Lab 14: Impulse-Momentum Theorem

In a collision, kinetic energy is often transferred into heat and sound energy, such that it is hard to measure and not conserved. In such cases, we use conservation of momentum to model the reaction of the collision. Momentum is defined as:

p = mass * velocity

We also know that momentum is caused by force over the time interval which the force acts. We call this impulse, which is defined as:

J = Force * time

Therefore, since the momentum of the system cannot change without forces, we know that, if friction and gravity could be disregarded, since the magnitude of force before and after a collision must be 0, there is no change in momentum before and after the collision. And unlike kinetic energy which could be transferred into other energy forms, momentum, given enough time, only transfers momentum to other objects. Thus, we say that momentum of a system is conserved, regardless of elastic or inelastic collisions (which just vary in force-time). This is shown by the Impulse-Momentum equation (the change of impulse is equal to the change of momentum):

F * Δt = m * Δv

...wherein the change in momentum mΔv could also be defined as:

m * (v₂ - v₁)

If we know the force and velocity of the moving object, then we could test whether this is true. But how is this possible?!

Once again, we produce our trusty gadgets: the motion sensor and the force sensor! As pictured above, we have two wheeled carts on a level track, making the effects of friction and gravity (hopefully) negligible. We put pieces of paper under the track until it is precisely leveled, showing less than 0.4° on the phone app, such that the cart does not roll on the track by itself. Then we mount a force sensor atop it with a rubber attachment to simulate an elastic collision. The rubber would bounce off the translucent plastic on the second cart, causing force over the collision time. We measure the cart with the force sensor to be m = 433 g. The motion sensor, attached to the other end of the track, measures the velocity and position of the track.

So... we hook the sensors up to Logger Pro, and give the red cart a push, being careful the cables don't contribute tension or get in the way of the sensors. Logger Pro gives us the standard force vs time, and velocity vs time graphs:

What we're interested in here is the change of impulse, which can be obtained by using the integral function to find the curve under the force curve during the collision. On the other hand, we find the change of momentum by finding the highest and lowest velocities right before and after the collision, and multiplying that by the mass. In our first experiment, we find that our impulse is 0.5358 Ns. Our Δv, as reported by Logger Pro analysis, is 1.282 m/s. Multiplied by our cart's mass of 0.433 kg, we get the figure calculated in purple in our data table. Although it reads impulse, it is actually momentum (but also impulse!), coming in at 0.555 kg m/s. This is under 4% error, a relative success by our standards--it shows a clear correlation.

The instructions require us to run a second experiment with additional mass, and a third inelastic collision without the mass, but having messed up the inelastic collision once, we ended up doing it with the mass on. It should be inconsequential. The higher mass supposedly demonstrates that larger momentum change is conserved, while the inelastic collision increases collision time. But by the time we got to the second experiment, we've realized that the force sensor has less chance of peaking if we pushed the cart slower, therefore our second and third experiments actually showed less velocity, and therefore less momentum change.

Without further ado, we've tied a 200 g mass to the back of the cart for our next two experiments, bringing the total mass of our cart up to 633 g. Our second push yielded:

This one was pushed much slower to ensure that the force sensor captures without problem, but also to ensure that the mass stays in place. As such, the Δv is only 0.479 m/s; multiplied by 0.633 kg, the resultant change in momentum is approximately 0.303 kg m/s. Using integral again, our impulse is 0.2994 Ns. This time there's less than 2% error, not bad!

The third trial, once again, is with the additional mass, and also with a nail attachment to the force sensor, and putty on the other end, in order to create an inelastic collision. The capture looks like this:

The cart was pushed faster, but ended up stuck to the putty at 0 velocity, so the change in velocity turned out about the same as the second trial, at Δv = 0.510 m/s. Once again, using 0.633 kg mass, we calculate 0.323 kg m/s momentum change. The impulse turned out to be 0.3210 Ns. Under 1% error.

The point here is that, regardless of the mass or material of the cart, momentum and impulse are conserved, thus proving that the Impulse-Momentum Theorem is true. One unexpected result from this experiment is that the collision time are all under 1/5 of a second, and neither mass, nor the putty, seemed to greatly affect it.

