Coriolis Force Experiment

The Coriolis force is one of the three inertial forces of classical mechanics that occur in a rotating reference system. It occurs precisely when a body moves in a rotating reference system and when this movement is not parallel to the axis of rotation or the vector of the angular velocity ω. The other two inertial forces in the rotating reference system, centrifugal force and Euler force, also act when the body is at rest in the rotating reference system. The Coriolis force is named after Gaspard Gustave de Coriolis (1792 - 1843), who discussed it in detail in a publication published in 1835. The formula for the Coriolis force is: F_c = -2*m*ω x v

It is therefore normal to the angular velocity vector ω (= axis of rotation) and the velocity vector v. Where do Coriolis forces occur in everyday life? The earth is a rotating reference system with a small angular velocity, but nonetheless. The direction of rotation of hurricanes, for example, is based on the Coriolis force.

Low pressure vortices rotate counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere.

Or let's take a very tall tower from which a ball is dropped. How will it move due to the Coriolis force?

In this case, there is a deviation to the east. This can also be explained as follows: Let's assume that the tower is on the equator. Then the base of the tower has a speed v_ground = ω_earth · r_earth. ≈ 1600 km/h due to the rotation of the earth. The top of the tower is slightly further away from the center of the earth and has a speed v_tip = ω_earth · (r_earth + h). If you now let the ball fall, it has a higher speed than would be necessary to keep up at a lower height. As a result, it overtakes the places located vertically below the starting point. The trajectory then looks something like the picture added but greatly exaggerated.

With a tower height of 100 m, the east deviation is only about 22 mm. Here is the exact derivation of the formula for the east deviation:

It's hard to believe, but even snipers have to take the Coriolis force into account.

Here's a quick calculation: distance to target = 800 m, projectile speed = 2000 km/h, shot exactly north at a geographical latitude of 45°.

The deviation of a considerable 5.9 cm due to the Coriolis force is therefore not to be ignored. A good sniper like Mark Wahlberg in the film calculates the necessary correction in his head in no time at all 😉

Note: At half the projectile speed, the deviation due to the Coriolis force is even twice as large. Because the Coriolis acceleration a is then only half as large, but the flight time t is twice as long. According to the formula s = 1/2 · a · t², this results in a doubling of the distance s!

We will investigate the Coriolis force or Coriolis acceleration using a turntable. A ball then rolls from the outside onto the rotating turntable and follows a curved path due to the Coriolis force

The following parts are required for this project/experiment:

1. large turntable
2. threaded rod and wooden strip
3. Arduino and IR proximity sensor for measuring the frequency of the turntable
4. Arduino and IR remote control for starting the metal ball
5. small servo motor for starting the ball
6. metal ball
7. gear motor + rubber wheel for driving the turntable
8. bent PVC pipe as a launch pad for the ball
9. pipe clamp for mounting the PVC pipe
10. smartphone for recording the ball's path
11. software "Tracker" for analyzing the ball's path

First, mount the wooden strip for the smartphone with the two threaded rods. Then drill a hole in the wooden strip exactly above the middle of the turntable. The lens of the smartphone camera then goes exactly above the hole. Once you have aligned the smartphone, draw a line from the center of the turntable to the edge so that it is aligned exactly parallel to the x or y axis in the smartphone. This will make it easier for you to later analyze the video with the Tracker software.

Then mount the bent PVC pipe at the end of the line and align the pipe so that it points exactly towards the center of the turntable, i.e. along the line. Fix the pipe using the pipe clamp and use a threaded rod. You then mount a wooden plate for the servo motor on this. In order to be able to start the metal ball with the servo, you have to extend the servo arm.

Attach the Arduino, the IR receiver and the battery to the turntable. When you now operate the remote control, the servo's rotating arm pivots and releases the ball so that it rolls through the curved tube and then onto the turntable.

Stick a small piece of reflective aluminum foil onto the edge of the turntable to record the rotation frequency. Place the Arduino with the proximity module to the side of the turntable.

Now when you turn the turntable, the IR proximity module should receive a signal for each rotation and the Arduino should display the frequency on the display.

The experiment then runs as follows: You start the video recording of the smartphone. Then you hold the rubber wheel of the gear motor to the edge of the turntable. The turntable will start to rotate. When a stable rotation has been established and the Arduino tachometer shows a constant rotation frequency, release the metal ball with the IR remote control. The ball then rolls over the turntable. Then stop the video recording again. Repeat the experiment for different motor voltages so that the turntable rotates at different speeds. For example, I set rotation frequencies between 0.17 Hz and 0.5 Hz.

We will now use the Tracker software to analyze the movement of the metal ball on the turntable. In my case, the ball in the videos leaves the PVC pipe in exactly the positive y direction, i.e. from bottom to top in the video. With the Tracker software, you can now play the individual images of the video frame by frame and mark the position of the ball. But so that the program knows the exact position, you have to specify a known distance at the beginning. To do this, I measured the distance from the center of the turntable to the exit of the PVC pipe. In my case, this distance was 184 mm.

In the Tracker software, you then also have to specify the coordinate origin. I placed the coordinate origin exactly at the location of the end of the pipe. The ball initially only rolls in the y direction and is then deflected to the right in my case.

I then let the software output the graph v_y(t) and note the speed v of the ball immediately after it has left the PVC pipe.

At the beginning, only the Coriolis acceleration acts in the x direction. Therefore, I use the tracker software to display the graph v_x(t). This should increase linearly at the beginning. Its increase k then corresponds exactly to the Coriolis acceleration a_c at the beginning of the movement.

We now know the starting speed v of the ball and the rotational frequency f. From this we can calculate the theoretical Coriolis acceleration a_c = 2*2*Pi*f*v. This theoretical acceleration is then compared with the experimentally determined acceleration in the x direction. Both values ​​should agree more or less.

You can also simulate the trajectory of the ball on the turntable under the influence of the Coriolis force and the centrifugal force using EXCEL. You can find the corresponding program here:

https://stoppi-homemade-physics.de/corioliskraft/

To do this, you first need to know the acceleration due to the Coriolis and centrifugal force in the x and y directions. If you know the acceleration, you can calculate the new speeds from the old speeds in individual time steps. If you know the old and new speeds in the x and y directions, you can determine the new position of the ball using the relationship v = delta_s/delta_t and a specific time step size. In the EXCEL program, I can enter the starting position of the ball, the starting speed of the ball and the rotation frequency of the turntable. You get a curved trajectory like in the experiment.

Finally, you can use the Tracker software to record the trajectory of the metal ball for a specific rotation frequency. Then you simulate the trajectory with EXCEL and enter the same frequency f and speed v as in the experiment. The theoretical trajectory obtained in this way should be very similar to the experimental trajectory. Eureka

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