Lab 7: Modeling Friction

Fortunately, we have not had to simulate class tension or class friction to model them. Instead, as physicists, we let small objects do the work for us and we stand above the fray. The only difference with this lab over the others is that we'll be doing 5 parts of relatively similar things. Tedium is a good way to model friction, indeed.

 

SECTION 1: Static Friction of Table

In this first part, we will fill a cup partially with water, and allow it to dangle over the side of the table with a string attached. The string will go over a pulley, and on the other side: blocks. We expect that the normal force between the blocks and the table will be the weight of the blocks. We also expect that the maximum static friction force will be the weight of the cup as it starts to gain enough leverage to move the blocks. Therefore, we expect a linear relationship as we start to add more blocks to the amount of water necessary to move them. And since the only thing standing between friction force and weight is the coefficient of friction, that's what we'll find.

The process involves slowly and carefully filling up that Styrofoam cup with a dropper, with a hand under the cup, and observing when the block starts to move. Ideally, the block moves just as it overcomes static friction. In practice, there are many flaws with this experiment. For one, the table and the blocks aren't of uniform smoothness, making it so that the placement of the blocks affect the results. Secondly, the angle of the blocks with the string changes the amount of force required to pull it. We also suspect that the table might be slightly slanted, perhaps with uneven legs. We found that pressing the blocks against the table slightly, then letting go, enables it to withstand far greater tension. These problems will influence our experimental results later on. We had to exercise our own experimenter's discretion in navigating these waters, and we find that our weeks of experience had not paid off. We were still raining wet behind the ears.

Nevertheless, we proceed to weight the cups.

And then weigh the blocks. And add more blocks, until we have 4.

Here's our data:

We launch LoggerPro to graph the maximum static force over normal force:

And since:

F_static = µ_s N
µ_s = F_static / N

The slope of the graph must be µ_s = 0.2188, as shown by the linear fit.

 

SECTION 2: Kinetic Friction of Table

If in the first part, we found static friction, it only makes sense that we now find kinetic friction. The process is similar. We hook up Logger Pro to a force sensor, which detects the weight that tugs on its little hook. We had a little trouble calibrating it, so that it zeroes the default atmospheric weight in both vertical and horizontal orientation. We use the same blocks used in Section 1, this time tied to the force sensor instead of the cup. As we pull on the block, LoggerPro should record the input from the force sensor, and we get a neat graph with force over time. The goal here is to find a relationship between friction force and mass, so we try to introduce little variance by pulling as steadily as possible. It is impossible to be perfectly steady, but that doesn't seem to affect our experiment much. Once again, we do this for 1-4 blocks.

We use statistical analysis in LoggerPro to calculate the mean force over a period of time. Since the blocks were pulled steadily, force should have been relatively constant. Then, we graph the force of the pull over the normal force of the blocks again. Once again, if:

 

F_tension = µ_k N
µ_k = F_tension / N

Thus, the coefficient of kinetic friction should be the slope of this graph:

Again, with a linear fit, we find that µ_k = 0.2395, as shown by the linear fit. Here, an close observation should reveal a fatal flaw in this experiment: The kinetic friction is larger than static friction, when it should really be a fraction of it. This result seems consistent across many groups, yet we don't know the exact cause.

 

SECTION 3: Static Friction of Track

The goal is slowly raise one end of the track, with a block on it, until gravity overcomes static friction, causing the block to slide. If we know the angle θ and the mass m of the block, then it should be possible to calculate friction. As you can see, the track is supported with a metal rod clamped to another one. The clamp can be loosened, allowing the support rod to be raised or lowered with ease.

Finally! By the way, this experiment suffers many of the same problems as the prior: uneven surfaces, block angle affects movement, air resistance might even play a part, vibrations of the block slipping could cause track to slip... We measured the height of our final track configuration to be approximately (22.6 ± 2) cm, and horizontal distance to be (98.0 ± 2) cm. Using the following formula, we could find the track angle, as well as a propagated uncertainty.