Lab 13: Magnetic Potential Energy

Thus far, we have stuck to energy forms with neat, predefined formulas to calculate using the work-energy theorem. In fact, all kinds of energy can be used in that way. This time, we demonstrate that this is true with magnetic potential energy, whose formula we derive. We do this by experimentally recording magnetic force under different circumstances. Magnetic force which repels two objects will have magnetic potential energy, since the objects, which have the inclination to move towards each other, is being forced apart by a distance. As before, we graph this relation between force and distance and integrate to find the work caused by the energy.

To do this successful, we must first reasonably eliminate friction. Pictured above is an air track with little holes connected to a blower, which pushes air through those holes. The red cart on the air track will approach 0 friction, such that it could be slid across the track unimpeded until it hits the end. At the ends of the air track and the cart are magnets which repel each other, causing the cart to bounce elastically from the end even if the magnitude of velocity overcomes the repelling force. However, first, we'd like to discover the relation between this magnetic force and distance, so to alter the role of gravity, we lift the air track to different angles and record how far the cart is held from the ends. The force of the magnet is, of course, mg sinθ. We label distance x, in meters.

The air track is lifted by placing a number of books under the other end. This rudimentary method to change the angle is counteracted by the advanced cellphone angle measuring apps that put Inspector Gadget out of business. The measured angles should be accurate relative to each other, but we expect an offset due to the geometry of the phone, and that the table isn't exactly flat. There is a ruler taped to the end of the air track, wherein the position of the magnet measured at 527 mm. This aided in measuring the distance, for all we had to do was measure the position of the magnet on the cart in the same way, and find the difference. We recorded 8 data points, although the first or last may not be as accurate, as we later found. The data is as follows:
 

Position (mm)	x (m)	θ (º)	sinθ
476	0.051	1.1	0.0192
487	0.040	2.7	0.0471
492	0.035	4.9	0.0854
497	0.030	6.9	0.1201
500	0.027	8.8	0.1530
502	0.025	10.5	0.1822
503	0.024	12.1	0.2096
506	0.021	14.0	0.2419

 Now that we've gathered the data, we can plot the graph to find the relationship:

We set a curved fit that we expect the data to conform to. The data doesn't exactly fit, but it is reasonably close considering the experiment. Here, we've found our magnetic force equation to be F = 3.168 x 10^-5 r^-2.690. We integrate and flip the signs to get the potential energy:

We can now test our magnetic potential energy against kinetic energy to see whether energy is conserved like we expect it to. If our calculations are close, then we should see the pattern of conservation. To do so, we attach a motion sensor to the end of the air track and measure its offsets from the magnet. We need to know the differences to accurately reflect distance. We also make sure the air track is horizontally flat, such that the cart doesn't move due to gravity.

We measure that the back plate or our cart, which the infra-red beam from the motion sensor would hit, is at position 433 mm. Thus, if we subtract it from the end position at 527 mm, so have 94 mm. This figure will be entered into Logger Pro to calibrate the motion sensor.

The thinking is this: If we give the cart a push, then record its movement with the motion sensor, we should get its position (distance) and velocity, which should be enough to calculate the potential energy and kinetic energy, respectively. Entering new calculated columns:

And summing up total energy, we get:

The cart hitting the end of the air track causes the dip in kinetic energy, and at the same time increases magnetic potential energy since the magnets approach each other. We can see that the energies are inverse of each other, causing the sum to be relatively even compared to the total energy in the middle of the track. The larger spikes in energy can be explained by gaps in the motion capture, causing Logger Pro to interpolate and guess what's there. The smaller spikes are likely explained by the previous inaccurate modeling of the force, possibly due to loose measurements of the angles. Also, despite the air track, friction isn't completely eliminated, nor air resistance.

Here is a snapshot of our data, exported to a CSV file, with the gaps taken out (which have 0.001 J kinetic energy). As you can see, the total energy is even.

