Lab 17: Moment of Inertia

Moment of inertia is what connects torque to angular acceleration. In this lab, we will calculate the inertia for a wheel, and then use it to determine how long a cart hanging off the wheel by a string would take to traverse a meter-long ramp.

The wheel shown above is steel cast in one piece with the cylinder axle protruding on both sides. While it would be difficult to calculate the inertia of a complex object in one go, we could imagine the object as three pieces: a solid disk, and two cylinders on the sides. Luckily, the inertia for both disk and cylinder about a central axis is:

I = 1/2 mr²

That means we need to calculate the mass and radius of each part. The total mass of the wheel is etched onto it, probably according to manufacture specifications, so M_total = 4.829 kg. The width is needed to figure out how much material there is of the disk relative to the cylinders in order to calculate mass. Unfortunately, we could only measure the diameter of the cylinder and width of the disk using calipers, since the calipers are too small, so the rest has to be measured with a ruler, which is less accurate. This might cause some error, but it's likely not significant. The radius of the disk is measured by figuring out the circumference of it with a string, and making the necessary manipulations. The measurements are as follows:

Disk	Actual Measurements	Measurement in meters
Radius	9.93 cm	0.0993
Width	15.6 mm	0.0156
Cylinder	Actual Measurements	Measurement in meters
Radius	15.3 mm	0.0153
Width	11.66 cm	0.0505

Finally, the mass of the disk portion itself is calculated as such:

And each cylinder:

And here's the calculations for the moment of inertia:

Now, our goal is to determine the time it takes for the cart to go a meter, but what we'll do is calculate it analytically, then experimentally, and compare the results. Experimentally, we use a stop watch, which should be good for a 4% error. Analysis requires more work. Ideally, a disk without any friction could be able to spin indefinitely. In reality, this is impossible, and so the disk eventually stops. Thus, before we add the cart, we should find the frictional torque (τ_friction) of the disk, because we know that the cart will not approach free fall. As we noted in the previous lab, torque is a function of inertia (which we have) and angular acceleration (which we don't have). We conceive that if we could figure out the distance a point on the disk moves over time, it would be possible to find the tangential acceleration, which could then be converted into angular. To do this, we use video capture in Logger Pro on a spinning disk. Then we go through each frame and note the position of the same point on the disk.

This gives us a velocity graph. A linear best-fit line will get us our tangential acceleration, and divided by the radius, we get the venerated angular:

a_tan = -2.106 m/s²
α_rad = -2.106 m/s² / 0.0993 m
α_rad = -2.121 rad/s²

With all the peripheral pieces, we calculate the crux of the puzzle:

τ_friction = I α
τ_friction = 0.02071 kg m² * (-2.121 rad/s²)
τ_friction = -0.04393 Nm

Now, we are ready to attach the cart. When we attach a cart, such that it causes some tension force, we envision that the force should affect the acceleration of the disk, so we need to take into account the cart to calculate a new acceleration, which could then be plugged into kinematic equations to solve for traversal time.

Note that since the cart is wrapped around the smaller cylinders for simplicity sake, we use the radius of the cylinders in our calculations. In order to minimize the effect of the frictional torque (because of possible disk defects that were brought to our attention, we raised the track angle to 48.5° and taped additional masses to our carts, such that the total cart mass becomes 0.904 kg. Plugging our numbers in, we get:

And plugging into kinematics:

However, our actual timed trials were closer to 8 seconds, meaning something went wrong. We speculate that something was caught on the axle, which caused the wheel to spin unevenly. This can be seen in our velocity graph: There seems to be periodic spikes. Due to time and equipment limitations, unfortunately, we can't tell for sure what the problem was. On the bright side, through the multiple trials to see what went wrong, we did get to observe some correlations between the cause and effects of our, ultimately unfruitful, rectifications.

Lab 16: Angular Acceleration

The term angular acceleration inspires in my imagination a drifter tackling sharp corners while carrying tofu, but today we're going to be doing something even more exciting. We're going to be intercepting Morse code from the Communists.

 

Here's an apparatus manufactured some time in the 1970s. The large steel cylinder contains a toroid that will pick up electromagnetic waves...

 

Okay, back to reality. Actually, the cylinder is a pair of polished disks with a tiny hole that will accept air from the machine, reasonably eliminating friction and allowing the disks to turn with minimal force. We will use a light mass with a string tied around a pulley connected to the disks to provide torque, then measure the angular acceleration experimentally.