θ = tan^-1( y / x )
θ = tan^-1( 22.6 / 98.0 )
θ = 13.0º

dθ = | (∂θ/∂y) | dy + | (∂θ/∂x) | dx
dθ = | [1 / (1+y²)] * (1 / x) | 2 + | [1 / (1 + x²)] * y | 2
dθ = 2 / [x (1 + y²)] + 2y / (1 + x²)
dθ = 2 / [98 (1 + 22.6²)] + 2 (22.6) / (1 + 98²)
dθ = 0.00475º

We found our θ to be 13º ± 0.00475º, but it could even be give or take a few, since consecutive runs seem to give different results. As before, sometimes the block would "stick" until much higher angles, before sliding with gusto. This figure accounts for measurement error, but not all the other errors, which would make the problem too complex. The purpose of this lab, after all, is mainly to model frictional forces.

We also found the mass m of the block to be (0.160 ± 0.001) kg. That should be enough to solve for static friction, since the block has no acceleration before it overcomes static friction:

m g sinθ = µ_s N

m g sinθ = µ_s m g cosθ
µ_s = tanθ

µ_s = tan13º
µ_s = 0.231

dµ_s = | (∂µ_s/∂θ) | dθ
dµ_s = sec²(θ) dθ
dµ_s = sec²(13º) 0.0000829 rad
 dµ_s = 0.0000873

Coefficient of static friction is µ_s = 0.231 ± 8.73 * 10^-5, therefore, at least, the possible measurement error should play only a negligible role in the problem.

 

SECTION 4: Kinetic Friction of Track

We found that if we tilt the track high enough, gravity should over come static friction. So if we tilt the track higher, we should be able to find the kinetic coefficient if we know the acceleration of the block. Well, that's what the motion sensor's for!

We hook up LoggerPro with a motion sensor, which sends an infra-red signal to an object directly in front of it, and counts the length of time it takes to bounce back. As such, the motion sensor would not work if the object is too close, if the signal reflects before the sensor is ready to receive it, or too far, since the signal will be too weak. Luckily, if we place the motion sensor at one end of the track, then it works fine if the object runs from the middle.

So first the track is tilted up to an arbitrary angle higher than in Section 3. Then we use the cell phone app Clinometer, which allows us to tilt the phone on one side to measure the angle of the track. Since we have already, in the previous section, and in the previous lab, figured out the side of the phone we need to use to get an accurate measurement, we can just trust what the app outputs. A few tests on different parts of the track did give slightly different results, so we say that the angle is θ = 20.5º ± 0.5º. The mass m is, once again, (0.160 ± 0.001) kg, as the block hasn't changed.

We put the block somewhere half-way down the track, and capture the movement with the LoggerPro via the motion sensor. Here's the resulting graph of velocity vs time, and acceleration vs time:

Only a portion of this graph is relevant. We know that a falling object, subjected to theoretically constant friction and constant gravity, should accelerate linearly. The first part of the graph is either due to the object being let go, or the motion sensor software not working properly as it is turned on. The last half of the graph is due to the block hitting the bottom of the track, thus velocity reduces to 0 m/s. What we are interested in is the middle section, where velocity is linear and acceleration is constant, relatively speaking. Theoretically, if we find the slope of velocity during that section of time, and the mean average of acceleration, it should be the same. However, the real world isn't so clean most of the time: Our velocity slope turned out to be 0.8574 m/s², and our mean acceleration turned out to be 0.8527 m/s². So we split the difference and call it (0.855 ± 0.0028) m/s². Is it right to be so practical? The alternative is to do multiple runs and take a standard deviation, and that would have been beyond the time frame of this exercise.

Now we have all the variables we need. Since the acceleration must be the same as the difference between the forces of gravity opposite kinetic friction, the equation should be:

F_kinetic = m g sinθ - µ_k m g cosθ = m a

Once again, canceling out the mass, we get:

g sinθ - µ_k g cosθ = a
µ_k g cosθ = g sinθ - a
µ_k = (g sinθ - a) / g cosθ
µ_k = tanθ - [a / (g cosθ)]

Let's get this over with!

