Chelci Houston Burroughs IB Physics IA
Cantilever Physics Lab Introduction
How does altering the length of the load arm of a simple cantilever affect the declination? A cantilever is a stick, bar, or beam anchored at one end that stretches out some distance from the support. If a load were placed at the free end, one would expect that there would be some bending, or flexion, of the cantilever. In this experiment, the length of a cantilever will be changed in order to determine how it affects its declination and hence the relationship between the two will be determined. Declination in this instance is the angular deviation from the normal, or in other words 90°. The load arm length, that will be altered for each trial will be considered as the independent variable, this measurement being in meters. And the declination being calculated for each length will be the dependent variable. The controlled variables include the length and width of the same meter stick used, the same tensioned lab clamp being used, the mass of the weight/load (1.50kg ±0.1 g), the temperature of the room in which these measurements taken, and the method used to calculate the declination which in this case was with the same protractor.
Hypothesis I hypothesize that as the length of the cantilever’s load arm decreases, the declination of it will also decrease. I predict this outcome because, when length is increased, the mass is given less support from its equilibrium point. So, the further the cantilever arm is away from
its equilibrium anchor, the less support its load has causing more stress and hence more declination. Design Materials/Apparatus List: Lab Clamps Table End Meter Stick (±0.001 m) 1.50 kg Bob/Weight (±0.01 kg) Protractor (±1°) Scotch Tape String Method: To measure the exact mass of the load a digital weighing scale was used To measure the control the 90-degree angle of release a protractor was mounted against the table end The cantilever was then lifted to line up directly with the 90-degree mark on the protractor and I used another subject to check for accuracy from the front and side of the apparatus The same meter stick and load were used throughout this experiment. After each test I waited a few seconds so that the pivot point would lose heat generated to lessen the affect of frictional forces and lose potential tension increases I applied many clamps to the anchored end of the cantilever to keep it as stable as possible Exact Procedure: 1. After all the required materials are acquired set up apparatus as such in Figure 1. Extend a meter stick over the end of a table as such to begin. Figure 1
2. Begin first measurement around 0.600m away from anchored end of the cantilever Remember to measure the length from the top of the load to the middle of the load. 3. Lift pendulum cantilever to the 90° (degrees) marking, so that the meter stick is parallel to the protractor. (Be sure to make measurement as precise as possible) check 3 or 4 times. 4. Tie a string around the load and wrap the 1.5 kg load along the center of the 0.500 m marking. Make sure that the center of mass is lined directly to the marking. Have a thick enough meter stick so that cantilever can hold the mass accordingly. 5. Record the declination of the meter stick. Declination calculation being your new angle subtracted from the normal (θ-90°) 6. Repeat steps 2-4 three more times with the same cantilever arm length in order to record an accurate measure. 7. Increase the length of the cantilever from the anchor can so that significant change can be observed. I recommend by about 0.05 m (±0.02) each time. 8. Record 10 different measurements, which will transfer to data points with 3 trials each. Measurement and Data Tables The data table below shows the raw data I collected during my experiment. This includes the varying lengths of the cantilever, and 3 separate trials for each of those. Length from Anchor Angle of Cantilever Angle of Cantilever Angle of Cantilever of the Cantilever Declination (+ 0.5°) Declination (+ 0.5°) Declination (+ 0.5°) (+ 0.001m) Trial 1 Trial 2 Trial 3 0.350 91.0 92.0 91.0 0.400 93.0 93.0 93.0 0.450 94.0 95.0 94.0 0.500 96.0 95.0 96.0 0.550 97.0 97.5 97.0 0.600 98.0 99.0 98.0 0.650 99.0 99.0 99.0 0.700 102.0 103.0 104.0 0.750 104.0 104.0 104.0 0.800 105.0 105.0 107.5
This data table below displays the length from the cantilever anchor, the average declination for each trial and the standard deviations of those declinations, which will be use in the graph comparison of length vs. declination. This table represents the values from smallest length to greatest length. (For a length of zero, there would intuitively be a length of zero, however the line of best fit would be skewed so this point was left out) The equation used to calculate the standard deviation was (max value-min value)/(2)
Length from Anchor of the Average Cantilever Declination Cantilever (+ 0.001m) (+ 0.5°) All Trials Standard Deviation 0.350 91.3 0.400 93.0 0.450 94.3 0.500 95.7 0.550 97.2 0.600 98.3 0.650 99.0 0.700 103.0 0.750 104.0 0.800 105.8
0.58 0.00 0.58 0.58 0.29 0.58 0.00 1.00 0.00 1.44
The data table below displays the measurement of the cantilever declination from the normal. In other words the normal of 90 degrees subtracted from the average cantilever declinations (θ-90°).
Length from Anchor of the Cantilever (+ 0.001m) 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800
Average Cantilever Declination from the normal (θ-90°) (+ 0.5°) All Trials 1.3 3.0 4.3 5.7 7.2 8.3 11.0 13.0 14.0 15.8
Below I have constructed a graph to show the connection or variation of declination as the length from the anchor increases in order to conjure a numerical expression.
Length Away from Anchored End vs. Declination of Cantilever Graph Comparison y = 32.509x - 10.313 R² = 0.9946 18.0
Cantilever Declination from 90 degrees (+ 0.5°)
16.0
14.0
12.0
10.0
8.0
6.0
4.0
2.0
0.0 0.000
0.200 0.400 0.600 0.800 Length of Cantilver Arm from Anchored End (+ 0.001m)
1.000
This Best fit Line automatically conjured by Microsoft Excel proved to be very much linear in pattern so one could conclude that declination is directly proportional to the Length from the anchor of the cantilever. However, I will show my own calculations below of the best fit line equation by using separate points I have marked above.
Conclusion: From the graph it can be seen that within the uncertainties in this experiment declination is proportional to length of the cantilever arm. Therefore I can also conclude that the altering of cantilever arm length is a main factor that affects the declination of the anchored cantilever. With a longer cantilever arm you will in turn have a larger angle of declination. For a shorter cantilever arm there will be a smaller angle of declination. The R value of this graph is approximately 0.994 which means that the line is very close to the accepted values. Evaluation: Looking at the graph I can see that the data points lie very closely to the best fit line conjured by Microsoft Excel. And the small error bars reflect the precision in my measurements. Therefore, I believe the most influential source of this error (with error bars) was the difference is the tightness/tension between the clamps and the meter stick. The anchored end was adjusted every time the distance changed. Therefore there could be variance in the tightness at that end, which could possibly effect the amount declination more or less. Also, a possible source of error comes when the lever arm begins to wear down and is stressed by the heavy weight. This would then increased the amount of declination because the wood is stretched down to its limits ad cannot fully snap back into the normal position. Possible Improvements and further experiment: In order to lower the error that comes with the variance in anchor tightness, next time I could maintain a specific number of twists to tighten the anchor for every trial. I would have to do a few test trials first before coming upon a reliable number or turns of the anchor screw. In order to lower the error in lever arm wear, I could use a different meter stick for every separate length. This meter stick would have to be of same dimensions and made of the same materials. A possible further experiment could include doing the same number of trials on a longer and thinner cantilever as to see more of a declination difference with each trial.
