frac{{13.6 gm/cm^3}}{{1 gm/cm^3}}\), which simplifies to \(h' = 13.6 cm\). Thus, the student can drink water from a glass up to a maximum depth of 13.6 cm. Therefore, the correct answer is (C) \(13.6 cm\).

Key concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the air above us in the atmosphere. It's what allows us to drink through a straw, among many other things. In our problem, the atmospheric pressure equals the pressure exerted by a column of mercury that's 760 mm tall, and the student can reduce it to 750 mm of mercury by sucking through a straw.
Simply put, atmospheric pressure is the natural pressure created by the air molecules all around us. It changes with elevation: it's higher at sea level and lessens as we go higher up into the mountains. This pressure is crucial in various daily activities and plays an important role in understanding fluid mechanics.
When we drink with a straw, we are lowering the pressure inside, causing the liquid to rise to fill that lower pressure space. Thus, knowing how to manipulate atmospheric pressure can help us in understanding fluid behavior.

Density of Mercury

The density of a substance is essentially a measure of how much of a material is packed into a given volume. Mercury, known for its density, has an impressive density of 13.6 gm/cm extsuperscript{3}. This means mercury is much heavier than most liquids, like water.
Mercury's high density is why it was traditionally used in devices like barometers to measure atmospheric pressure. In such cases, even small changes in atmospheric pressure lead to significant shifts in the mercury column's height. This property is used in our problem to help convert the change in pressure that the student can create into the depth of a column of water the student could drink up.
Understanding the density of mercury also demonstrates how it can create a much higher pressure with the same height compared to other liquids. Thus, its density is a key factor when comparing the pressure between two different substances.

Pressure Conversion

Pressure conversion is often necessary in fluid mechanics to compare and relate different kinds of pressures exerted by various columnar liquids. Our original problem required converting a pressure change in mercury (10 mm) to a corresponding height in water.
The conversion involved here uses the formula: \( h' = h \times \frac{\rho_{Hg}}{\rho_{H2O}} \),where \( h \) is the height of the mercury column, and \( h' \) is the height of the water column. By using the densities ( and ), we can relate the height of the mercury column to that of water. This conversion shows how much of one liquid can be equaled by a specific amount of another.
So, always keep in mind the importance of being able to convert pressure between styles or measurements: it helps in understanding relative forces between different systems.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at rest due to gravity. It's a key concept in fluid mechanics and is responsible for the climbing water in the straw when the atmospheric pressure decreases.
The depth of the fluid plays a crucial role in hydrostatic pressure: the deeper we go, the more pressure there is. This pressure can be calculated as \( P = \rho \cdot g \cdot h \),where \( \rho \) is the density, \( g \) is the gravitational acceleration, and \( h \) is the height of the fluid column. What's fascinating is that this pressure depends only on the density, gravity, and height, irrespective of the shape of the container.
Understanding this concept helps decipher why, even with a small change in pressure, such as through the act of sipping a straw, significant fluid movement can be achieved. Hydrostatic pressure is a foundational idea that explains much about how fluids behave under various conditions.