Time	Position	Velocity	Acceleration	Kinetic Energy	Potential Energy	Total Energy
1.65	0.625	-0.268	0.109	0.013	0	0.013
1.7	0.608	-0.252	0.216	0.011	0	0.011
1.8	0.587	-0.248	-0.111	0.011	0	0.011
1.85	0.574	-0.254	-0.067	0.011	0	0.012
1.9	0.561	-0.255	0.008	0.011	0	0.012
1.95	0.548	-0.252	0.058	0.011	0	0.011
2	0.536	-0.247	0.058	0.011	0	0.011
2.1	0.513	-0.251	-0.096	0.011	0	0.011
2.15	0.498	-0.259	-0.011	0.012	0	0.012
2.2	0.486	-0.251	0.059	0.011	0	0.012
2.25	0.473	-0.248	0.046	0.011	0.001	0.011
2.3	0.461	-0.247	0.029	0.011	0.001	0.011
2.35	0.449	-0.246	0.027	0.011	0.001	0.011
2.4	0.437	-0.245	0.041	0.01	0.001	0.011
2.45	0.424	-0.243	0.071	0.01	0.001	0.012
2.5	0.412	-0.238	0.117	0.01	0.001	0.011
2.55	0.4	-0.232	0.187	0.009	0.002	0.011
2.6	0.389	-0.221	0.317	0.009	0.003	0.011
2.65	0.378	-0.205	0.573	0.007	0.004	0.012
2.7	0.368	-0.172	0.999	0.005	0.007	0.012
2.75	0.359	-0.108	1.461	0.002	0.011	0.014
2.85	0.358	0.08	1.573	0.001	0.013	0.014
2.9	0.365	0.154	1.15	0.004	0.008	0.012
2.95	0.374	0.196	0.677	0.007	0.005	0.012
3	0.385	0.215	0.332	0.008	0.003	0.011
3.05	0.396	0.221	0.168	0.009	0.002	0.011
3.1	0.407	0.227	0.112	0.009	0.002	0.011
3.15	0.419	0.232	0.076	0.009	0.001	0.011
3.2	0.431	0.235	0.043	0.009	0.001	0.011
3.25	0.443	0.236	0.018	0.009	0.001	0.011
3.3	0.454	0.236	0.007	0.009	0.001	0.011
3.35	0.466	0.237	-0.007	0.009	0.001	0.011
3.4	0.478	0.235	-0.01	0.008	0.001	0.011

Lab 12: Conservation of Energy

An oscillating mass-spring is a relatively closed system, so the total energy should be preserved even if kinetic and potential energies alternate. For this lab, we look at how to use calculus to derive a formula for all the energies in play, and then measure them in Logger Pro to see that total energy has, in fact, been preserved!

Although the math is long, we will discuss it a little bit to provide a context. The above diagram shows a spring of arbitrary length L, hanging on some apparatus at height h above ground. The mass of the spring weighs to be m = .100 kg. When we do physics problems, we tend to represent these object with a single point, but in real life the spring takes up quite a bit of space. Therefore, since gravitational potential energy depends on height, the potential energy of the spring actually varies from top to bottom. To solve this problem, we see that it is theoretically possible to slice this spring horizontally into a number of pieces, each with a portion of the mass of the entire spring, we call dm. Each dm slice is also the overall mass m divided by overall length L, times the height of the slice, or change in height dy. We can sum up all the slices to get an approximation of potential energy, with it being more accurate as the slices approach infinity, such that:

Note that this is the same as integrating, which we can do to solve for gravitation potential energy:

In class, we had derived the kinetic energy in this manner as well, but for the sake of brevity (and my precious time), I present to you the formulas. Ta-da!

Now, the set up is easy: We hang a lengthy weighted (weights are .600 kg) spring on a stand, over a motion sensor on the ground. The motion sensor is, of course, hooked up to our trusty Macbook running Logger Pro, so that we could record the position, and the corresponding velocity, of the spring, by pulling it and letting it go. The hardest part was aligning the motion detector, and making sure the spring stays over it, and making sure the weights don't fall and damage the motion detector, causing a lawsuit!

What was really helpful was taping a piece of paper under the weights so that it makes it easier for the motion sensor to pick it up. Here's out data:

All those wavy lines... The only thing missing now is the spring constant k, and the unstretched height of the spring. Although we installed a force sensor to our spring, we didn't actually use it. Finding k was simpler measuring the spring at rest without any mass (96 cm from ground), and the spring at rest with the mass (72 cm from ground), and finding the difference, such that:

kx = m_weightsg

k = m_weightsg / x

k = (0.6 kg * 9.8 m/s²) / 0.24 m
k = 24.5 N/m

We also do a run in Logger Pro with the spring at rest with mass, just to see how far the motion detector figures that height to be. Since we didn't zero or calibrate the motion sensor, there is expected to be an offset. This will help us find elastic potential energy, which requires that we find the stretch of the spring. With all the variables known, we start making calculated columns in Logger Pro with the formulas we've discovered, as shown above. We substitute y for the position data, and v for the velocity data. Once completed, we sum up the potential energies of the mass and spring, both elastic and gravity. We also sum up the kinetic energy of the mass and spring. Finally, we get the total between the two as well. If theory stands, this final total should be relatively stable, since all the variable energy should be conserved at any given time.

