Physics in Your Cup of Hot Chocolate

By Randy Stefan Tanuwijaya

If you are someone who loves coffee, like me, then you will love it even more when you learn that there is a fun and delicious experiment you can do with your cup of coffee called the Hot Chocolate Effect. All you need for this experiment is a cup, hot water, a metal spoon, and instant coffee or cocoa powder – all of which can be obtained from your own kitchen. And the best part about this experiment is that you can enjoy a great cup of coffee afterwards.

The Experiment

Unlike a regular cup of coffee, start by pouring hot water into the cup first, leaving a little room for the coffee powder to be added later.
Lift your mug by holding the handle and tapping the bottom with a metal spoon. Listen to the knocking sound carefully and remember the tone.
Pour the coffee powder into the cup and stir vigorously.
Tap the bottom of the cup continuously while the liquid is swirling. Note that the starting note will be lower than the previous note but will increase with each tap until it is restored.
Stir the liquid again and you can lower the pitch one more time and repeat the observation.
Remember to enjoy your cup of coffee while it is warm!

The Explanation

What is the possible explanation behind this simple phenomenon? The answer lies behind the propagation of sound in a liquid and was explained in 1982 by Frank S. Crawford [1], an American physicist and a talented musician from University of California, Berkeley.

The phenomenon can be described using simple physics you might have learned from your science class. Consider the cup as a closed-end organ pipe. The frequency heard is the fundamental frequency, i.e. the lowest frequency and longest wavelength pattern the pipe can produce, in which the wavelength equals four times the height of your container. By substituting $\lambda = 4h$ into $v=\lambda f$ , you will get:

$f=\frac{v}{\lambda}=\frac{1}{4}\frac{v}{h}$

Here $f$ is the sound frequency, $\lambda$ is the wavelength, $v$ is the speed of sound, and $h$ is the height of the cup. The height and shape of the cup does not change as you tap, so it must be the speed of sound in the liquid that is increasing!

Then why is the speed of sound increasing? It is because the bubbles trapped in the coffee powder are released into the liquid when you vigorously stirred the coffee. Sound travels faster in liquid than in the air. Therefore, the bubbles in liquid slow down the overall speed of sound, and lower the fundamental frequency. More precisely, the air bubbles reduce the bulk modulus (see text box) of the liquid by making it more compressible or springy. As time goes by, air bubbles in the liquid will escape gradually. Therefore, the speed of sound is slowly rising – along with the pitch of the sound produced!

Bulk modulus is also known as the incompressibility of a material. It defines how resistant a material is to compression. For example, air is less resistant than liquid, so air has a lower bulk modulus. The speed of sound in a material is given by:

$v=\sqrt{{\frac{K}{\rho}}}$

where $K$ is the bulk modulus and $\rho$ is the material density. The change in material density is negligible in our experiment.

A very simple yet exciting and fun effect, isn’t it? There are many other variations of this effect you can explore by yourself! For example, try substituting the powder with other soluble substances like powdered milk or baking soda. Try to adjust the water temperature or use a taller container. You may also try it with soft drinks and beer as well. The next time you are enjoying your favorite beverage, there might be another interesting physics phenomenon waiting to be discovered!

Reference:

[1] Crawford FS. The hot chocolate effect. Am J Phys. 1982;50(5):398-404. doi:10.1119/1.13080