µ_k = tan20.5º - [0.855 m/s² / (9.8 m/s² cos20.5º)]

µ_k =  0.281

dµ_k = | (∂µ_k/∂a) | da + | (∂µ_k/∂θ) | dθ

dµ_k = | 1 / (g cosθ) | da + | sec²θ - (a / g) secθ tanθ | dθ

dµ_k = 0.0028 m/s² / (9.8 m/s² cos20.5º) + 0.00873 sec(20.5º)[1 - (0.855 m/s² / 9.8 m/s²) tan20.5º]
 dµ_k = 0.000305 + 0.00964
dµ_k = 0.01

Coefficient of kinetic friction is µ_k = 0.281 ± 0.01. Looks like the difference in acceleration had a minor effect compared to our careless measurement of the angle. Note, still, that our coefficient of kinetic friction is larger than our coefficient of static friction--it should be smaller! This could be for all the reasons above, but ultimately, we're not sure why this is.

 

SECTION 5: Kinetic Friction over Inclined Slope

This last section is similar to Section 4, except we're going to modify the experiment a bit. Instead of allowing the block to slide down the track, we're going to combine this with the first section and tie a weight to it over a pulley, so that the block slides upwards. To do this, we must move the motion sensor to the bottom of the incline, since the block will be moving to the top.

If the experiment is successful, then the coefficient of kinetic friction should be the same, or similar, to the previous part. However, given the large errors previously discovered, this may not be the case. As with before, our block is the same, so mass m is (0.160 ± 0.001) kg. The angle θ was measured again, and we found that the track has slipped a bit to 21.5º ± 0.5º. Oops! On the other end of the string, we used weights of known mass, which we'll designate as m₂, of 0.15 kg. It took a couple tries to get the timing down, since the block was getting pulled too fast, but pretty soon we got our graph, once again, of velocity vs time, and acceleration vs time:

As before, our velocity slope (0.5312 m/s²) is a bit different than our acceleration mean (0.5450 m/s²). So, once again, we split the difference and get a = (0.538 + 0.0070) m/s². The difference, this time, is that, with multiple masses, the situation is getting a bit hard to picture, so we'll be using free body diagrams.

This is a model of the experiment (made in Photoshop) showing all the forces. The pulley changes direction of the forces, so, to simplify our free body diagram, we could imagine the string straightening out, with the weights (wg in the model) pointing outward parallel to the slope. In real life, gravity points downwards, but insofar that the weights w interact with the system, such that it pulls the string in the direction before the pulley, we are justified in doing this. Next, we notice that there are 3 forces at an angle, and it would be much easier to set up the equation if we have less angular forces to break up into x and y components. Thus, we tilt the model, using different axes, such that f_k, T, and wg are all horizontal, and N is vertical, and mg is the only force at an angle. Once again, although gravity really points downwards, what matters is that the forces are preserved relative to each other, and not their absolute orientation. If we do this, our free body diagram looks like this:

We want to find µ_k, so we are only interested in the horizontal component:

F = (m + w) a = wg - µ_k mg cosθ - mg sinθ
µ_k = [(wg - mg sinθ - (m + w) a] / [mg cosθ]
µ_k = [(0.15 kg - 0.16 kg sin(21.5º))(9.8 m/s²) - (0.31 kg) 0.538 m/s²] / [0.16 kg (9.8 m/s²) cos(21.5º)]
µ_k = [0.8953 N - 0.1668 N] / 1.4589 N
µ_k = 0.7285 N / 1.4589 N
µ_k = 0.5

The result here is much different than the result in the previous part, despite measuring the kinetic coefficient of the same surface. We can speculate that the pulley might have its own friction. By eye, the block in this part seems to move faster than the previous part, but that is admittedly unreliable.

dµ_k = | (∂µ_k/∂a) | da + | (∂µ_k/∂θ) | dθ + | (∂µ_k/∂m) | dm
dµ_k = | (m + w) / (mg cosθ) | da + | [(wg - ma - wa) / mg] secθ tanθ + sec²θ | dθ + | (w / m²) secθ - wa / m²g cosθ | dm
dµ_k = 0.2125 s²/m (0.0070 m/s²) + 1.507 (0.05) + 5.952 (1/kg) (0.001 kg)
 dµ_k = 0.0014874 + 0.07535 + 0.005952
dµ_k = 0.0828

So it turns out that the track slipping in between experiments made quite an effect! Despite our unreasonable results, we have found that it is possible to model friction, and have identified some areas that deserve closer attention if we want more accurate results.