Mathematics mechanics 2 - Sample Essay
Balance a standard laboratory metre rule on your index fingers horizontally as far apart as possible. Now slide your fingers towards one another maintaining the rule horizontal, and observe what happens. Does the same thing happen each time you try it? Do your fingers ever come together other that in the centre? Preliminary Experiment This was an initial experiment carried out to observe the motion of the fingers as they slide towards one another. The first observation noticed was that each finger moves consecutively, never at the same time.
Another observation noticed was that after the first finger (A) moved, the first move by the second finger (B) was considerably greater. Each consecutive move would decrease in distance until the fingers would meet. Also, the fingers always came together in the centre of the rule. The measurements of the preliminary experiment were recorded from the centre of the rule: Move Number with Distances From the Centre of the Rule (m) 1 2 3 4 5 6 A 0. 36 0. 19 0. 07 0. 02 0. 01 0 B 0. 27 0. 13 0. 04 0. 02 0. 01 0 Modelling Assumptions 1.
Rule is constant (Consistent density throughout) – this will help in the model in assuming the centre of mass is exactly in the centre of the rule. 2. Fingers are both the same – this will help in the model in assuming the dynamic and static frictions are the same for both fingers. 3. Width of fingers is 2 cm – this will help to allow theoretical number of moves as with this limit the model will be infinitely long and will never reach 0. 4. There is no impulse – this will help in creating the model as an impulse would include a velocity and alter the forces 5.
Pint point masses – although the width of the finger is accounted for to produce theoretical number of moves, to simplify the model the fingers nee to be pin points. Manipulating the model 1. Representing the problem visually in a mathematical form Vertical equilibrium: RD + RS = Mg Equate moments about Mg: 0. 5RD = 0. 5RS 2. Calculating the frictions The first step in getting the model is to calculate the co-efficient of static friction (i?? S) This involved a simple experiment in which the rule needed to be raised at an angle on the finger until the rule was just about to slide.
The result obtained was that: ? = 23i?? i?? 1i?? The maths involved in obtaining a value for i?? S is as follows: F = MgSin? i?? R = MgSin? i?? MgCos? = MgSin? i?? = MgSin? MgCos? i?? = Sin? Cos? i?? = Tan? The various values for i?? S can now be calculated: Max i?? S 0. 45 (2 s. f. ) Min i?? S 0. 42 (2 s. f. ) Av i?? S 0. 40 (2 s. f. ) The next step in getting the model is to calculate the co-efficient of dynamic friction (i?? D) To calculate this value we needed the first distance moved by finger A in the preliminary experiment Vertical equilibrium: RD + RS = Mg.
0 Below are the theoretical results conveyed on graphs: Conducting the ExperimentTo test the model an experiment must now be conducted and involves the following: Apparatus * A standard laboratory metre rule The index fingers on a pair of hands Diagram Method 1. Place both fingers at 0. 5 m from the centre either side. 2. As slow as possible, slide apply force into the centre of the rule horizontally. 3. Record distances from centre at switching points 4. Repeat 10 times Steps taken to reduce experimental error There was an observer to measure the switching points The observer also made sure that the rule was always horizontal Results The results of the experiment in table form:
Moves and Distances ( Min B 0.The results of the experiment shown on graphs: Comparisons Theoretical Experimental As you can see, the experimental values are all within the bounds of the theoretical values calculated. This shows that the model worked accurately and accounted for all the values. But, as you can see, the range in the values in the model is considerably wider then that of the actual experiment. For this reason the model needs to be refined. Revision of the process.
To improve the model, the value for the angle measured to calculate the static friction could have been more accurate. The effect of this would be that the bounds of each distance would be smaller and closer to the real experimental results. If the angle was measured to an ac 42 Therefore, the graphs would now look like: These graphs are now much closer to the experimental values than previously calculated.