Method One
I will set up the apparatus as shown above and to the specifications on the Practical Cover Sheet. Then I will place two stools at either side of the apparatus, on one of the stools I will place a light connected to a lab pack and one the other I will place a large flat aboard with an attached piece of white paper. This board will be attached to the stool. The light will then be turned on and the shadow of the string will then be focussed cleared on the board by moving the stools either back or forward. I will then draw the vertical line down from the pivot, and then trace the line that the string makes to the vertical from the shadow that forms on the paper; I will repeat this for different angles to the vertical, by moving the clamps up and down to adjust the tension in the string and therefore the angle to the vertical. I will record the force displayed on the spring balance for each different angle.
Method Two
I will set up the apparatus as shown in the diagram, but then I will “blu-tack” a protractor to the head of the nail, so that it is in line with the vertical of the ruler. I will then adjust the clamps so the angle is changed and I will record the angle and the force that is displayed on the spring balance.
Chosen Method
After testing both of the above methods I have decided to use method two with the attached protractor, as this is the easiest to set up and also the easiest to record readings for. The main problem with this method, though is the fact that because I will be using a protractor, parallax error will be a factor, but I will try to reduce this as much as possible, by having the protractor at a decent eye level and having it completely vertical (not bend in any one direction). Method One was very fiddly to set up and time consuming to take readings because the line had to be drawn onto the paper and sometimes this would move the board and I thought that the error would be too great to consider using this method for my investigation. Repetitions need to be taken to draw accurate conclusions from and to produce reliable readings.
Equipment
For the diagram:
- 2 Nails
- 1 Ruler
- 1 Clamp stand
- 3 Bosses
- 3 Clamps
- 1 Spring Balance
- 2 50g Masses
And Also:
- 1 Protractor – As explained above this will be blue tacked to the head of the nail to which the ruler is attached and then used to measure the angle from the vertical.
- “Blu-tack”
Prediction
I predict that the values for the force will all be very close together and there will not be a great deal of difference between each of them, except for at the extreme ends of the angles such as 0° and close to 180°.
Hypothesis
The principle of moments states that for an object to be in equilibrium the sum of the forces around any point must be equal. This means that if an object is in equilibrium (i.e. balanced) then the sum of the anticlockwise moments must be equal to the sum of the clockwise moments, about that point. This means, for my experiment that the sum of the clockwise moments (of the ruler and the fixed masses) will equal the sum of the anticlockwise moments (i.e. the tension in the string) around the pivot (the nail). From this I can work out the tension in the string because I will know the moments of the ruler and the fixed masses and I also know the distance of the string to the pivot.
Moment of Tension = Moment of Ruler + Moment of Masses
(Where Moments are equal to the force multiplied by the perpendicular distance from the line of action from the pivot, the pivot being in this case the nail to which the ruler is attached)
The tension could also be calculated by resolving the forces perpendicular to the ruler.
The arm holding a weight in this manner is an example of a class three level as the effort (the muscle) is on the same side of the pivot as the load and it is closer. This makes the Mechanical advantage of the lever less than 1. This gives the body greater control over the arm.
Below, over the next few pages it is shown how I developed my initial hypothesis into my final calculation that I used to calculate values so that I could relate back to them when my investigation is complete and also have some pattern of results that I can compare with.
Simplistic View
This diagram above is a simplistic view of what we are trying to measure. From Module 1 we can use the principle of moments to calculate the force, F if the Weight of the “arm”, W is known and also if the mass, M is known. The mass of the ruler is 35.45g and the mass is 100g.
