Physics Internal Assessment
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Introduction
A cantilever is an inflexible structural element, a projecting beam, that is fixed at one end and extend horizontally. As I go my day to day business, I come across several infrastructures that represent Cantilever like bridges designed to handle road, box girder made from prestressed concrete and trusses built from structural steel. All this tells how importance cantilevers are in mechanical engineering. They are a significant feature in modern architecture. Big bridges and rail traffic that uses trusses built from structural steel, often create amazing and interesting scenes and makes one wonder how they were constructed. However, time structures may collapse. Well to me this how also brought about many questions amongst them being how does a cantilever act?
The load L, due to gravity, will pull the Beam as it applies a downward force on it, making it to bend slightly. The force exerted by the load is equal to the force acting at point Z. The Beam distributes the load in different ways. On part Y the Beam is stretched indicating that the load is under tension and in part X of the Beam, the Beam is compressed. Cantilever takes the load and redistribute up the Beam by compression and down through the post. As the distance between the load and point x reduces, the downward force acting on the Beam reduce hence the deflection of the Beam reduce. If the stretching at Y and compression at point X is exerted for a longer period, the Beam may be deflected permanently.
The length of the distance affects the vertical length of deflection f; therefore, there is a relationship between the two lengths. The distance a is proportion to deflection f hence they two will make a linear graph when a graph is drawn.
When no force is applied to the Cantilever, it displays a horizontal position. It is because the action and reaction forces are equal. When a specified force is applied to the Cantilever, a vertical depression occurs.
The length of the vertical depression varies with the distance between the fixed point and the point where the force is applied. As the distance between the fixed point and the point where the force is applied increases, the vertical depression also increases. Applying a force for a long time at a point further away from the fixed point may lead to deformation of the cantilever beam.
f ∝ ab
f= (kab) equation 1
log(f)=log(kab)
log(f)=log(k)+log(ab)
Log (f) =log (k) +blog (a)…………equation 2
To practically find the relationship between the reflection length f and the distance a, I measured the deflection distance f in centimeters(cm): 5,10,15,20,25,30,35,40,45. My Cantilever is of 0.2cm thickness. It helped me ensure that I can observe any change in deflection.
With this research, the engineer will be able to know the relation between the distance and the deflection length. As they are making the type of Cantilever, they will ensure that the downward force exerted at the Beam is less than the upward one exerted at the other end. It will able the Cantilever to last for a long time.
Assumption
The force applied by the load is equally applied at the other end of the Beam, but in the opposite direction, this makes our force to be constant. Our two variables, f and a are proportion to each other indicating that there must be some form of the logarithmic, linear relationship between the two: log(f)=log(k)+blog(a).
Variables
We have different types of variables from the experiment, that is, the independent variable and the dependent variables. The independent variables are variable that causes variation in other variables. Dependent variables are variable whose variation is caused by other variables.
For this experiment, the independent variable will be the intervals of the distance between the fixed point and the point where the force is applied. Distance interval: 5cm, 10cm, 15cm, 20cm, 25cm, 30cm, 35cm, and 40cm.
The dependent variable will be vertical depression. The changes in the distance intervals cause variation.
The experiments also have various controlled variables, the variables that should be kept constant, these are;
| Controlled Variables | Reason for Controlling these Variables |
|
Length of the Cantilever | A change in the length of the Cantilever causes a variation in the vertical depression. Increasing the length of the Cantilever causes an increase in the vertical depression and vice visa, hence it crucial to maintain a constant length. |
|
The thickness of the Cantilever | An increase in the thickness of the Cantilever causes a decrease in the vertical depression and vice visa. It is, therefore, necessary to maintain the same thickness. |
|
Mass of the slotted-mass | Different masses of the slotted-mass apply different forces of the Cantilever, therefore causing a variation in the lengths of the vertical depression. |
| Length of the Mental Beam Suspended from the table | An increase in the length of the mental Beam causes an increase in the mass of the suspended Beam, which in turns causes an increase in the force of the Beam hence a high vertical depression is produced. To avoid this ensures that the metal beam is firmly connected to the G-clamp. |
| The material of the Cantilever | Different materials have a different rate of expansion, therefore, bringing about variation in the height of the vertical depression. |
Apparatus
| Apparatus | Quantity | Properties |
| G-Clamp | 1 | – |
| Wooden Ruler | 1 | Length: 100cm |
| Slotted mass | 1 | 100grams |
| Metal Ruler (Beam) | 1 | Steel Ruler, 100cm, thickness: 1mm |
| A Vernier Caliper | – | (Error margin): ± 0.01cm |
| Masking tape | 1 | – |
| Table | 1 | Normal height |
| An electrical balance | – | (Error Margins): ± 0.1 grams |
| String | 1 | – |
Procedure (Measurements)
- Since the metal ruler is calibrated, ensure that the length of the metal beam attached to the table is 10cm and the spare 90 cm is suspending in the air.
- Use an electronic balance in measuring the mass of the slotted- mass. Maintain the constant weight of the slotted-mass throughout the experiment.
