Lab 20: Conservation of Linear and Angular Momentum

There comes a point when the basics have all been established, and the only thing left to do is to combine concepts and find new ways to apply them. This lab is like that, marking the end of a chapter with one final inelastic collision to slow us down, as if saying, "Hold yer' horses", while pulling on the reigns. However, this petty obstacle will be soundly defeated like Excalibur upon a vegetable.

In the first part of this lab, we have a steel ball being released from a ramp, leaving it horizontally at some velocity. Sounds familiar, right? And then hits a ball-catcher clamped on top of a pair of disks with diminutive friction. Still familiar? What is the final angular velocity? Well, there are two parts to this problem, clearly delineated, both of which have been covered in previous labs. The first part is almost identical to a part of the lab we've completed on trajectories (Lab 5), and the second is very similar to the past few labs which have used this "Rotational Dynamics Apparatus ME9279-A".

Thankfully, Professor Wolf has offered to do the experimental part of the lab--an excellent opportunity to show off his decades of classroom wisdom. First, he magically produced this ramp from the depths of the catacombs, clearly a sacred treasure of the Mt. SAC Physics Department, unavailable for use to the common practitioners, for example, in Lab 5.

Before anyone could take a good picture, he lays paper and carbon paper on the floor over the projected trajectory, as if he had inherited Goku's instant transmission. Then, with a measured hand, he centers the ball over the ramp, making sure not to exert any excess energy, so that the yang of the ball perfectly coexists with the ying of the ramp, creating a state of absolute universal synergy, in the absolute stillness which could only truly be achieved by accomplished classroom experimenters, called t₀.

The ball hits the paper, producing a round mark, indicating the lack of spin caused by nervousness when releasing the ball.

Measuring the distance with a dowsing rod plumb bob.

With all the dimensions measured, we can now figure out the velocity of the ball as it exits the ramp. Recall that we cannot exactly figure out the velocity from using the work-energy theorem because we lack information on the frictional coefficient of the ramp. Such an arcane artifact undoubtedly lack manufacture specifications, which would probably require top level clearance at the physics equivalent of Hogwarts. Instead, we can use kinematics to solve for air time and work backwards to get initial velocity. The ball is sufficiently heavy as to minimize the effects of air resistance. Of course, for Professor Wolf, solving physics problems of this level is a rote exercise, not even fit to be called manual labor. As such, the measurements were produced automatically, without any hint of hesitance that would plague the average student trying to figure out what to do next. And here are the measurements, ±0 (okay, maybe ±1 mm or so):

While the next part of the experiment is being set up, we solve for v₀, the speed of the ball as it exits the ramp:

Okay, so we need to figure out the inertia of the disks, again for this lab, if we are to apply the conservation of angular momentum principle to solve for angular velocity. Since the air pushing through the disks nearly eliminates friction, we are able to solve for inertia using the tried and true method of using a hanging mass and Logger Pro to record angular acceleration.

By the time we had solved for v₀ though, Professor Wolf had long finished measuring the diameter of the torque pulley (50 mm, so the radius is 0.025 m), and the hanging mass (0.0247 kg), and obtained the descending and ascending accelerations from Logger Pro (α_down is 5.934 rad/s² and α_up is 5.339 rad/s², so α_average is 5.6365 rad/s²).

All that's left to do is to plug in the numbers and get the inertia of the disks.

 

Due to the aforementioned instant transmission technique, we find that the ramp has been teleported, and screwed, next to the rotational disks!

Now for the final act--the Prestige. If we release the same ball off that, presumably, same ramp, then it should enter the ball-catcher attached to the disk at the same velocity we previously calculated, and due to conservation of angular momentum, we should be able to predict the angular velocity of the spinning disks.

But there are a few variables missing! On behest of an anonymous student amidst the crowd, who was probably planted there by the Secret Order of the Physics Department, Professor Wolf measures the diameter (19 mm, so radius is 0.0095 m) and mass (0.0283 kg) of the ball, and the distance from the center of the disks to the center of the ball (0.07 m). One could tell from the ruler already attached to the ball-catcher that this measurement was preordained, yet the suggestion comes from the crowd! This is a conspiracy no less than JFK!

