If you put a pendulum under the influence of a flow, a drag force will move the pendulum creating an angle with respect to the vertical. As the angle is related to the drag force, and because the drag force is related to the velocity of the flow, you can determine the velocity of the flow by only measuring the angle of the pendulum.

You could measure the angle with a protractor, but what we decided to do is to attach a large, light ball to the top of the pendulum and to register the rotation of the ball with a computer mouse. Using a MATLAB program, we transformed the registered rotation (in pixels) to flow velocity (in meters per second).

The balance of forces can be represented by a right triangle, from which the following equation is obtained:

$$\tan\theta=\frac{F_a}{W}$$ (1)

where $\theta$ is the angle of the pendulum with respect to the vertical, $F_a$ is the drag force, and $W$ is the weight of the pendulum.

The drag force of an object surrounded by a stationaty flow is defined by the following equation:

$$F_a=\frac{1}{2}C_dA\rho v^2$$ (2)

where $C_d$ is the drag coefficient, $A$ is the area of the drag body, $\rho$ is the density of the fluid, and $v$ is the flow speed.

Combining equations (1) and (2), the we obtain:

$$W\tan\theta=\frac{1}{2}C_dA\rho v^2$$ (3)

Solving for the velocity, we find:

$$v=\sqrt{\frac{2W\tan\theta}{C_dA\rho}}$$ (4)

If we consider that the fluid and the pendulum do not change with time, it can be supposed that the following parameters can be considered constants, and that they may be included in a constant:

$$K\equiv\sqrt{\frac{2W}{C_dA\rho}}$$ (5)

Then the equation for the velocity is:

$$v=K\sqrt{\tan\theta$$ (6)

If the value of the constant $K$ is known, then only the angle of the pendulum with respect to the vertical is necessary to obtain the velocity of the flow.

The difficult part was to build a device that permits the free movement of the pendulum, while at the same time, allowing the mouse to register the rotation of the ball. We built a frame with wooden poles to house the ball, which rested on skate bearings. A piece of cardboard with a hole that allowed the optical sensor of the mouse to register the rotation of the ball was attached to the top of the frame. Depending on where and what kind of optical mouse you buy, this digital anemometer (or currentmeter) could cost around 10 bucks.

References:

- Hernández-Walls, R., Rojas-Mayoral, E., Baéz-Castillo, L., & Rojas-Mayoral, B. (2008). Design and calibration of an inexpensive digital anemometer Physics Education, 43 (6), 593-598 DOI: 10.1088/0031-9120/43/6/005