Taking moments about the pivot:
Sum of Anticlockwise moments = Sum of clockwise moments
F x 5cm = (W x 17.5cm) + (Mg x 35cm)
F x 5cm = (0.03545 x 9.81 x 17.5cm) + (0.1 x 9.81 x 35cm)
F x 5cm = (6.086) + (34.335)
F x 5cm = 40.421
F = 40.421 / 5
F = 8.08 N
More Complex View
In this case the fact that the muscle (i.e. the string) isn’t vertical is taken into account and so the angle which the string makes to the ruler is taken into account (in this case theta is equal to 50O)
Applying the principle of moments about the pivot:
Sum of Anticlockwise moments = Sum of clockwise moments
Fsinθ x 5cm = (W x 17.5cm) + (Mg x 35cm)
Fsin50 x 5cm = (0.03545 x 9.81 x 17.5cm) + (0.1 x 9.81 x 35cm)
Fsin50 x 5cm = (6.086) + (34.335)
Fsin50 x 5cm = 40.421
Fsin50 = 40.421 / 5
Fsin50 = 8.08
F = 8.08 / sin50
F = 10.55 N
Even More Complex View
In this case the fact that the string isn’t vertical is taken into account and so the angle to the ruler is taken into account. And also because the ruler is not vertical the components of the weights are taken into account. (In this example the angle, ψ = 50)
Applying the principle of moments about the pivot:
Sum of Anticlockwise moments = Sum of clockwise moments
FsinΣ x 5cm = (Wsin ψ x 17.5cm) + (Mgsin ψ x 35cm)
FsinΣ x 5cm = (0.03545 x 9.81) sin50 x 17.5cm) + ((0.1 x 9.81) sin50 x 35cm)
FsinΣ x 5cm = (4.662) + (26.302)
FsinΣ x 5cm = 30.964
FsinΣ = 30.964 / 5
FsinΣ = 6.193
F =
F = 9.51 N
Most Complex View
In this example all of the above is taken into account and also the fact that the hole for the string is off centre and so this angle is taken into account when calculating the angle that the tension force must be turned through when calculating FsinΣ.
Sum of Anticlockwise moments = Sum of clockwise moments
FsinΣ x 5cm = (Wsin ψ x 17.5cm) + (Mgsin ψ x 35cm)
FsinΣ x 5cm = (0.03545 x 9.81) sin50 x 17.5cm) + ((0.1 x 9.81) sin50 x 35cm)
FsinΣ x 5cm = (4.662) + (26.302)
FsinΣ x 5cm = 30.964
FsinΣ = 30.964 / 5
FsinΣ = 6.193
F =
F = 8.69 N
Using the Symbols in the example above, I have calculated the following results using the same method.
NB: In Reality the arm would never get to 180° and so the force would be zero (and it almost is 10-15 is extremely small, and so this tells me that the graph will have an asymptote at ψ = 180°) In this example though, as explained before because the string is off centre of the ruler and the 5.7° has to be taken off then the lowest angle Ψ that will be possible to calculate will be (180 – 5.7) which is 174.289°,and this will result in the force no being able to measure as we will be diving by zero and so the asymptote will be at Ψ = 174.289°.
Looking at the calculated value for 174° shows me that my assumption was correct because 170° gives a value for the force as 14N, but 174 gives a value of 125N, which is increasing very rapidly to infinity:
As can be seen from the above graph, the relationship between the force and the angle is not linear, and it is always changing for each different angle.
So far all I have looked at is changing the angle. The other thing to change is the weight of the fixed masses, or the weight of the ruler, because from the principle of moments these two factors affect the force on the string. As I already have a basic outline of what the patter of the graph is for a ruler of mass 35.345g and fixed masses of mass 100g, I could conclude that change the mass of either of these grater than the above values would translate the graph upwards (i.e. in the positive y direction), and as a result the force on the string would be greater, whereas is the mass of the fixed masses were lowered the graph would be translated downwards (i.e. in the negative y direction) and as a result the fore4c on the string would be lowered. The below graph shows the same experiment as before but this time for 3 different masses:
Please note that only the weight of the masses have been changed and not the weight of the ruler, but if the mass of the ruler were changed the graph would look the same as above except that the proportions would be different (i.e. the lines would be closer together because the weight of the ruler is less than the weight of the masses). If time constraints permit, I will perform an experiment to prove that the above data is correct.
