Animated Solution for Physics - Properties of Solids and Liquids: The normal density of a material is $\rho$ and its bulk modulus of elasticity is $K$. The magnitude of increase in density of material, when a pressure $p$ is applied uniformly on all sides, will be

Imagine you have a block or a sphere of some material. It has a certain initial volume $V$ and an initial density $\rho$. Now, imagine you take this material and plunge it deep underwater, or put it in a pressure chamber. You are applying a uniform pressure $p$ on it from all possible directions.

What happens physically? The material gets squeezed. Its volume decreases by some amount, let's call it $\Delta V$. But here is the crucial part: the amount of stuff inside the material hasn't changed. The mass $m$ remains perfectly constant. If you pack the same mass into a smaller volume, the density must go up! Our goal is to find exactly how much this density increases, which we will call $\Delta \rho$.

Bulk Modulus

To relate the applied pressure to the change in volume, we need a property of the material that tells us how "squishy" it is. That property is the Bulk Modulus, denoted by $K$.

The definition of Bulk Modulus is the ratio of volumetric stress (which is just the applied pressure $p$) to the volumetric strain (the fractional change in volume). Mathematically, we write this as:

Why the negative sign? Because an increase in pressure (positive $p$) causes a decrease in volume (negative $\Delta V$). The negative sign ensures that the Bulk Modulus $K$ is a positive quantity.

We can rearrange this equation to isolate the fractional change in volume:

Keep this equation safe; it is the key to unlocking the problem.

Now, we need to bridge the gap between volume and density. We know the fundamental definition of density:

Since we are dealing with small changes, a great mathematical trick is to take the natural logarithm of both sides. This turns division into subtraction, making it much easier to differentiate:

Now, let's differentiate this equation. Remember, the mass $m$ is constant, so its derivative is zero.

This is a beautiful result! It tells us that the fractional increase in density is exactly equal to the fractional decrease in volume.

We are in the endgame now. We have two crucial pieces of information:

1. $-\frac{\Delta V}{V}=\frac{p}{K}$ 2. $\frac{\Delta \rho }{\rho }=-\frac{\Delta V}{V}$

By simply substituting the first equation into the second, we get:

To find the absolute increase in density, $\Delta \rho$, we just multiply both sides by the initial density $\rho$:

And there we have it! The magnitude of the increase in density is directly proportional to the initial density and the applied pressure, and inversely proportional to the Bulk Modulus. This perfectly matches option (b).