Screaming Into the Void: Ace Your Exam, With a Side of Hot Coffee

Finals. Not an easy time for any student, or professor—especially in physics and astronomy! I would wager that by graduation, every STEM student has spent tears and been compelled to scream. This is totally normal, given the time, energy, effort, and stress toll of learning to critically think and solve the unsolved.

Speaking of energy, time, and effort, we all know that sound—screaming, in this very specific case—takes energy to produce. We can think of it in biological terms, but also in terms of the energy-carrying physical waves we produce when we scream. Sound waves contain energy.

Sound waves are longitudinal pressure waves, or variations in air pressure. In this case, they’re caused by a person’s vocal folds vibrating at specific frequencies, which creates alternating regions of high and low pressure in the larynx. These waves, modified by traveling through the larynx, pharynx, and finally the oral cavity, propagate through the air. If you’ve studied acoustics, or even the waves part of introductory physics, this puzzler might have crossed your mind during a finals cram session: If I were to scream long enough, could I heat up the cup of coffee in front of me?

A silly question? Maybe. Sound interesting? Definitely.

Where do we begin? With any Fermi problem (estimation problem), the first step is to restate the puzzle into a potentially answerable physics problem. For starters, watt’s the goal? We want to heat a cup of coffee… using sound waves. So, in essence, we’re trying to see if there is enough energy in sound to heat the water in coffee within a measurable time frame.
(While this might seem like a silly exercise, this type of problem is exactly what many physicists are paid to do in the workforce. In industry, companies will hire physicists for their ability to take ill-formed, impossible to look up questions and answer the deepest question of all: Is it possible?)

There are many ways to approach this. First, let’s declare some basic assumptions to a) help narrow the complexity of the problem and b) try and simplify the calculations to something manageable. This is really an energy problem, so first let’s estimate how much energy it takes to heat a cup of coffee. From Intro Physics II, we might remember that the energy E_cup to heat water looks like:

E_cup=m∙C_coffee∙ΔT,

where ΔT is the change in temperature of the coffee, m is the mass of coffee, and C_coffee is the thermal heat capacity of the coffee.

Let’s assume that the thermal properties of coffee are the same as water. Given that coffee is almost entirely water, this is likely an accurate assumption. If anything, the organic material in the coffee has a lower thermal mass than water, since water has such a large value. So

C_coffee = C_H2O= 4.186 kJ/g/K

Let’s also assume that the coffee is cold from a long study session, about 20°C, and that we really want it around 90°C. And finally, we need to estimate the amount of coffee. A cup is about 8 oz., or about 240 mL. We can use that to get the mass of coffee since the density of water is about 1 kg per liter at standard temperature and pressure.

Taking everything together, we can calculate how much sound energy we’d need to make the coffee warm:

E_cup=m∙C_coffee∙ΔT=(0.240 kg)∙(4.186 kJ/kg∙K)∙70 K≅70 kJ.

Now that we know the energy needed to warm the coffee, we need to consider how much energy a person’s screaming could, in theory, impart to the coffee. From introductory physics we might remember that we can calculate the power of sound per unit area, which is the sound intensity level, using the equation

L_I=10∙log₁₀(I/I_o),

where I is the sound intensity in decibels (W/m²) and I_o is the reference sound intensity 10^-12 W/m². Therefore, the power delivered to the cup would be the sound intensity level times the area of the absorber. As a helpful hint, a loud, but not painful, scream is about 100 decibels.

Remember that the farther someone is from a sound, the quieter it is, since sound decreases as the inverse square of the distance. And, finally, the last assumption I’ll give you is this: Let’s assume that the cup is right in front of us, so the sound waves don’t go anywhere except into the cup (Fig. 1).

What other assumptions do you need to make? Improving the diagram might be helpful. Then you’re ready to tackle these three puzzlers.

Puzzler 1: How long would it take to warm up that coffee?
Puzzler 2: lf you wanted it to be warm in an hour,
how many people would you need to scream simultaneously?
Puzzler 3: Assuming that it would take a “long”
time for one person to warm the cup, what other assumptions do we need to make? lf we allowed black body radiation of the cup, how much longer would it take?