We have previously determined the inertia of the disks. Now we can determine the inertia of the ball. With its radius, we needn't consider it as a point mass, but a solid sphere rotated about a parallel axis:

So the total inertia: I_total = 0.00121331888 kg m²

We have all the pieces. So, according to my calculations...

Muhahaha. What are Professor Wolf's experimental results, using the rotational sensor?

1.775 rad/s! Mine are faster, I win. Actually...

Results (rad/s)		Error
Analytical	Experimental	Absolute (rad/s)	Relative (%)
1.931	1.775	0.156	8.8

Almost 9% error. Drat! Well, given the multiple parts in this lab, there are actually many different sources of error. For one, the ball does not exactly leave horizontally from the ramp into the ball catcher. Also, the ball could have rolled inward after landing. The lab tables, as noted in prior labs, are not necessarily flat (the table could have been slanted to one side). The measurements for the height and distance the ball landed on the floor from the table could be affected by incongruences in the table, or the possibility that the ramp did not point perpendicularly out from the table. Also, since the ramp was not actually screwed into the rotational device, it could have shifted while the ball rolled. Also...

Ehh, time to call it a day.

Lab 19: Angular Momentum

If a ruler held from a fixed pivot swings down and hits a mass of putty, how high can it continue to swing? Or alternatively, how high can this shoe swing after picking up that ball of gum?

Or how wide can my mood swing after being denied this sandwich because the school store had a line and there was only one worker there?

These are things that may or may not involve angular momentum. For this lab, we will try to analytically predict the swing of the ruler, and eat that sandwich while no one's looking. Then, we will compare our predictions with results from a video capture. In order to figure this out expediently, I swiftly weighed our ruler and putty. We need to find the length of the meter stick... before the pivot.

So we used the ruler to measure itself recursively, and pondered whether it would be possible for it to measure it measuring itself, and so on... However, that idea had to be put to rest, for the sooner we set up the experiment is when I get to pass off the tough stuff to my math wiz lab partner and eat my sandwich, killing two physics problems with one putty. Fortunately, my appetite was sated. Unfortunately, my lab partner did not do the correct math the first time. In any case, I present our measurements:

Measurements
Axis shift from center of mass to pivot	0.493 m
Mass of ruler	0.088 kg
Mass of putty	0.015 kg

There are three steps to this problem. First, as the ruler is held up and starts its descent, gravitational potential energy converts into rotational kinetic energy. At the bottom of the swing is when the ruler moves the fastest, since all of the energy has been converted. Second, when the ruler hits the putty, angular momentum is conserved; however, since it's an inelastic collision, the angular speed changes. The lower speed clearly lowers the energy that could be converted back into gravitational potential. So our three steps are:

1. GPE → rot KE
2. L₀ = L_f
3. rot KE → GPE

The moment of inertia of the ruler is as a bar rotating about its center of mass, but shifted to one side (using the parallel axis theorem). Because the ruler is not swing exactly at an edge, it would be more reasonable to use the inertia formula for swinging about the center than swinging about an edge. We begin converting energy to figure out the speed of the ruler just as it reaches the bottom:

Then, we use angular momentum to figure out the speed of the ruler just after it makes contact with the putty:

Now we find the height from the kinetic energy of the ruler converted back into gravitational potential:

The numbers are plugged in. Voila!

To be honest, that seems a bit high, but we are assuming that there is no rotational friction in the pivot or air resistance, so it probably makes sense that the numbers do not match personal experience. There is usually friction, whether between objects or air, whenever an object swings about another, which is why a playground swing does not act as an ideal pendulum. How convenient would that be for the parents! Push once, and, "I'll be back in half an hour, son!" Anyways, now to set up and do this experimentally:

The clay is wrapped with some tape to aid it in sticking on the ruler.

