I predict that as the weight increases on the ruler the more it will sag and the more it will sag each time, demonstrated by this rough graph:

There should be a strong positive correlation between the sag and the force

I predict this because of the rule of compression and tension; as more force is exerted vertically onto the end of a ruler, the top face becomes tense and the bottom face becomes compressed. The explanation of this will be in my conclusion. I also make this prediction using the formula of moments:

Moment of a force (Nm) = force (N) x perpendicular distance from the pivot (m)

This will also be explained in my conclusion.

(The diagram of my method is shown above next to my predicted graph shape). A wooden metre rule will be clamped onto a table using a G – clamp with a certain distance sticking out from the table. Another metre rule will be clamped with a retort stand at right angles to the floor so to measure the sag of the other ruler. The weights will be added to the ruler at three different lengths they will be added in one hundred gram amounts for each length (of the ruler) to the end of the ruler with some string and sticky tape (100g to 1000g). Before this is done however a reading will be taken from the vertically mounted ruler to find out how many cm will need to be taken away from the results to give the correct sag of the ruler. As the weights are added, care will be taken to make sure that the weights are not added and the ruler then dropped suddenly so as to reduce the risk of the ruler splintering.

Apparatus Used:

* Wooden metre rule

* G – Clamp

* A second ruler with which the sag will be measured

* Weights ranging from 1 to 10 Newton’s, in increments of 1 Newton (therefore 10 of them)

* String and tape to put around the ruler with which the weights will be tied

Each reading will be taken three times so as to make the results as accurate as possible. An average will then be taken to make the investigation more accurate and to take into account all three results for each point equally. It all will be presented in a table.

There are certain factors which affect this investigation, they include: the length, thickness and material of the ruler, the accuracy of the weight of the weights being used, how far along the ruler the weights are situated and any human error involved. To make this investigation a fair test the only variables will be the force exerted on the ruler and the distance along the ruler that force is.

All my results will be to one decimal place as well as the averages taken from each set of three results.

A preliminary experiment was carried out initially to determine whether there was a relationship worth investigating between the sag of the ruler and the amount of weight added to the ruler. It was determined that there was.

The results for this experiment are as follows, which were taken to the nearest half cm, (the reading for the sag was taken every 200 grams and the length of the ruler was 90cm – the results were compensated as described earlier):

Force (N)

0

200

400

600

800

1000

Sag (cm)

0.0

4.5

9.0

11.0

13.0

19.5

A graph showing these results:

To conclude my plan; these results lead me to believe that there was a relationship between the sag of the ruler and the force added to the end of it worth investigating and these results formed the basis of my investigation.

Results/Evidence:

The experiment was carried out and three sets of results were taken for each quantity of weight and an average was taken for each set of three results.

The results are shown in the table on the following page:

This table can be summarised by these three graphs that show the average sag of the ruler at 90cm 70cm or 50cm against the force in Newton’s:

It’s very evident that there is obviously a strong positive correlation between the sag (or depression) of the ruler used and the amount of force exerted on it. So therefore this experiment of mine will not have to be repeated because the relationship is as expected for all of these graphs.

Conclusion:

I conclude, generally, that there is a strong correlation between the sag and the force exerted on the ruler and that this point is helped proven by the line of best fit running through the points on my graph; they are straight and therefore indicate that the sag is directly proportional to force exerted on the ruler. The gradient of the line differs slightly in all of the graphs in that it (the gradient of the line) becomes greater as the length of the ruler increases (i.e. it’s at its steepest at 90cm of ruler length). This is because as the length of the ruler decreases it becomes increasingly difficult to bend it. This can be explained using the rule of compression and tension as mentioned in my plan: as more force is exerted vertically onto the end of the ruler, the top face becomes tense and the bottom face becomes compressed; the amount of tension is directly proportional to the amount of compression so as one increases the other increases at the same rate; this explains why there is a strong positive correlation shown by the straight lines of best fit show on my graphs. This can be also be illustrated by the formula of moments:

Moment of a force (Nm) = force (N) x perpendicular distance from the pivot (m)

This formula shows that if the distance from the pivot were to decrease the amount of force required to obtain the same moment would have to increase. This demonstrates my point: as the graphs show that the smaller the length of the ruler becomes smaller more weights are required to obtain the same sag.

