Lab 7: March 22, 2017: Modeling Friction Forces

Lab 7: Modeling Friction Forces

Amy, Chris, and John 

March 22, 2017

Today's lab was to model friction forces in five different experiment, in both flat surface and sloped surface scenarios. Compare results obtained in experimentation with derived values for kinetic and static friction.

Theory/Introduction:

Static and Kinetic Frictional Forces

(1) Static Friction (on a flat surface):

Frictional forces have two different characteristics, one for non-moving cases (static) and move (and possibly accelerating) cases (kinetic). The magnitude of static friction scales along with the applied force that opposes it until it reaches the maximum static friction point where your object starts to move and convert over to a constant kinetic frictional force (shown in picture above).
Static frictional force can be modeled mathematically by:

The less than or equal sign is used since the force of static friction scales along with the force applied. If the force applied is 0, your static frictional force is 0 and the value continues to match the force applied until it reaches the maximum static friction point.

This maximum point of static friction can be modeled mathematically as:



where N is the normal force that the surface exerts on the object that counteracts the gravitational force.

Fig. 1. Free body diagram for the
static friction setup.

M:





m:





We model this by attaching a wooden block to a string with a pulley over the side of our table. There is a flat wood panel we lay on our table and the wooden block has a linoleum surface that touches the wood panel. On the other side of the pulley we have a mass that is hung and we increment the amount of mass until the wooden block starts to move. The amount of mass required to just get the  block moving allows us to calculate the point where the force applied starts to tip over the maximum threshold of static frictional force.

(2) Kinetic Friction (on a flat surface):

Going back to the graph of frictional forces, once the object starts to move the amount of kinetic frictional force is a constant value modeled by:



Acceleration of the system can be equal to zero when the object moves with constant velocity. The force applied would be equal to the force of kinetic friction. In our lab we measure this quantity by attaching the string on the wooden block to a force sensor. We then pull this force sensor with a constant velocity to measure the amount of force which allows us to calculate the kinetic frictional force.

(3) Static Friction (on a sloped surface):

Static friction on a sloped surface reaches a maximum when the x component of the gravitational force is big enough to just start to move the object. We model this portion of the lab by tilting the wooden panel (used in both experiments 1 and 2) up until the block just starts to move and we record this angle. With the angle, we can draw a free body diagram to calculate the forces acting on the object and solve for static frictional force.

Fig. 1. Free body diagram of block on
sloped surface with static friction.

(4) Kinetic Friction (on a sloped surface):

Fig. 3. Kinetic friction on slope setup
with motion sensor taped to top

As mentioned previously, kinetic frictional force is a constant value. Once an object is moving, if the force that moves the object is greater than the kinetic frictional force, the object will accelerate (in this case, down the slope by the x-component of gravitational force). We measure this acceleration of the block down the tilted sloped using a motion sensor (shown in Fig. 3). With the motion sensor recording position, we can derive equations and calculate for a value of acceleration.

(5) Predicting the Acceleration of a Two-Mass System:

Fig. 4. Two-mass system used in
experiment 5

In a two mass system (shown in fig. 4 similar to the one used to find static friction in experiment 1), if the gravitational force of the hanging mass is heavy enough, we can accelerate the wooden block. We attached a motion sensor at the other end of the table and we find the motion of the block while its being accelerated by the weight of the hanging mass. We find acceleration of the block the same way we did in experiment 4.
We can also solve for the acceleration of the system using free body diagrams and calculations of sum of forces. We make the assumption that the mass of the pulley and string are negligible and the pulley has no friction.
After having done this, we can compare our experimental value with our calculated.

Experimental Procedure:
With our wooden panel and wooden block with linoleum surface setup, we attach a string to the hook on the block. The string goes to the edge of the table over a pulley that ties onto a hanging mass. The mass of the wooden block is measured and we add mass in small increments (we used a couple 20 gram masses initially and then added 5 gram masses until the system moved) until the block just starts to move. We took the mass just before the final mass as the mass for our gravitational force calculation. This process was repeated with an additional 200 gram masses added to the wooden block, up to 600 grams (so the process was repeated 4 times).

With these four data points we graphed them with force of static friction (which we know is the gravitational force of the hanging mass from our FBD earlier) as the y and the Normal force as the x. The linear fit of these data plots gives us the coefficient of static friction:

Fig. 5. Wooden Block and Force
Sensor setup

(2) For this portion we use the force sensor attached to the wooden block by a string (shown in Fig. 5). We first calibrate our force sensor using a 1 kg hanging mass (which we know to be 1 kg * 9.8 m/s^2 = 9.8 N). Once our force sensor is calibrated, we pull the wooden block horizontally at a constant velocity to find the force which equals the kinetic friction force. We repeat this process similarly to the first experiment where we run four different trials with an extra 200 gram mass each time. We plot this data and extrapolate the coefficient of kinetic friction by finding the slope of the line.

(3) We used a phone application to find the angle from the horizontal of the wooden panel. As we lifted the panel higher and higher, we took the angle measurement when the wooden block just started to move. Using this angle measurement, we are able to calculate the coefficient of static friction by using our free body diagram.

(4) We have a motion sensor taped to the top (ascending side of the panel) and have the angle measurement reading through the phone app. As the panel is lifted high enough for the wooden block to accelerate downwards, the angle is recorded and the graph for the motion of the block is used to find the acceleration.