There are only power strips on one side of the table, and we didn't want to move the entire set up, so we had to do some creative ad hoc wiring...

We use a video capture to record the swing of the ruler. Through a string of bad luck, we could never set up the camera correctly. For one, the preview didn't work, so we couldn't check the clarity beforehand. Second, higher definition seemed to cause problems. Maybe I should bring my own camera to physics classes next year!

As you can see, Logger Pro shows a paltry 0.1321 m for the final height, about a third of our analytic calculations. We should've brought some Viagra! If we count the blurry dots on the meter stick though, which presumably represent 5 or 10 cm marks, the blue dot waaay to the right is about 3-4 notches up, suggesting that the scale is probably calibrated badly. Also, since the capture was so blurry and overexposed, we had to make some guess-timates on where the end of the ruler was. These problems could contribute to the disparity, but I still think that friction isn't eliminated in the pivot. If we lubed it up, the stick would've probably gone higher. Just saying!

Lab 18: Moment of Inertia of a Triangle

In Lab 16, we used a rotary device that pushes air through a pair of disks, minimizing frictional torque, and allowing the disks to spin for a long time, almost unimpeded. We were able to discover torque from the tension of the mass and angular acceleration, setting up the equation to solve for inertia, but we did not go through with it. In Lab 17, we used a disk that did not push air through it, so we had to solve for frictional torque. Here, we will use the air disk device again to simplify calculations, and use measured angular acceleration to actually solve for inertia of the disks, and any object attached to them. In this case, we use a uniform metal triangular plate rotated about its center of mass.

There are two ways to affix the triangle on the rotating mechanism. We will call the first the "vertical" orientation:

And this we call the "horizontal" orientation:

Our model presents this triangle very graciously.

Before we experiment, it is possible to solve for the inertia of the triangle analytically. We find it easiest to solve for the inertia from one edge, then work backwards with the parallel axis theorem to find the inertia about the center of mass:

Whew! So, apparently, the inertia only depends on the mass and one of its dimensions. So let's get out those calipers and start measuring.

Length	0.149 m
Width	0.098 m
Mass	0.455 m

So analytically, our moments of inertia are:

I_vertical = 0.455 kg * 0.098² m² / 18
I_vertical = 0.000243 kg m²
I_horizontal = 0.455 kg * 0.149² m² / 18
I_horizontal = 0.000561 kg m²

Now that we have in mind what to look for, let's get the experiment underway. As before, we will first find the inertia of the disks by measuring angular acceleration with a mass hanging off a torque pulley. We need to make a few measurements:

Hanging Mass	0.025 kg
Pulley Radius	0.025 m

Everything is hooked up to Logger Pro. We let it record the rotational motion and use the slope of the velocity charts to figure out acceleration:

Disks by itself, without triangle.

Triangle in vertical orientation.

Triangle in horizontal orientation.

And as before, we average the descent and ascent acceleration to cancel out any remaining effects of friction that the disk has, even with the air blowing through it.

α (rad/s2)
Disk by itself.
down	2.1040
up	2.2780
average	2.1910
Vertical orientation.
down	1.9430
up	2.1020
average	2.0225
Horizontal orientation.
down	1.7350
up	1.9380
average	1.8365

Finally, we have all the pieces to calculate the inertia experimentally:

We can simplify this by replacing all the variables that remain constant:

Now, we can just substitute the angular acceleration we found earlier to find the total moments for each system. The inertia of the triangle by itself are then derived by subtracting the inertia of the disks.

The results are surprisingly similar. It's always daunting to set out to mathematically prove something, since any error either calls into question the entire concept in one's knowledge, or sends him in furious exercise to try to match what he knows as theory with reality. In this case, it worked out okay...

	Results (kg m2)		Error
	Analytical	Experimental	Absolute (kg m2)	Relative (%)
Ivertical	0.000243	0.000233	0.000010	4.3
Ihorizontal	0.000561	0.000540	0.000021	3.9

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.