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.