If certain values are put into this formula then this can be illustrated even more fully; for example, referring to the table: in the (i) 5 Newton column the average for “Sag at 90cm” is 9.1 cm (therefore 0.091m) and then looking say in the (ii) 7 Newton column at “Sag at 50cm” the average here is 2.5 cm (therefore 0.025m). Therefore (i) 5 Newton’s x 0.091 m = 0.455 Nm and for (ii) 7 Newton’s x 0.025m = 0.175 Nm. This shows that the moment is greater for 5 Newton’s of force than for 7 Newton’s yet the sag is less demonstrating my point.

What happened was as predicted in my plan – there is a straight line of best fit which runs through the majority of the points demonstrating a strong positive correlation between the sag (cm) and the force (N) exerted on the ruler. This compares very closely with my predicted graph shown in the plan. The amount of sag becomes less (as mentioned earlier) as it requires a greater moment to increase the sag but this is not given because the force exerted is the same for all the different lengths of the ruler.

Evaluation:

The procedure I used seemed to collect the data that was expected in the plan, generally but there are reasons as to why this project could well be not as accurate as it could have been; these will be discussed in this evaluation.

Looking at the three different graphs there is a definite strong positive pattern as discussed previously between the force and the sag and this seems to be consistent suggesting that my results are quite reliable. If the results were totally accurate however the line of best fit would run through the origin (at zero on both axes). It appears not to do this on the graph showing the sag at 50 cm; this is likely to be due to how accurately I took my results (whether I took a slightly false reading as a result of a) parallax error and how many times I took them. To make them more accurate I could have taken them more times and used a more accurate way of reading from the (sagging) ruler to the vertical ruler (for measuring); like for example using a spirit level to make sure that when reading across that it is being taken exactly horizontally and therefore a lot more accurately. There are also a few of rogue results that are likely to be because of inaccuracy and which probably have also influenced the accuracy of my results. Two of these are more pronounced than the others on the graph showing the sag at 90 cm at the points for 8 Newton’s and for 10 Newton’s.

The evidence this investigation has gathered could well be wrong and inaccurate as only one type of cantilever (the ruler) was analysed in it and therefore could be biased. The ruler was a certain width, length and breadth, and my conclusion could only be true for wooden objects. If this were the case then I would have to test other substances are more likely to be used for building bridges such as steel or concrete (with pre-tensioned steel infra structure): if concrete on its own was used as a cantilever then it would simply fracture and break if large forces were forced on it. So to overcome this problem when the concrete is made it is set round tensioned steel; once the concrete has set, the pressure is released and the steel remains tense because the concrete holds it like that. This stops the concrete from fracturing when bent because even if it is bent downwards, referring to the rule of compression and tension, the tense steel compresses the top face even though it is bending and it as the same effect on the bottom face except that it is a lot more compacted than the top. The experiment was a fair test as much as possible because the only variables which changed was the force put on the ruler and the distance the force was placed along the ruler.

When carrying out this investigation certain difficulties were encountered: it was hard to set up the vertically mounted ruler for measurements and the sagging ruler at an accurate length, there was also difficulty in measuring the sag as the ruler with the weights on tended to oscillate up and down and therefore making the sag harder to read.

If I were to repeat the experiment then I would change my method slightly; I think I would use a steel ruler as well as the wooden one and take readings using a spirit level so to increase the accuracy of my investigation for reasons already explained.

To make the investigation more accurate and to obtain more evidence I would have to have collected more data for all the different lengths. This would make me surer of my conclusion as it would show that my conclusion can be more widely applied.