- Ensure that no other solid supports the metal beam apart from G-Clamp and the table.
- Using the wooden ruler, measure the initial vertical displacement. Use a set square to avoid parallax error.
- Use the formula (Displacement between the point of suspension and the floor –initial vertical displacement) to calculate the initial vertical depression of the metal beam.
Data Collection
- Move the string 5cm along the metal beam. Whenever the string slips downwards, put the masking tape.
- Use the wooden ruler to measure the vertical displacement from the end of the suspending Beam to the floor.
- Repeat procedure 6 and 7 for other distances: 10cm, 15cm, 20cm, 25cm, 30cm, 35cm, and 40cm.
- Repeat the data collection procedure 6, 7 and 8 for trial two and trial 3.
Data Processing
- Using the formula: Vertical depression (cm) = (Displacement between the point of suspension and the floor – Vertical displacement) to calculate the vertical depression.
Data Analysis
The table shows the vertical distance (cm) and distance (cm)
| The vertical distance between the floor and the end of the Beam in cm | ||||
| Distance f (cm) ±0.05 | Attempt 1 | Attempt 2 | Attempt 3 | Mean displacement m (cm) |
| 0.0 | 39.5 | 39.5 | 39.5 | 39.5± 0.0 |
| 5.0 | 38.7 | 39.3 | 37.6 | 38.5±0.9 |
| 10.0 | 36.5 | 36.8 | 35.2 | 36.2 ± 0.8 |
| 15.0 | 34.0 | 33.6 | 33.0 | 33.5±0.5 |
| 20.0 | 31.0 | 30.6 | 29.9 | 30.5 ± 0.6 |
| 25.0 | 28.2 | 27.2 | 26.8 | 27.4±0.7 |
| 30.0 | 23.1 | 22.2 | 22.1 | 22.5±0.5 |
| 35.0 | 21.1 | 20.4 | 20.1 | 20.5±0.5 |
| 40.0 | 19.5 | 18.5 | 18.5 | 18.8±0.5 |
| 45.0 | 18.3 | 16.7 | 16.5 | 17.2±0.9 |
Calculation
To find the mean displacement for row 3, I used:
To get the mean for row 4;
= 36.2
To calculate the range of the mean value I used
For row 4;
= 0.8
Table 4 shows vertical depression of the Beam and the distance in cm.
| Distance f (cm) ±0.05 | Attempt 1 | Attempt 2 | Attempt 3 | Mean depression (cm) |
| 0.0 | 30.5 | 30.5 | 30.5 | 30.5± 0.0 |
| 5.0 | 31.3 | 30.7 | 32.4 | 31.5± 0.9 |
| 10.0 | 33.5 | 33.2 | 34.8 | 33.8 ± 0.8 |
| 15.0 | 35.5 | 36.4 | 37.0 | 36.3± 0.8 |
| 20.0 | 38.5 | 39.4 | 40.1 | 39.3±0.8 |
| 25.0 | 41.3 | 42.8 | 43.2 | 42.4± 0.9 |
| 30.0 | 46.4 | 47.3 | 47.9 | 47.2±0.8 |
| 35.0 | 48.4 | 49.1 | 49.9 | 49.1± 0.8 |
| 40.0 | 50.0 | 51.0 | 51.5 | 50.8± 0.8 |
| 45.0 | 52.3 | 52.8 | 53.1 | 52.7±0.4 |
Calculation
The vertical depression the first row was calculated using
Distance between the point of suspension and the floor – Vertical displacement
70cm – 39.5cm = 30.5 cm
Calculation of the error range for the Vertical displacement
(±0.05) + (±0.05) = ±0.10 = ±0.1 cm (1 decimal place)
Calculating the mean depression for row 3
= 33.8cm (1 decimal place)
Calculation of the error range for row 3
Graphical Representation
Graph 1
Showing the quadratic relationship between distance and mean depression
Graph 2
Showing the quadratic relationship between distance and mean depression
Graph 3
Showing a linear relationship between distance and mean depression
From the data collected, graph one the line intersects all the point, graph 2, the line intersects only four and graph 3; it intersects two-point only. It shows that the data supports a cubic relationship which is the opposite of our hypothesis. It could be due to deformation, as seen in Table 4. The depression distance increase from one attempt to the next. Attempt 1< Attempt 2 <Attempt 3. I, therefore, may conclude that the Beam was an inelastic body since it does not return to its original form after deformation.
I, therefore, need to make a logarithmic, linear relationship graph that can to prove the hypothesis all the relationships are not in line with it.
