Animated Solution for Physics - Properties of Solids and Liquids: An object is located at $2\text{ km}$ beneath the surface of the water. If the fractional compression $\Delta V/V$ is $1.36\%$, the ratio of hydraulic stress to the corresponding hydraulic strain will be ............... . (Take, density of water is $1000{\text{ kg m}}^{-3}$ and $g=9.81{\text{ ms}}^{-2}$)

Imagine you are standing on the deck of a ship, looking down into the vast, dark expanse of the ocean. Now, picture an object sinking deep into this abyss, reaching a staggering depth of 2 kilometers. At this depth, the water doesn't just rest against the object; it crushes it from every conceivable angle. This crushing force, distributed over the object's surface area, is what we call hydrostatic pressure, and it acts as the hydraulic stress on the material.

In this problem, we are tasked with finding the ratio of this hydraulic stress to the corresponding hydraulic strain. But what does this ratio actually represent? In the realm of elasticity, the ratio of volumetric stress to volumetric strain is defined as the Bulk Modulus ($\beta$). It is a measure of how resistant a substance is to uniform compression. A high Bulk Modulus means the material is incredibly stiff and hard to compress.

Before we dive into the calculations, let's gather our known variables and ensure they are in the correct SI units. This is a crucial step where many students make silly mistakes.

- Depth of the object, $h=2\text{ km}=2000\text{ m}$ - Density of water, $\rho =1000{\text{ kg/m}}^3$ - Acceleration due to gravity, $g=9.81{\text{ m/s}}^2$ - Fractional compression (Hydraulic Strain), $\frac{\Delta V}{V}=1.36\%=\frac{1.36}{100}=0.0136$

Notice how we immediately converted kilometers to meters and the percentage into a decimal fraction. This sets a solid foundation for our math.

First, we need to calculate the hydraulic stress, which is simply the hydrostatic pressure at that depth. The formula for hydrostatic pressure is beautifully simple:

Let's substitute our values into this equation:

Multiplying these together, we get:

This is the immense pressure acting on the object, trying to squeeze it into a smaller volume.

Now, we return to our core objective: finding the ratio of hydraulic stress to hydraulic strain, which is the Bulk Modulus ($\beta$).

Substituting the stress we just calculated and the strain given in the problem:

When we perform this division, we get approximately $1442.6\times 10^6$, which we can elegantly rewrite in standard scientific notation as:

And there we have it! The ratio of hydraulic stress to hydraulic strain is $1.44\times 10^9{\text{ N/m}}^2$. This perfectly matches option (d). The sheer magnitude of this number tells us a fascinating physical story: it takes an enormous amount of pressure (nearly 20 million Pascals) to compress this object by just a tiny fraction (1.36%). Physics isn't just about numbers; it's about understanding the resilience of the materials that make up our world.