Results
These are the results that I actually collected while doing my experiments. I repeated the experiment three times because I though this would provide me with a good range of results and I hoped this would also ‘iron-out’ any anomalous results. These results can also be seen plotted graphically over the next few pages.
Actual Method Used
- I set up the apparatus as shown on the diagram and then let the ruler hang loosely at 0° to the vertical.
- I attached a protractor to the head of the nail with blue tack making sure not to get any blue tack on the shaft of the nail as this is where the ruler is attached and the blue tack could interfere with my investigation.
- I then lined up 0° on the protractor with the vertical line that runs down the centre of the ruler and through the pivot.
- I then moved the clamp that is attached to the spring balance up and this in turn pulled on the string on moved the ruler up a few degrees or more.
- I would the look on my protractor, making sure that I am directly in line with it to reduced parallax error and I would then record the angle to the vertical.
- Then the value on the spring balance would be recorded, again making sure to reduce parallax error, so my results can be recorded with precision and be reliable.
- I then repeated this for different angles to the vertical, until I get to the top (i.e. almost 180°).
- This will be repeated 3 times, so I can draw accurate conclusions.
Analysis of Results
The three graphs below show the actual results that I gained while implementing my investigation. As you can see they are all quite alike and all look something like what my theoretical results looked like when plotted graphically.
All of the above graphs have the same basic curvature, and they all start of rising very steeply and the curve then almost levels off and starts receding a little and then shoots sharply up again. This can be seen on all three graphs and is better shown when all three sets of results are plotted together as shown on next page.
As you can see from the graph the results for each experiment are very similar, with only a few anomalous results being obvious. These anomalies can be attributed to one of many factors which will be discussed in my Conclusion and Evaluation Section. There is a very strong agreement between the results and this tells me that my results are reliable and the right precision was attained when performing my investigation.
To relate back to my original predictions I have plotted a graph like the one below, but also with the theoretical predictions on so it can been see how they differ and if they follow the same trends and patterns. This graph can be seen on page 18.
The graph above is of actual results as well as my theoretic results. This graph shows how well my readings agree with my predicted readings. As can be seen the same basic pattern in the same (i.e. the peaks and falls are all in the right places). The only major difference is that my theoretical set of results is shifted down in the first half of the graph but then it rises earlier in the last half. The difference can be attributed to many things which will be explained in later sections (see evaluation and conclusions section).
Evaluation + Conclusions
I think that the way I conducted my experiment efficient and suitable, as you can see from my results, they quite close to the original predicted values and they follow the same pattern. But as always there is room for improvement, which is explained in more detail below.
As the graphs show, my actual results that I obtained strongly agreed with my prediction. There were of the same basic shape, and even though they don’t match exactly I’d say that the experiment was a success. On the next page is a hand drawn graph of all my results that I actually obtained. I plotted the points all on the same set of axis so that I could draw a line of best fit through them and this could then be used to make accurate conclusions.
Suitability of Procedures Used
I think that the method I used was quite suitable for this investigation as it produced consistent and reliable results as can been from the graphs and the fact that they are so much like the graph I predicted back in the Hypothesis section. The use of the protractor was quite fiddly to see the angle, but I coped, and also the blue tack didn’t hold it tightly enough and so if I were to repeat this investigation again, I would consider are more permanent or rigid way of securing this on the head of the nail. Due to friction at the nail, and also on the string that is rubbing the nail, and also in the spring balance, the readings for one force could have many angles, and sometimes the force would be significantly different for two angles that are the same.
As the graphs show, my actual results that I obtained for my experiment strongly agreed with my prediction. Friction both of the nails (one touching the string, and one to which the ruler pivots on) was not taken into account in my predictions; this is a reason why my actual results could have differed from my predicted results.