And we see, in light blue, that the total is conserved, that despite the peaks and valleys of the other forces, the sum is relatively constant. The slight error could be accounted for by inaccurate spring constant measurement, that the height isn't exactly calibrated with the motion sensor, and the spring itself isn't perfect (it could have flopped during the experiment). But in all honesty, this is as good as we should expect.

Lab 11: Work-Kinetic Energy Theorem

According to theory, the work done to move an object turns into kinetic energy, therefore, if we limit the amount of air resistance and friction, we should be able to find a direct identity relationship between work and kinetic energy.

To do this, we need a motion and force sensor, since kinetic energy is a function of velocity, and work is a function of force. We deduce the kinetic energy after capturing velocity and figuring out the mass of the system, applying the formula for it. We find work by noting the integral of force and, by plotting both against the distance, we could compare work and kinetic to see if they are indeed the same, such that:

W = F * d = (m v²) / 2

The force sensor is clamped to the far end of the picture above, connected to a spring and a weighted cart over a track. The motion sensor is on the near end. First, we measure and zero the sensors with the spring uncompressed. Then, we pull the cart, such as to extend the spring an arbitrary amount--this will be recorded as positive distance. Thus, the chronological order of events will appear in reverse when the graph is plotted against position. When released, the stretched spring should be at maximum force, reaching zero at uncompressed (neutral) position,while velocity increases. Work is converted to kinetic energy.

This is our data:

The data points of negative position are crossed out, telling Logger Pro to ignore what happens after the spring reaches uncompressed position, and begins to be compressed. We create a calculated column for kinetic energy. And we use Logger Pro to find the area under the force curve between certain limits, and analyze the value of kinetic energy at that point. To illustrate, here are a few points. Note the similarity between the values:

Please excuse the moire, the resolution of the phone doesn't capture computer screens well. More importantly, note that work and kinetic energy as reported by Logger Pro diverges as time goes by, revealing the faults in our experiment. First, our spring isn't ideal, and hangs to one side off the track, even at rest, which could distort the zeroing of our sensors. Second, there are other forces at work in reality, such as friction. Our measurement time is brief, exacerbating error. Nevertheless, we see clear signs of a correlation, so we consider that the relationship between work and kinetic energy confirmed.

Lab 10: Work

I'm all for practical learning, but physics in itself feels like an exercise in reinventing the wheel--over and over and over again. Everything we prove is something that has mathematically modeled hundreds of years ago. That's why I aspire to be an engineer instead of a physicist--to create something of practical use to people who don't live inside an underground Switzerland lab. And honestly, we're too busy trying to jump through a bunch of hoops within a predefined time limit during a lab to actually sit down and think about all the relationships and why they are--much easier to see the symbolic relations without distractions like how to enter a constant in a proprietary software called Logger Pro.

But this lab isn't just work, it's about work. More specifically, it's about defining work in a physical context, and how physics work coincides with the common understanding of work. We measure work in three ways: walking up stairs, running up stairs, and pulling a weighted backpack up a distance with slippery rope. Since the physics definition of work is just:

Work = Force * Distance

... if we know the carrying weight and the displacement, then there should be no problem in calculating it. The hardest part is actually doing the experiment.

First we measured the vertical height of each step to be 17 cm, with a total of 26 steps, for a total height of 442 cm. The first task, then, is to walk up the stairs in an arbitrary pace. As with all labs before, we use the small plastic timer, which does it's job. What it doesn't capture is the precise time or place in which the stairs start or finish, as this requires a manual button-push, introducing human error. For the run--self-explanatory!

	Time (s)
Walk	16.08
Run	4.96
Rope	27.82

Next, we use a rope and pulley system tied to a balcony at the top of the stairs (same height) to lift backpacks weighted 9 kg.

If the incline of the stairs is 30º, then it should only take half the force to traverse it compared to climbing vertically.

Note that these are not the stairs we climbed, this is for illustration purposes only!

