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.