- There is no impulse – this will help in creating the model as an impulse would include a velocity and alter the forces
- 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
- Representing the problem visually in a mathematical form
Vertical equilibrium: RD + RS = Mg
Equate moments about Mg: 0.5RD = 0.5RS
- Calculating the frictions
The first step in getting the model is to calculate the co-efficient of static friction (μS)
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: θ = 23º ± 1º
The maths involved in obtaining a value for μS is as follows:
F = MgSinθ
μR = MgSinθ
μMgCosθ = MgSinθ
μ = MgSinθ
MgCosθ
μ = Sinθ
Cosθ
μ = Tanθ
The various values for μS can now be calculated:
The next step in getting the model is to calculate the co-efficient of dynamic friction (μD)
To calculate this value we needed the first distance moved by finger A in the preliminary experiment
Vertical equilibrium: RD + RS = Mg
Equate moments about A: 0.86RS = 0.36Mg
Therefore, RS = 0.36Mg
0.86
= 0.61535 N
Therefore, RD = Mg – RS
= 0.85465 N
At this point the motion switches so the frictions are equal:
FD = FS
μDRD = μSRS
μD = μSRS
RD
μD = 0.61535μS
0.85465
- The final model
For switching FD = FS so the letter F can be used for both frictions
xRD = yRS
x F = y F
μD μS
x = μDyF
FμS
x = μDy
μS
The final models:
Max x = 0.32y
0.40
Min x = 0.29y
0.45
Av x = 0.31y
0.42
Theoretical Results
Below are the theoretical results in a table form with the possible boundaries and the average values:
Below are the theoretical results conveyed on graphs:
Conducting the Experiment
To 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
- Place both fingers at 0.5 m from the centre either side.
- As slow as possible, slide apply force into the centre of the rule horizontally.
- Record distances from centre at switching points
- 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:
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 accuracy of 0.5º, then the model and graphs would be as follows:
Max x = 0.31y
0.41
Min x = 0.30y
0.43
Av x = 0.31y
0.42
Therefore, the graphs would now look like:
These graphs are now much closer to the experimental values than previously calculated.