F = mg sin30°

However, note that the distance is doubled. Therefore, since work is force times distance, there should be no change in work over different angles:

(1/2)F * 2h = F * h

The difference then, comes down to mass. Climbing stairs is equivalent to carrying the entire body weight, which is much heavier than pulling up a 9 kg backpack. The work required for each activity (assuming 160lbs body weight, or roughly 72.6 kg) is:

 

	Mass (kg)	Force (N)	Distance (m)	Work (J)
Walk	72.6	711.48	8.84	6289
Run	72.6	711.48	8.84	6289
Rope	9	88.20	4.42	390

Therefore, while lifting things target a specific muscle for working out, running would be preferred for burning calories. Speaking of calories, apparently a calorie is defined as "the amount of energy required to warm 1 g of air-free water from 14.5 °C to 15.5 °C at a constant pressure of 101.325 kPa (1 atm)". The definition requires the ambient temperature to be at 15 °C, so given that assumption (the day was much hotter!), the conversion is approximately 0.2389 calories per joule of energy. We could also figure power using the formula:

Power = Work / Time

The results are:

	Work (J)	Time (s)	Calories	Power (w)	kcal
Walk	6289	26.08	1502.46	241.14	1.50
Run	6289	4.96	1502.46	1267.94	1.50
Rope	390	27.82	93.17	14.02	0.09

The last column is kilo-calories, which is commonly referred to as a "calorie" in nutrition, so one would need to burn 1000 actual calories in order to burn what is commonly referred to as a calorie. Considering that climbing these stairs only burnt 1.5 kcal, a person would need to climb the stairs roughly 634 more times in order to burn off the 952-calorie Kentucky Club Salad from the WOW Cafe. Wow. Looks like I need to stop ordering 1180-calorie Texas Toast Burgers.

Lab 9: Hurricane Speed and Angle

The Hurricane is a common amusement park ride created by the Allan Herschell Company in the 1940s. It consists of a central support pillar with levers extending out to rides, which will spin and, using the behaviors of centripetal acceleration, cause the rides to left up. The angle that the ride lifts, as it turns out, depends on the speed by which it turns. The real version also uses pneumatic cylinders to oscillate the height, but we are modelling this without the hydraulics.

Since we don't have enough rulers to measure the Hurricane itself, we use our own rotating apparatus: a tripod tied to a motor and a stick, and a string tied to a mass on the other end. It sounds dangerous, but it's quite impressive in person.

The motor can be controlled to spin at a certain volt, which implies a constant speed if unimpeded. As it spins, the mass, a rubber stopper connected by a string at one end of the horizontal bar, rises at an angle. The first step to setup any experiment is to measure all the relevant angles and hypothesize on the relationship, so we could have an idea on what to expect.

Now we draw a model with the relevant dimensions:

Using the model, we draw a free body diagram:

And we're set to calculate for the relationship between angular velocity ω and the angle θ. Because there is centripetal acceleration on the horizontal access, we say that (where r is radius):

F_x = m a = m r ω²

Resolving the x and y components of the forces, we have:

x: T sinθ = m r ω²

y: T cosθ = m g

Dividing them, we get:

tanθ = r ω² / g

A word about the radius r: Since the width of the top horizontal bar is the radius of the system with the mass at rest, and the mass only swings outward, we could say that the radius is the radius at rest r₀ plus the length that forms a triangle between the string at rest and the string that is swung out at a certain angle θ. Thus...

r = r₀ + L sinθ
tanθ = ω² (r₀ + L sinθ) / g
g tanθ = ω² (r₀ + L sinθ)
g tanθ / (r₀ + L sinθ) = ω²
ω = sqrt[g tanθ / (r₀ + L sinθ)]

With the theoretical relationship armed in hand, we are ready to start experimenting to see if it confirms our analysis. To find the speed of the mass, we use a basic timer to time 10 revolutions, to improve accuracy.

As the motor spins, we expect that the string will form an angle outward, but this also raises the height of the mass. If we know the difference in height of the mass compared to the position at rest, then it is possible to solve the horizontal component. To measure the height, we place a piece of folded paper clamped to a metal bar under the apparatus.

We slowly raise the height until the mass just hits it as it swings around.

Studiously measuring the height:

And we do this 8 times...

Time (s)	Rev	Height (cm)	Velocity (ω)	Angle (θ)
39.68	10	50.0	1.5835	9.0125
32.88	10	62.8	1.9109	24.6816
29.53	10	76.2	2.1277	34.3175
26.98	10	95.5	2.3288	45.0257
25.14	10	106.6	2.4993	50.3369
22.05	10	114.1	2.8495	53.7027
19.40	10	142.5	3.2388	65.3757
18.50	10	148.7	3.3963	67.7657

 

We get the velocity ω by dividing 2π radians per revolution by T, the period, or seconds per revolution. The angle θ is found by subtracting the measured height from the overall height h of the apparatus, then dividing it by the length L of the string, and taking an inverse cosine of the result.