Throughout all of my experiments, the patterns that the graphs I produced showed the same basic patterns, the only different was the magnitude of the graph, but even this was very close. The main area where the results differ is when the angle is approaching round about 140°. At this point in my predictions the force starts rising rapidly, whereas in my actual results, the same trend is carried on until about 165° and then the graph starts rising very rapidly, until it is too big for the spring balance.
I think that because friction was not taken into account in my initial predictions is why the real results differ slightly from my calculated results. The friction played a big part in the investigation, because uses moving parts, and objects pivoting. This is the main factor and there are also other factors explained in sources of error and the other sections below
Anomalous Results
There is only one very visible anomaly in my results and there is also one other. Looking at the hand drawn graph you can see that there is one point for an angle of 5° that gives a force of 9.1 N. This is obviously anomalous as it doesn’t fit the trend of all the other results. Because of this anomaly I have ignored it when drawing my line of best fit on the graph, there is also one other result which I consider to be anomalous, although it is not as extreme as the afore mentioned result. This point occurs at an angle of 64° and gives a force of 9.7N. These two anomalous results can be attributed to many things which are explained in the other sections (below).
There are two sections on my graphs where the actual values are significantly different from the calculated values. These sections can be seen clearly on the graphs that I have plotted. I attribute these sections to the friction that was not taken into account in my predictions, but I also feel that more readings should have been taken around this area to gain a better insight into what was happening; this is especially true towards the large angles.
Why My Results Were Slightly Different From My Calculated Results
I think the main reasons why my results were slightly different from the results that I had calculated before are:
- The accuracy of the weight of the fixed masses. The fixed masses that were used in my experiment were quoted a value of 50g each but when I measured this on a mass balance there was quite a bit of difference but it was hard to actually quote an accurate value because the mass balance is so sensitive that even slight vibrations and even air movement can displace the scale, and adjust the display. If I had an accurate value for the weight of theses fixed masses then this would have had an effect on my theoretical results and they might have agreed with my actual results more.
- The accuracy of my results could have also played a part in this. This will be better explained in the sections below on ‘accuracy’ and ‘sources of error’.
- The friction I my experiment was, I think the main reason why my calculated results and my actual results were not exactly the same. This is because there were so many places where friction could occur and did occur in my experiment and I think that this did have a profound effect on my results. It is easily noticeable when taking readings because the ruler can stick in a certain position and when moved it can have a wide range of readings on the spring balance i.e. every angle has a range of readings on the spring balance, and it is not as if this follows a measurable pattern or trend, it just seems to be random, and can get annoying in the investigation when the slightest knock can affect the reading of force on the spring balance or alternately changes the angle but not the force. There is friction at both nails - the one for the string and the one and the one for the ruler. The nail that the ruler rotates around is a min area for friction because the hole in the ruler is quite tight and so the ruler could stick on it and affect my results. The nail which the spring moves up and down on will also drag on the string and cause friction which could affect my results. There could also be friction in the spring balance which stops it from extending properly.
Sources of Error
There are quite a few sources of error in my investigation and anyone of these could have contributed to my two anomalous results.
The main sources of error in my practical experiment were the accuracy of the measuring devices that I used to measure the angles and also the forces. These devices would have been better if the accuracy was improved i.e. more divisions, or a higher guarantee from the manufacturer that they are accurate.
Another main source of error could have been human error because everyone judges the scales differently even when trying to reduce parallax error by being straight on with the measuring device. Even if the scale was read correctly the reading could have been recorded wrong by me and could have contributed to my anomalous results.
I think that another source of error could have been how accurate the protractor and the spring balance was, as the build quality of all the devices involved could vary and this could contribute a lack of integrity of my results.
The fact that I was using a spring balance might have made zero errors possible because it was quite hard to place the line accurately at the zero mark, and this could have altered my results slightly. On the same note the protractor was about an inch away from the ruler and this could have altered my results because there were angles involved. The protractor had to be this distance away because the ruler was being pulled sideways towards the protractor and at this distance the two did not touch.
