Measuring Propeller Thrust Using a Kitchen Scale, LEGO and an Arduino

In this experiment, everything literally revolves around a propeller; specifically, I want to determine a propeller's thrust as a function of electrical power (P = U · I) and rotational frequency (f). For the latter, I am using my Arduino tachometer based on an infrared reflection sensor.

For a change, I am not measuring the thrust using a bending beam and HX711 load cell, but rather via a double-sided lever made of LEGO Technic parts and a standard scale. The electric motor is powered by my beloved homemade laboratory power supply.

For this physics experiment, the following items are required:

1. a powerful electric motor
2. a propeller adapter compatible with the motor shaft
3. a propeller
4. an Arduino (e.g., Nano)
5. an IR reflective optical sensor module for Arduino
6. aluminum adhesive tape
7. Lego bricks
8. a kitchen scale
9. an adjustable power supply for the electric motor

The simple Arduino code is attached. The IR reflective optical sensor is used to measure the time interval between two incoming pulses, from which the rotational frequency f can be determined.

As mentioned, I determine the rotational frequency f using my DIY tachometer. To do this, I attached a piece of aluminum foil to each of the two propeller tips. Consequently, the display shows 2 · f rather than f.

So, let’s rev the motor up nicely... At U = 6.1 V and I = 2.78 A, the scale registered a mass of 108 g. Based on the formula F = m · g, this corresponds to a force of 0.108 · 9.81 = 1.0595 N. In this instance, the rotational frequency was f = 234 / 2 = 117 Hz.

We are dealing with a double-sided lever here. Therefore, the force displayed on the scale does not generally correspond to the thrust itself, because the principle of moments applies: F1 · L1 = F2 · L2, or F1 = F2 · L2 / L1.

To determine the thrust, we therefore also need the lengths L₁ and L₂ of the two lever arms. Specifically, L₁ = 20 cm and L₂ = 10 cm. As a result, the scale actually displays twice the amount of thrust. So, for 108 g, it is 1.0595 N ÷ 2 = 0.53 N.

The graph of F = F(f) looks very much like a parabola, which is what one would expect. If the relationship F = k · f² actually holds, then plotting F against f² should yield a straight line. Naturally, I checked this immediately. Voilà—a beautiful straight line...

Finally, I plotted the graph of F(P)—that is, thrust as a function of motor power. This graph, too, was approximately linear.

As always, however, I am not satisfied with just the experimental results; I also want to be able to explain them physically and compare them with theory. Specifically, the issue at hand is the quadratic dependence of thrust F = k · f² on rotational speed.

Let us consider a propeller blade inclined at 45° moving at a speed v. In the laboratory frame of reference - my kitchen - the propeller moves at speed v while the air particle remains stationary. If, however, we switch reference frames by "jumping" onto the rotating propeller, the propeller is at rest, and the air particle approaches it at speed v. The air particle is then reflected. Due to the 45° inclination, the particle leaves the propeller horizontally.

If we then switch back to the laboratory/kitchen frame of reference, we must add a velocity vector v pointing vertically upwards: Thus, in the laboratory frame, the air particle undergoes a momentum transfer of Δp = m · Δv = m · √2 · v, directed diagonally upwards and to the right.

However, the law of conservation of momentum applies; therefore, the propeller receives a momentum transfer of –Δp directed down and to the left. For forward thrust, however, we are interested only in the x-component of this momentum transfer: Δpx = m · √2 · v · cos(45°).

This represents the momentum transfer from a single air particle. Naturally, the propeller collides with a vast number of air particles. Let us assume a particle density of n. Per second, it sweeps through an air volume of V · f, encountering n · V · f air particles within that volume.

Now, let’s consider the thrust force F: force is defined as the momentum transfer per second. We know the momentum transfer per air particle and the number of air particles encountered per second. Multiplying these two terms gives us the total momentum transfer per second.

For the propeller speed, we must naturally use an average value, as not all particles strike the propeller tip. We can use the speed at the midpoint to the propeller tip, that is v = 1/2 · ω · r. In the final, more complex formula, we see that the rotational frequency f appears twice. This confirms the experimentally observed proportionality F = k · f², Eureka!

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