Animated Solution for Physics - Properties of Solids and Liquids: A solid sphere of radius $R$ made of a material of bulk modulus $k$ is surrounded by a liquid in a cylindrical container. A massless piston of area $A$ floats on the surface of the liquid. When a mass $M$ is placed on the piston to compress the liquid, the fractional change in the radius of the sphere, $\delta R/R$, is ......

Visualizing the System

Imagine a beautifully crafted physical experiment. We have a rigid cylindrical container filled to a certain level with an incompressible, non-viscous liquid. Submerged deep within this liquid rests a solid sphere of radius $R$. This sphere is not made of rigid, unyielding stone; rather, it is characterized by a finite Bulk Modulus $k$, meaning it can shrink under uniform pressure.

At the top of the liquid column, a massless piston of cross-sectional area $A$ seals the system. Initially, everything is in a state of perfect, serene equilibrium. The pressure throughout the liquid is uniform, and the sphere maintains its natural radius $R$.

Now, let us introduce a disturbance. We gently place a block of mass $M$ on top of the massless piston. Gravity immediately pulls this mass downward with a force:

Since the piston is massless, it does not absorb any of this force. Instead, it transmits the entire force directly onto the surface of the liquid. This downward force acting over the piston's cross-sectional area $A$ creates an excess pressure $\Delta p$ at the liquid surface:

According to Pascal's Law, this increase in pressure is transmitted undiminished to every single point within the fluid. Consequently, our submerged solid sphere experiences this excess pressure $\Delta p$ acting uniformly and radially inward over its entire surface. This uniform squeezing action is what we call volumetric stress.

How does the material of the sphere respond to this uniform squeezing? This is where the Bulk Modulus $k$ comes into play. The Bulk Modulus is a fundamental material property that measures a substance's resistance to uniform compression. It is defined mathematically as the ratio of volumetric stress to volumetric strain:

Here, $\Delta V$ is the change in volume, and $V$ is the original volume of the sphere. The negative sign traditionally associated with bulk modulus is omitted here because we are interested in the absolute magnitude of the fractional change.

By rearranging this definition, we can express the fractional change in volume (volumetric strain) as:

Substituting our expression for the excess pressure $\Delta p=\frac{Mg}{A}$ into this equation yields:

This equation tells us how much the volume of the sphere shrinks. But the question asks for the fractional change in the radius of the sphere, $\delta R/R$. We need a mathematical bridge to connect volume to radius.

We know from basic geometry that the volume $V$ of a solid sphere of radius $R$ is given by:

To find how a small change in radius affects the volume, we can take the natural logarithm of both sides:

Now, let us differentiate both sides. The constant term $\ln \left(\frac{4}{3}\pi \right)$ vanishes, leaving us with a remarkably simple linear relationship between the fractional changes:

This is a beautiful result! It tells us that for any sphere, a small fractional change in volume is always exactly three times the fractional change in its radius. This makes intuitive sense because volume is a three-dimensional quantity ($V\propto R^3$).

We can now easily solve for the fractional change in the radius, $\frac{\delta R}{R}$ (where $\delta R=\Delta R$):

Substituting our expression for the volumetric strain $\left|\frac{\Delta V}{V}\right|=\frac{Mg}{Ak}$ into this relation, we arrive at our final elegant formula:

This is the exact fractional change in the radius of the sphere when compressed by the mass $M$.

Let us analyze the physical dependencies of our final result: 1. Direct Proportionality to $M$: A larger mass creates greater pressure, leading to more compression and a larger fractional change in radius. 2. Inverse Proportionality to $k$: A material with a higher Bulk Modulus is stiffer and resists compression more effectively, resulting in a smaller change in radius. 3. Inverse Proportionality to $A$: A larger piston area spreads the force over a wider region, reducing the excess pressure transmitted to the liquid and thus reducing the compression of the sphere.

Common Pitfall: A frequent mistake students make is setting $\frac{\Delta V}{V}=\frac{\Delta R}{R}$ directly, forgetting the factor of $3$ that arises from the three-dimensional nature of volume. Always remember to use logarithmic differentiation when relating fractional changes of power-law variables!