The ruler had a vertical line down it, going through the pivoting nail so I could accurately line up the 0° mark on the protraction with this line. But because the protractor was only attached to the nail by blue tack, any knock or movement could have moved the 0° mark and my results could have been affected by this.
The spring balance had been used before for different experiments and, because it contains a spring this could have been slightly deformed, and this would alter my results slightly as the extension of the spring would not be uniform.
The clamps that that held the string and balance and nails to the clamp stand had a tendency to move and this could have affected my readings, by moving the various components, and thus changing friction and angles among other things.
The string that held the ruler to the spring balance although considered to be inextensible in my calculations is in reality extensible and this could have been a source of error, even if only a marginal one.
Overall the sources of error that I can think of are:
- Zero errors on spring balance
- Zero errors on protractor
- Extensible String
- Human Error (parallax and recording)
- Build Quality of measuring devices (protractor, spring balance)
- Sideways pull of ruler
- Movement of clamps
- Spring in Spring balance
Accuracy
The things that made my practical experiment inaccurate were:
- As explained in the ‘sources of error’ section above, human error must have played a part in making my experiment less accurate than it should have been, such as the reading scales on the spring balance.
- The accuracy of the protractors and spring balance was a source of inaccuracy in my investigation as the angle was only measured to 0.5°, and the accuracy was probable less because the protractor was an inch or so from the string, and therefore the angle was harder to view and record. The force was only measured to 0.05N and this could be improved by using a spring balance with a different scale i.e. one which displays forces to a higher degree of accuracy say 0.01N.
Limitations + Improvements
The main limitations in my investigation where the measuring devices that were used, and the accuracy to which they measured. These factors though have been explained above.
Overall, even taking into account the factors explained in sources of error, and the other sections I think that the measurements were taken as accurately as possible with the equipment use, but if I were to repeat this experiment again I would:
- Change the spring balance so that it has better scale (i.e. more accurate) to improve accuracy
- Change the protractor so the scale would be improved and find a way to affix it securely to the nail, and try to have it closer to the string.
- Take more results overall, and repeat the investigation more times to gain a wider insight and to produce more reliable evidence. I would also take more results in the range 130°-160° as results in this range were lacking in my investigation, and more readings in the final stages (i.e. near 180°).
- Find some way to fix the things to the clamp stand because the clamps that were used had a tendency to move and this is what could have made the ruler be pulled sideways.
- Put lubricant on the nail to which the ruler pivots on so it is allowed to hang freely, either that or to make the hole bigger / nail thinner.
Reliability
Overall I think my results were quite reliable, and they linked back well to my initial calculations. I think there is room for improvement as there is in any case but overall I think that my investigation was conducted reliably and to as high a degree of accuracy as the involved apparatus permitted. The comparison between the actual results and calculated results was quite good, and when I take into account the thing that I missed out in my prediction – friction, I’d say that the results are very reliable, even taking into account the sources of error and other factors. I would say that with the equipment that was available the measurements taken as accurately as possible and this produced reliable results.
Final Thoughts
The uncertainty of my results due to friction, and other sources of error made my investigation less reliable and this uncertainty of the readings leaks through to my graph which is what I draw my overall conclusions from. Even taking this into account I consider my results to be reliable and accurate and I can say that my results agreed with my predictions to a certain extent and the evidence is there to back up my theory.
The relationship between the force and the angle is not linear, and is hard to explain without the use of a graph. This is because of the arm and the way it is set up and how the muscles in the arm function.
To conclude I think that given the situation and the equipment provided, I made the best use of it and even through there was room for improvement, my collected results were of a high standard and fit almost the same standard as shown when comparing to my calculated values.
Data Sources and References
The books that I used while researching my investigation where:
- Physics 1 (Cambridge University Press: Advanced Sciences)
- Health Physics (Cambridge University Press: Advanced Sciences)
