Animated Solution for Physics - Properties of Solids and Liquids: A small sphere falls from rest in a viscous liquid. Due to friction, heat is produced. Find the relation between the rate of production of heat and the radius of the sphere at terminal velocity.

Imagine dropping a tiny pebble into a tall jar of honey. At first, it accelerates under gravity, but very quickly, it settles into a slow, steady, and majestic crawl. This steady speed is what we call terminal velocity.

But have you ever wondered what happens to all that lost gravitational potential energy? It doesn't just vanish into thin air (or thin honey)! Instead, it is transformed into heat due to the friction between the falling sphere and the surrounding fluid molecules.

In this article, we will explore the fascinating physics behind this energy conversion and derive a surprising scaling law: how the rate of heat production depends on the size of the falling sphere.

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When a solid sphere of radius $r$ and density ${\rho }_s$ falls through a viscous fluid of density ${\rho }_L$ and viscosity $\eta$, three forces immediately engage in a tug-of-war:

1. Gravity ($F_g$): Pulling the sphere downwards.

2. Buoyancy ($F_b$): Pushing the sphere upwards, equal to the weight of the displaced fluid.

3. Viscous Drag ($F_v$): Opposing the motion. According to Stokes' Law, for a sphere moving at velocity $v$:

As the sphere accelerates, the viscous drag grows. Eventually, the upward forces perfectly balance the downward pull:

Substituting our expressions:

Solving this for the terminal velocity $v_T$ yields:

Notice that $v_T\propto r^2$. A larger sphere falls much faster than a smaller one!

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Rate of Heat Production

The rate of heat production, $\frac{dQ}{dt}$, is simply the rate at which mechanical work is done against the viscous drag force. In physics, the rate of doing work is power ($P$):

Using Stokes' Law for the viscous force at terminal velocity ($F_v=6\pi \eta r{v}_T$):

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Now, let's substitute our expression for $v_T$ back into the heat rate equation:

Expanding the squared term:

Simplifying the constants:

Since $g$, $\eta$, ${\rho }_s$, and ${\rho }_L$ are constant parameters of the system, we arrive at our beautiful final relation:

This is an incredibly strong scaling law! If you double the radius of the sphere ($r\to 2r$), the rate of heat production increases by a factor of $2^5=32$! This explains why larger particles falling through fluids generate significantly more thermal disturbance than tiny microscopic ones.