Note: That's not a pizza box.

This process will be repeated in future labs to derive inertia. For this lab, we will determine the factors which affect angular acceleration. The relevant formula is τ = Iα, but what is that in terms of actual objects? First, we need to accurately measure the diameter and mass of the top steel disk, bottom steel disk, an alternative aluminum disk, a large and small torque pulley, and the hanging mass. We will swap these items out to see their effects. We use a caliper for the diameters and, obviously, a scale for the masses.

A few minutes later and carpal tunnel, we've recorded all our data:

Item	Diameter (mm)	Mass (g)
Top steel disk	126.2	1361
Bottom steel disk		1348
Top aluminum disk		464.5
Small torque pulley	24.9	10
Large torque pulley	49.8	36.3
Hanging mass	N/A	25

 

Since we need air to be pumped through the disks, we need to plug power into the machine, made by Pasco, according to the lab worksheet--their motto (on the website) is amusingly "Science is served!" This is all connected back to Logger Pro. Everything is connected to Logger Pro. We set the sensor settings to correspond to the disks, so now it will measure the θ that the disk moves. For some reason, the acceleration graph isn't reliable, so we use measure the upward and downward slope of the velocity graphs, corresponding to the descent and ascent of the mass on the string, and average the downward and upward accelerations. We repeat this 6 times, once for each condition. Experiments 1-3 measure the effect of different mass. Experiments 1 and 4 measure the effect changing torque pulley diameter or weight. Experiments 4-6 measure the effect of changing the weight of the disks. Here are our graphs:

Ex 1: Hanging mass, small torque pulley, steel disk

Ex 2: 2X Hanging mass, small torque pulley, steel disk

Ex 3: 3X Hanging mass, small torque pulley, steel disk

Ex 4: Hanging mass, large torque pulley, steel disk

Ex 5: Hanging mass, large torque pulley, aluminum disk

Ex 6: Hanging mass, large torque pulley, steel + aluminum disk

We are able to use the bottom disk by clamping the air hose on the side of the machine:

Using a pair of linear fits, we record downward and upward accelerations, and calculate average acceleration for each. Here's the resulting chart data:

Expt #	Hanging mass (g)	Torque pulley	Disk	αdown (rad/s2)	αup (rad/s2)	αaverage (rad/s2)
1	25	small	top steel	1.0830	1.1920	1.1375
2	50	small	top steel	2.0830	2.4990	2.2910
3	75	small	top steel	2.5970	4.3280	3.4625
4	25	large	top steel	2.1100	2.3160	2.2130
5	25	large	top aluminum	5.9550	6.4720	6.2135
6	25	large	top steel + bottom steel	0.7944	1.5610	1.1777

Ignoring minor variation, we observe a linear relationship between the hanging mass weight and the average acceleration. We also notice a linear relationship between the diameter of the torque pulley, but not its weight--the larger pulley is roughly 3.5 times heavier than the smaller. Finally, there is an inverse relationship between the weight of the disk and its acceleration. The top steel disk is about 3 times heavier than the aluminum disk, and acceleration with the steel is about 3 times slower.

This seems consistent with what we know about angular acceleration. If:

τ = rT = Iα

We see that, if all else is equal, increasing the radius and tension (represented by the hanging mass) of the torque pulley should proportionally increase acceleration. Since inertia is mr² of the disks, increasing the mass of the disk should have an inversely proportional effect on the acceleration.

Lab 15: Two-Dimensional Collision

In reality, hardly anything occurs in one dimension. Luckily, the calculation for two-dimensional collisions are similar--it only requires that we consider the axises separately. That sounds like twice the work, and it would have been, if it weren't for the wonderfulness of technology. In this lab, we demonstrate the conservation of momentum, and calculate how much kinetic energy is lost, in a two-dimensional collision. And we do it twice to note the different behaviors depending on the conditions:

1. Lighter aluminum ball hits heavier steel ball
2. Heavy steel ball hits another steel ball of equal mass

We do this on a smooth glass surface, limiting the effects of friction and unpredictability of uneven surfaces. This surface is specially constructed for this, after all. Overhead is a video camera, the same one we used to capture coffee filters last month! We hit the ball in a glancing angle so that they go off into different directions, which makes calculating two-dimensional momentum more interesting. Unfortunately, since my pool skills are not up to par, I let my lab partner handle that, while I masterfully click the "Capture" button in Logger Pro.

