Animated Solution for Physics - Properties of Solids and Liquids: A large number of water drops, each of radius $r$, combine to have a drop of radius $R$. If the surface tension is $T$ and mechanical equivalent of heat is $J$, the rise in heat energy per unit volume will be

Imagine a large number of tiny water droplets, each of radius $r$, floating around. Now, picture them merging together to form one single, massive drop of radius $R$. What happens physically? Well, the total amount of water, or the volume, stays exactly the same. But here is the catch—the total surface area shrinks! And since surface area holds energy, this reduction means some energy has to be released. This released energy appears as heat.

Let's write down the math for the volume conservation. If we have $n$ small drops, their total volume is $n\times \frac{4}{3}\pi r^3$. This must equal the volume of the single large drop, which is $\frac{4}{3}\pi R^3$.

Canceling out the common terms, we get a beautiful relation:

This gives us the number of small drops, $n$, as:

Now, let's look at the energy. The surface energy of a drop is its surface area multiplied by the surface tension, $T$. So, the initial energy of all $n$ drops is $n\times 4\pi r^{2}T$. The final energy of the large drop is $4\pi R^{2}T$.

Since the area decreased, the initial energy was higher. The energy released, $\Delta E$, is simply the initial energy minus the final energy:

Let's substitute the value of $n$ we found earlier into our energy equation. Substituting $n=\frac{R^3}{r^3}$, the $r^2$ cancels out partially, leaving us with $\frac{R^3}{r}$. So, the released energy is:

We are almost there. The question asks for the rise in heat energy per unit volume. First, to convert the mechanical energy $\Delta E$ into heat, we divide by the mechanical equivalent of heat, $J$. Then, to find it per unit volume, we divide by the total volume $V$, which is $\frac{4}{3}\pi R^3$.

So, our target expression is:

Let's carefully plug in $\Delta E$ and $V$:

The $4\pi$ in the numerator and denominator beautifully cancel out. The $3$ from the denominator's fraction flips up to the top, giving us $\frac{3T}{J}$. Inside the bracket, we divide each term by $R^3$.

$\frac{R^3}{r}$ divided by $R^3$ leaves $\frac{1}{r}$. And $R^2$ divided by $R^3$ leaves $\frac{1}{R}$. And there we have it! The heat energy per unit volume is:

Before we wrap up, think about the reverse process. What if a single large drop shatters into millions of tiny droplets? In that case, the total surface area increases, which means energy must be absorbed from the surroundings. This causes a drop in temperature, which is exactly why spraying water creates a cooling effect! Physics is everywhere around us.