In this case, the bridge represents the steel spring. When a load (mass) is added at the (horizontally) middle point of the bridge, its sag is similar to the extension of a recoil steel spring while a force is exerted to the lower end of this spring. As in the Hooke’s Law, it is true if the elastic limit of a spring is not exceeded, this spring always returns to its original position once the force is being removed; same for a bridge, the bridge returns to its ‘original axis’ when the load is removed.
Graphically, I predict that the sag increased constantly would be directly-proportional to the increase of mass, i.e. a straight slope on the graph heading to the top right-hand corner, until it reaches its elastic limit (the yield point).
Varying distance
The sag of a bridge also affected by the change in length. The length is the distance between one of the two pivots and the load in the middle, which means there are two sets of identical rotations, one is clockwise and one is anti-clockwise, are taking place at the same time, this is because of moments.
Fig. 3 The motions when a sag occurs
The moment of force or the turning effect depends on both the size of force and how far it is applied from the fulcrum. It is measured by multiplying the force by the perpendicular distance from the pivot, i.e. Moment = F × d.
Ignoring one of the two identical turning effects, we are going to investigate how sag is affected by the distance from the pivot to the load.
Fig. 4 The moment acting on the left half of a bending bridge
As Moment = F × d, therefore, I predict that the moment of the turning effect would vary in the same proportion when distance varies. Same for the sag of a bridge, which would vary in proportion as distance varies.
Method of Obtaining Evidence
Apparatus list: 3 clamp stands, a wooden bridge, one or more metre rule(s), a rubber band, sellotape, a needle, metal weights (preferably 50g and 100g each to allow varible mass up to approximately 400g)
Set the apparatus as follows: -
Fig. 5 Settings of the experiment
Measure and make sure ha = hb, d1 = d2 and, d3 = d4. Clamp stand should be supporting the bridge loosely to enable sags.
Then, use a clamp stand and clamp a metre rule perpendicular to point M, where the mass would be added, to measure sags of the bridge (see Fig. 6).
Fig. 6 How to find sag
Record the reading of h0, while no mass is added. Keep the distance between the two clamp stands (pivots) constant, in which I chose 850mm, sellotape a needle at the middle point (point M) of the wooden bridge to enable it to point on the metre rule and observe the readings; put a rubber band round the bridge at the same point where the needle is (M), i.e. 425mm from one pivot to the load. Then add masses (metal weights) (at least 6 different ones, 6 different massesrange from 100g to 400g) and record the reading of h1 for each of them. Then work out the sags by substracting h1 from h0.
After the investigating the relationship between the sag and mass of the bridge, the next step is to investigate the relationship between the sag and distance (between the pivot and the load).
Keep the settings of the apparatus as Fig. 5 , choose a mass (I chose 300g) to be constant. At each time, adjust the distances between the clamp stands and measure the sag, similar to what have been doing before.
Record how sag varies when you change the distance between two pivots. Meanwhile, the distance from one of the two pivots to the load is exactly half of it.
Results
Fig. 7 Graph: S against m
Fig. 8a Graph: S against d Fig. 8b Graph: S against d2
Fig. 8c Graph: S against d3
Analysis
From the result of experiment “varying mass”, I chose the distance between the two pivots (clamp stands) 850mm, which the distance from pivot to load is 425mm. I tried 6 different masses ranging from 100g to 400g, and found the sags for each of them. I then plotted a graph of sag against mass (Fig. 7). I drew a best-fit which is formed diagonally, and found that the best-fit is a straight line.
The result of “varying mass” shows that sag of the bridge is directly proportional to mass, until it reaches the elastic limit of the bridge. The straight line shows Sαm ,while a mass constant k is the mass needed for sag; therefore,
m = k S
k = m ÷ S
where the mass constant is represented by the gradient (assume that there is no error).
In the second experiment on “varying distance”, I chose a constant mass 300g. I tried 9 different distances ranging from 450mm to 900mm between two pivots, i.e. 225mm to 450mm between one of the pivots and the load (point M).
Sag has been increasing as distance increases. However, when I then tried to plot a graph of S against d (Fig. 8a), I could not then draw a straight best-fit line through them, instead, I drew a parabola.
Therefore, sag is not directly proportional to the distance. I drew another graph of S against d2, I found that the best-fit is also a parabola (Fig. 8b).
Therefore, sag is not dirctly proportional to d2. I drew another graph of S against d3. This time I was able to draw a roughly straight line. From that I can prove that S is directly proportional to d3, i.e. Sαd3.
Conclusions
In this project investigated how sag changes when mass varies when distance between load and pivot is constant and, how sag changes when distance between load and pivot varies when mass of load is constant.
I predicted that Sαm, sag of a bridge is directly proportional to the mass applied on it, which this idea was taken from Hooke’s Law focusing on the extension of a spring, is directly proportional to the stretching force. After doing the “varying mass” experiment, I found that Sαm. I predicted correctly.
I also predicted that sag is the moment, multiplying force and distance; as
I predict that sag, which is the moment (= F × d) of the turning effect would vary proportionally when distance varies. Same for the sag of a bridge, which would vary proportionally as distance varies. After doing the “varying distance experiment, I discovered that my prediction is only partly correct. Sag is not directly proportional to distance, in fact, sag is directly proportional to distance3, Sαd3.
Evaluation
This investigation of the sag of a bridge has been done smoothly. Inaccuracy of results could be made by many factors. It was difficult to make an accurate point M (the middle point) and keep both sides absolutely identical to each other. It was in fact impossible to sellotape the needle and apply a mass at the same point, M. Therefore, measurements could not be read accurately and formed errors.
