Vernier Video Analysis: Studying Conservation of Angular Momentum
British mathematician Roger Penrose once stated, “Sometimes it’s the detours which turn out to be the fruitful ideas.” I am always searching for innovative ways to connect what students are learning in my physics class with the real world.
In late April, during a visit to the Staten Island Target, a family member yanked my arm so we could “see what cool stuff they had in Bullseye’s Playground,” the area of seasonal products right by the entrance. Amongst the plastic pitchers and 4th of July decor, I spotted the Target plastic Water & Sand Mill, priced at just $1.
In my physics class, we were studying rotational dynamics. My mind started to spin, pun intended. I thought, “my students could use the mill, along with some PlayDoh, to model conservation of angular momentum.” I scooped up every last mill and headed to the toy department to buy some PlayDoh. A new lab activity was born. It was that detour at Target that gave birth to a fruitful lab activity.
Conservation of angular momentum explains why ice skaters start to spin faster when they draw their arms inward. When the ice skater goes into the tuck position, the skater’s rotational inertia decreases, and, as a result, the skater’s rotational velocity increases.
Physics Background
For rotational motion, as with linear motion, conservation of momentum is an important and useful principle. For a mass system rotating about an axis with no external forces acting and negligible dissipative forces, angular momentum is conserved.
With the angular momentum of a rotating system represented by the letter L, and no outside forces acting (no external torques) then L = constant (or ΔL/Δt = 0 or dL/dt = 0). Now angular momentum can be expressed as the product of the rotational inertia and the angular velocity, or L = Iω. At a point on a wheel a distance R from the center of rotation, the angular velocity equals the point’s tangential velocity divided by the radius R or ω = v/R so that L = Iv/R.
For a mass that can be considered concentrated at a specific distance R from the center of rotation, the rotational inertia, I, is equal to I = mR2. Therefore, substituting for I above, we can write the angular momentum of the rotating mass as L = rmv.
For a rotating non-concentrated mass (not a single R), an equivalent rotational inertia, Isys, can be expressed for the rotating body where L = Isysω. Conservation of angular momentum can be expressed as Li = Lf .
The objective: Determine the theoretical mass of a blob of playdoh, using conservation of angular momentum.
Procedure
1. Measure the mass and diameter of your Water/Sand Mill. You will assume that the Water/Sand Mill is a solid, rotating disk with rotational inertia, I = ½ MR2.
2. Set up your Water/Sand Mill and mobile device as shown in the demo.
The angular collision that occurs is between the falling PlayDoh (“ a particle/rmv”) and the Water/Sand Mill (“as a rotating object”). The angular momentum of the falling PlayDoh is transferred to the Water/Sand Mill. Therefore, your equation for conservation of angular momentum is:
L Playdoh = L Water/SandMill + PlayDoh
4. Drop the PlayDoh blob from a height of no more than 5 cm. Observe what happens. The blob will, most likely, bounce out of the cup. If so, you may assume that the final angular momentum of the PlayDoh is zero.
5. Prepare to record the motion of the falling PlayDoh and the subsequent rotating Water/Sand Mill.
6. Repeat for two additional trials.
7. When you are finished, bring your PlayDoh to me. I will record the mass. You will only receive the mass after you have completed your calculations.
Data Needed
Using Vernier Video Analysis, we need to determine the velocity of the PlayDoh just before it’s angular collision with the Water/Sand Mill and the the angular velocity of the Water/Sand Mill just after the angular collision.
Students used Vernier Video Analysis to determine the two velocities needed.
Calculations
The goal of the experiment was to calculate/determine an experimental value for the mass of the Playdoh. The calculations are shown below.
Error Analysis
There are a number of sources of error in this experiment. For example, You assumed that the Water/Sand Mill is a rigid rotating disk. The Water/Sand Mill is clearly not. Does your assumption result in an experimental mass of the PlayDoh that is too high, too small, or just about right? Justify your answer.
Sample Student Response and Mathematical Proof
“The rotational inertia of the Water/Sand Mill is greater than that of a rotating disk because, compared to the rotating disk, the Water/Sand Mill has more mass distributed farther away from the center of rotation. Because of this, the experimental mass of the PlayDoh that I found is smaller than the actual mass.”
