Cantilever Arm Length Vs. Maximum Reflection

Cantilever Arm Length vs. Maximum Deflection

Introduction

As an avid swimmer and diver throughout my childhood, I have always been interested in improving my technique. Mathematical and physical relationships are present in almost every aspect of daily life, but diving in particular incorporates many aspects of physics including force, harmonic motion, momentum, elasticity, and so much more. Because of this, I chose to investigate one of these relationships. In my experience, standing at the edge of a long diving board allowed me to jump very high whereas a shorter diving board did not. Knowing that a diving board is a form of cantilever beam (a rigid structural element anchored at one end to a support from which it protrudes), I asked the

All I need to construct the cantilever was a meter stick and a clamp. The procedure from this point however, proved to be more challenging. Very few cantilever-building instructions are available to the public online, and those that were available required equipment such as optical fiber sensors, a hydraulic testing apparatus, and extensometers to name a few, none of which are available at my public school. Because of this, I was forced to create a solution that would allow me to measure the deflection of the meter stick cantilever. To do this, I used an identical meter stick and clamped it directly next to the cantilever so they were exactly level. This way, when I applied the mass to the cantilever, the other meter stick stayed stationary at the same height as the cantilever’s starting height. I then used a ruler to measure the amplitude of the deflection.

Figure 1 Figure 2

Another problem that presented itself was attaching the 1 kg mass to the cantilever. Ultimately I devised a string loop to hang the mass 1 cm from the end of the meter stick.

Considering a diver doesn’t apply their whole force to the very edge of the diving board, I did this to help simulate a more realistic model. Once the cantilever setup was complete, I adjusted it so that the arm length was 50 cm, hung the mass, then measured the deflection. I repeated this process 4 times for a total of 5 trials. I then

Figure 6

As previously mentioned, the error bars are so small that they are not visible when viewing the whole data set. Additionally, we can see that Logger Pro has identified the quartic relationship between the data points with the equation f(x)=-.0000114x4+.00323x3-.333x2+15.2x-256 Logger Pro also gives us the correlation, which in this case is 1.00 (an exact correlation). Beyond this, there is not much to analyze on the graph. Instead, we will now compare it to the theoretical values of the problem. To do this we will use the deflection equation =pa26EI(3l-a) solving for for each arm length (represented by l) and using E=15 and I=ab312where a=6.5 mm (thickness of meter stick/cantilever beam) and b=25.5 mm (width of meter stick/cantilever beam). The resulting points are:

(50,2.94) (60,5.11) (70,8.14) (80,12.2) (90,17.4)

We then graph these points and use Logger Pro to find the equation of the relationship.

Figure 7

The equation given for this graph is g(x)= -.00000025x4-.0000933x3-.0078x2+.314x -