(5) We set up our two-mass system with the block attached by string to a pulley that hangs over the table with a hanging mass. We hang a mass sufficiently heavy enough to cause acceleration to the block. We have a motion sensor at the other end of the table that detects the motion of the block very similarly to experiment 4. This gives us another graph of the motion and from which we can get a measurement for the acceleration of the block. We use this experimental value and compare it to our calculated value once again and analyze our results.

Data:

Fig. 6. Our static friction value from experiment 1.

From our experiment 1, we graphed the mg of the hanging mass vs the Normal force. As outlined in the Theory/Introduction, graphing these variables gives us a constant slope value which represents the coefficient of static friction. The slope given by the linear fit on LoggerPro shows that this value is 0.400 (shown in Fig. 6).

For experiment 2, our data looked like:

Fig. 7. Our data for experiment 2. The slope represents our value for kinetic friction coefficient.

Very similarly to the way we graphed in experiment 1, We graph the normal force vs mean of the force obtained from the force sensor pulling at constant velocity. Our slope value, which is the coefficient of kinetic friction, obtained here was 0.2769 +/- 0.0048.

For experiment 3, the angle we obtained for the point at which the block just begins to move was 33 degrees. 

Fig. 7. The acceleration we obtained from experiment 4.

From our experiment 4, The acceleration value we obtained through LoggerPro was 4.071 m/s^2 (not shown in Fig. 7 but this was data that we recorded).

Fig. 8 The acceleration we obtained from experiment 5.

For our experiment 5, we calculated the acceleration value the motion sensor picked up in the two-mass system. The acceleration value we obtained was 1.5m/s^2 (shown in Fig. 8).

Calculated Data:

Using the equations presented in the Theory/Introduction for part 1, we calculated for hanging mass weight (which we equated to our force of static friction) and also calculated for the Normal force (the contact force from the wood that went against the gravitational pull). The same process was done for experiment 2 in calculating our coefficient of kinetic friction.

For experiment 3, the equations listed above were used and our calculations for coefficient of static friction came out to be:



For experiment 4, the slope of the graph of velocity vs time gave us an acceleration value of 4.071m/s^2. The angle we used was 33 degrees. Our calculation for coefficient of kinetic friction was:



For experiment 5, the acceleration value the motion sensor gave us through our slope of the velocity vs time graph was 1.5m/s^2. The hanging mass we used was






Assuming T1 = T2, and the values of 0.183 kg for the block and 0.095 kg for the hanging mass, we can calculate for the coefficient of kinetic friction.



this gives us a value of 0.287.

Conclusion:
We had tons of discrepancy in data. The error presented in the experiment was substantial to the point where our calculated values for coefficients of static and kinetic friction varied. This was a precursor going into this lab, we could repeat this procedure multiple times and still come out with radically different numbers.

Gathering all our values for friction coefficients:

	µ (k or s)
Experiment 1	0.400 (s)
Experiment 2	0.2769 (k)
Experiment 3	0.65 (s)
Experiment 4	0.154 (k)
Experiment 5	0.287 (k)

Looking at the data, it seems the values for kinetic friction coefficient did not vary quite as much as static, although there are very little data points to compare to. It almost seems as though experiment 3 varied greatly from the first experiment's results. We thought that the reason for this discrepancy had to do with the speed at which the board was lifted as well as initial placement of the wooden block. We got varied angle values each time. The same goes for the first experiment, there were cases where the block just by itself would not move until we placed more than 120 grams of masses on the hanging mass and some case where it moved before even 80 grams. We went with a single value as instructed but I suspect this is a huge source of error in our values because we could not come up with on mass that made our system go.

For our kinetic friction value, the data obtained from experiment two and five are similar. The value obtained for four however is significantly different and we believe it has to do with our motion sensor. As the block was sliding down it initially started too close to the motion sensor and the amount of time it had to capture the velocity of the sliding block gave us erratic results.
For 2 and 5, it was a much more controlled environment where we did not have to oush our block for our system to move.. Experiment two merely required a well calibrated force sensor and as constant a pull in the horizontal direction as possible. While there is most likely human error present in the execution of experiment 2, judging by the relative high correlation value in our graph (0.9977) the data we collected at least had consistency. Experiment 5 had the mass accelerate not as fast as it did in experiment 4 and we got much more accurate readings of data on LoggerPro. Of course, similar to our experiment 1 setup, it could be that the amount of mass required to get the hanging mass moving could have been more or less just out of randomness. 
Although we started the experiment with as clean a surface as possible, the data collection still came out very messy and lacked a pattern. 

While our values came out with rather extreme values of uncertainty, the trend of coefficient of static friction being greater than kinetic friction was at least true. In our initial lecture, Professor Wolf explained the graph we first introduced for static and kinetic forces, it shows that force of kinetic friction compared to the maximum static friction force, the kinetic is always less. Our data sufficed that pattern at least, as the lowest value of static friction we obtained (0.400) was still greater than our biggest value of kinetic (0.287).
Further research showed that force of kinetic friction isn't necessarily "constant" as it is making small changes rapidly but has a mean value that represents a constant pattern. 

This would explain the slight differences we had in values for experiment 2 and 5 but the results we got from experiment 4 were mostly likely error in our motion sensor equipment. If we had a more accurate setup, possibly a longer surface to slide on or a more precise motion sensor, the value may have been closer to values obtained in the other 2 experiments.

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