There are only two of these set up around the room, so we borrowed this awesome looking USB flash drive to copy the recordings onto our own lab computers:

It's decapitated, ahh!! Before, we analyze the data in Logger Pro, first thing's first: We need the mass of our balls--no innuendo here. And no table scales either, since we need more accuracy. The balls keep rolling off the scale, so we use another weight to hold the ball in place.

The brass weight labeled '50 g' only weighs 48.5 g. My trust for lab equipment would have gone right out the window, but our room has no windows. Bad joke. Putting the steel and aluminum ball on the center hole of the weight, we were able to measure 115.5 g and 72 g, respectively. Subtracting the extraneous weight, the resulting weights are:

• Aluminum ball: 22.5 g
• Steel ball: 67 g

Armed with this miraculous data in hand, we are now ready to traverse the unknown... number of frames, one by one, carefully noting the position of the ball, from the initial flick to some time after the collision. By doing this, Logger Pro generates a graph with the x and y positions and velocity (by noting the distances between positions per frame). We can set an initial position, so Logger Pro knows how to offset the graph. We can also set a frame of reference, so Logger Pro could anchor the position to a known, real unit of measurement.

Our square glass plate measures 58 cm at one end, so we enter this into Logger Pro. And we end up with these two avant-garde, contemporary pieces of work which transcend notions of a single instance of time.

Using the position of each plot, Logger Pro generates two separate graphs, each with the x and y positions of each ball. In order to later solve the kinetic energy lost, we analyze, in Logger Pro, the slope of each ball before and after the collision. We'll leave that for now.

We could take a protractor to the angles of the trajectories in our capture above, or we could do some trigonometric calculations on the graphs, in order to find the x and y total momentum and total kinetic energy for each condition. Or, we could leverage technology to do the work for us. Choosing smart work over hard work, we create new calculated columns in Logger Pro. The total momentum per axis is just all the momentum summed, so we have:

p_x = m₁v_1x + m₁v_1x + m₂v_2x
p_y = m₁v_1y + m₁v_1y + m₂v_2y

And since kinetic energy is scalar, we use the Pythagorean Theorem for both balls, such that:

KE_TOTAL = 0.5 m₁(v_1x² + v_1y²) + 0.5 m2(v_2x² + v_2y²)

We plot these in a new graph against time, and get:

The initial slope of the results arise from Logger Pro interpolating and filling in gaps of data. Otherwise, it appears as if x and y momentum are relatively conserved. Other sources of error could be the imprecise nature of noting position from the captures; since the balls take up more than a single point, identifying the exact same point on the ball between the frames is difficult. The little bumps in the graph seem to reflect the little bumps in the capture trajectory. Kinetic energy is stable after the collision. Since we don't have enough data before the collision here, we turn back to the previous graphs with slopes to calculate the loss in kinetic energy. First, note that:

KE_loss = (KE_i - KE_f) / KE_i

Secondly, the slope of a position graph is the velocity. Dumping all the velocities into a spreadsheet table, we get:

Velocity	Aluminum Hits Steel	Steel Hits Steel
Cue Ball Initial X	0.482600	0.545900
Cue Ball Initial Y	0.000243	-0.075130
Cue Ball Initial	0.482600	0.551046
Cue Ball Final X	0.046430	0.159600
Cue Ball Final Y	-0.167400	-0.177400
Cue Ball Final	0.173720	0.238627
Object Ball Final X	0.109900	0.279600
Object Ball Final Y	0.045370	0.086380
Object Ball Final	0.118897	0.292639

 

And then, calculating for the kinetic energy:

Kinetic Energy	Aluminum Hits Steel	Steel Hits Steel
Total Initial	0.002620	0.010172
Cue Ball Final	0.000340	0.001908
Object Ball Final	0.000474	0.002869
Total Final	0.000813	0.004776
ΔKE	0.001807	0.005396
KELOSS	69.0%	53.0%

That's more energy loss than I expected. Nevertheless, we conclude that, in two-dimensional collisions, and perhaps regardless of any dimension, momentum is conserved, but not kinetic energy. Admittedly, it would be hard to conceive what mass and velocity would even mean in the context of a 9-dimensional collision. Maybe we shouldn't get too ahead of ourselves...

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