Table 3
| Distance (cm) ∆������ = ±0.10 | Log (a) | Mean depression(cm) | Log (f) |
| 5.0 | 0.69897 | 31.5± 0.9 | 1.49831055 |
| 10.0 | 1 | 33.8 ± 0.8 | 1.5289167 |
| 15.0 | 1.17609126 | 36.3± 0.8 | 1.55990663 |
| 20.0 | 1.30103 | 39.3±0.8 | 1.59439255 |
| 25.0 | 1.39794001 | 42.4± 0.9 | 1.62736586 |
| 30.0 | 1.47712125 | 47.2±0.8 | 1.673942 |
| 35.0 | 1.54406804 | 49.1± 0.8 | 1.69108149 |
| 40.0 | 1.60205999 | 50.8± 0.8 | 1.70586371 |
| 45.0 | 1.65321251 | 52.7±0.4 | 1.72181062 |
Graph 4: showing a logarithmic, linear relationship between distance and mean depression
log(f) = blog(a) + log(k)
log(k) = 1.426 (y-intercept)
n = 0.1541(gradient)
log(f) = 0.7913log(d) + 1.426(Equation 4)
f = kab
k = 101.426 = 26.7
f= 26.7a0.1541 (Equation 5)
The line of best-fit does not intersect with any point. Therefore, there is not an error range in the logarithmic equation. To verify the equation, we calculate the percentage difference between the measured and calculated depression. If the percentage difference is small, then the equation is accepted.
Table 3
|
Distance(cm) a | Measured depression f1/cm | Calculated depression (f2) 26.7a0.1541 | Difference z/cm z = |f1-f2| | % Difference z/% z’ = ( )(100) |
| 5.0 | 31.5± 0.9 | 34.2 | 2.7 | 8.57% |
| 10.0 | 33.8 ± 0.8 | 38.1 | 4.3 | 12.7 |
| 15.0 | 36.3± 0.8 | 40.5 | 4.2 | 11.58% |
| 20.0 | 39.3±0.8 | 42.4 | 3.1 | 7.89% |
| 25.0 | 42.4± 0.9 | 43.8 | 1.4 | 3.3% |
| 30.0 | 47.2±0.8 | 45.1 | 2.1 | 4.45% |
| 35.0 | 49.1± 0.8 | 46.2 | 2.9 | 5.91% |
| 40.0 | 50.8± 0.8 | 47.1 | 3.7 | 7.28% |
| 45.0 | 52.7±0.4 | 48 | 4.7 | 8.91% |
The equation is small, and therefore it is accepted.
Conclusion
This experiment was to determine the relationship between the distance and the depression. There is a logarithmic relation between the two. Forces exerted on the Beam of a cantilever, the upward and the downward force, are equal then the distance and the length of the vertical depression are directly proportional. We collect data on the by measuring the vertical distance from the floor and record in table 1. Then calculate the vertical depression in table 3. Afterwards compare different relationships between the distance and the vertical depression and represented them in graphs. The data was then converted to logarithmic form and plotted. All the variables were calculated, and the following equation formed:
log(f) = 0.1541log(a) + 1.426.
f=26.7a0.1541
After plotting the logarithmic graph, it was noticed that the line of best fit did not touch the error point. It may be because the Beam was not elastic and It had deformed. Therefore, the research question How does the vertical depression of a cantilever respond to a change in the distance at which an external force is applied to the Cantilever has been well illustrated and proven. There was a change in the vertical depression of a cantilever as the distance course by external force increases.
Evaluation
Additional Suggestions
The distance and vertical depression concept can be used in sport diving from cliffs. It can be used to find out the best place to stand on the diving board.
Merits, Demerits and improvements
| Sources of Errors | Evidence | Improvement |
| Beam Deformation Beam deformation occurred during the second trial and the third trial. | As the time of the experiment increased, it causes the deformation of the Beam, thus causing an increase in vertical depression. | A new identical beam should be used after every trial. The Beam should be identical in every aspect. |
| Parallax Error. | This factor contributes to the inaccuracy level in the measurement. | For every trial, take an eye-level reading at a constant point and distance. |
| Strength | Effect |
| The metal ruler Beam had a thickness of 1mm. | The small thickness was helpful in the reduction of the proportion between errors and the vertical depressions. It was thus making the result more accurate. |
Bibliography
Internet
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Mathewson, Glenn. 2020. “Cantilevers In The 2015 Code”. Professional Deck Builder. https://www.deckmagazine.com/design-construction/framing/cantilevers-in-the-2015-code_o.
Books
Wilcox, R. M. (Ralph McIntosh). n.d. Theory And Calculation Of Cantilever Bridges.
Rogers, Jonathan, and Mark Costello. n.d. Cantilever Beam Design For Projectile Internal Moving Mass Systems.
Findley, C. F. 1887. Cantilever Bridges. [Lieu de publication non identifié]: [éditeur non identifié].
Crews, John H, K. N Shivakumar, and I. S Raju. 1987. Factors Influencing Elastic Stresses In Double Cantilever Beam Specimens. Hampton, Va.: National Aeronautics and Space Administration, Langley Research Center.
Ratcliffe, James G., 2010. Sizing Single Cantilever Beam Specimens For Characterizing Facesheet/Core Peel Debonding In Sandwich Structure. Hampton. Va.: National Aeronautics and Space Administration, Langley Research Center.