A similar set of data derived in class, from the experimental angle θ, then using the analytic relationship calculated earlier to reach the velocity ω:

We graph the analytically derived velocity ω_theory against the experimental velocity ω_expt to see if there's a correlation:

The regression line coefficient is 0.96483. The slope is 1.0135, meaning that we are under 2% error. That isn't too bad considering all the measurements we had to take, each of which could introduce error. What's interesting is that we find that velocity depends wholly on radius, angle, and gravity, and not on mass. But perhaps that isn't too surprising given that the pendulum period equation also excludes mass, presumably for the same reason.

Lab 8: Centripetal Acceleration

The commonly believed centrifugal force has long been rendered fictitious. Although it is true that an object spun in circular motion tends to fly outwards, it must be made clear that this is not a force externally acted upon an object, but comes from the object's moment of inertia. Instead, an object confined into circular motion must be acted upon by an external force pulling it towards the center. As such, centripetal force is not some independent physical phenomenon, but the force named for causing circular motion; this force could be tension (when the object is spun by a string), or normal (for example, in banked curves). It might be more clear, then, to call it centripetal acceleration (and also centrifugal acceleration that an object causes about a system) caused by other forces. Since centripetal does not refer to an independent force, we conclude that there must be some relationship such that we could find centripetal acceleration without introducing any additional variables.

If we consider an object swung uniformly about with a string, its tendency to move in a straight line (in one direction) due to inertia is perpendicular to the string. In order to determine this relationship, we sample some initial point in the flight path, and call the velocity direction and magnitude v_i. Then, we sample another point and call this the final velocity v_f, such that the angle formed in between them is θ:

Note that both v_i and v_f have the same magnitude, but different directions. And since they are both exactly perpendicular to the points that form the angle θ, the angle between them must also be θ. We can place them together to calculate a vector difference, or Δv.

The change of this velocity over the change of time is the definition of acceleration:

 a = Δv / Δt

To find Δv, we consider that the vector diagram forms an isosceles triangle because both v_i and v_f have the same magnitude. Therefore, we can divide θ down the middle, forming 2 identical right angles, with v_i and v_f being the hypotenuses. Then, we can say that half of Δv is v*sin(θ/2), as illustrated by the following:

Therefore:

Δv = 2v sin(θ / 2)

And since time is the arc length divided by speed, we have (where r is radius):

Δt = r θ / v

We now have all we need to find the relationship. Acceleration restated becomes:

a = [2v sin(θ / 2)] / [r θ / v]

a = (v² / r) [sin(θ / 2) / (θ / 2)]

And since we're interested in instantaneous acceleration:

a = v² / r

This means that if we devise an experiment testing the acceleration and velocity squared of an object, we should note a linear relationship, and that relationship (slope) should be the radius. And we're going to do just that!

Pictured above is a heavy iron plate used for esoteric training by Shaolin monks. We attached an accelerometer to it, so that when we spin it, the accelerometer would pick up the acceleration and record it into Logger Pro. Also required are 5 pieces of masking tape and a messy desk. In order to get the velocity, we use rudimentary (compared to the accelerometer) handheld timers, timing 2-4 revolutions of the plate. If we divide the number of revolutions by the time, we should get the angular velocity (rev/s). On that note, we know that angular velocity ω is:

ω = v / r
a = r ω²

Here's the data from the timers, the calculations, and the data recorded by the accelerometer (wow, it's that much easier to copy your own spreadsheet directly into a blog instead of messing with copying and pasting from those Macbooks):

Time (s)	Rev	Velocity (ω)	Acceleration (m/s²)
4.4	2	0.455	1.40
2.5	2	0.800	5.00
3.7	2	0.541	2.20
1.4	2	1.429	12.90
1.6	4	2.500	37.41

 If we plot angular velocity against acceleration, this is what we get:

The linear fit has an pretty good correlation, meaning the experiment is pretty consistent. The slope is 0.1493, so we should expect the radius to be about 14.9 cm, if our calculations were correct.

Measuring the plate with a ruler, we find the radius to be about 15 cm. That confirms our results.