What happens when you take expensive equipment and do silly things with it? Awesome stuff is the correct answer—this is exactly what the Slow Mo Guy did and here is the full video. Yes, I love the Slow Mo Guys—who doesn’t. But what I really love is seeing cool things that no one saw before. That is exactly what happened when they used a high speed camera with a spinning Compact Disc. It turns out that these discs shatter when you spin them too fast—with a high speed camera, you can see the propagation of material failure as the disc spins.
It’s such a good video, let’s do some analysis.
How Fast Was the CD Spinning?
They break several CDs, but let me look at the second CD they shattered recorded at 61,960 frames per second. I tried to use the higher speed video, but I was getting repeating frames—not sure why. The guys claim the critical spin rate for a CD is 23,000 rpm. Let’s see if we can get the actual spin rate for this shattering CD. Yes, I am going to use Tracker Video Analysis—it’s free and awesome (do I have to keep on saying this)?
After loading the video into Tracker, I can set the scale by assuming the diameter of a CD is 12 cm. Next, I can put the origin in the center of the CD and track some mark as it spins (before exploding). Here is what I get.
From this, I get an angular velocity (slope of the angle-vs-time graph) of 3.914 x 10 3 radians/second which is 37,375 rpm. This is quite a bit higher than 23,000 rpm (but of course you know that). Ok, one small issue. I am going with the assumption that this video clip shows all the frames and that each frame is indeed 1/61,960 seconds.
Analysis of Flying Stuff
Really, this is like a classic physics question. It goes something like this.
With the following question:
The question is often in a multiple-choice format, but you get the idea. A common answer is that the ball will move away from the center of the circle when the string breaks.
Let’s look at the CD. What does a piece of the CD do after it breaks free? Here is a plot of the piece’s trajectory (x vs. y).
I know what you are thinking: “oh look, the path of the piece is sort of curving.” Well, actually—no. The vertical axis is in a much smaller scale than the horizontal axis. This data includes some time before the piece broke off and then after that, the piece was spinning as it was flung off. The piece broke off at almost the exact bottom of the CD and so it mostly only had a horizontal velocity. Here is a plot of the x-position vs. time for that piece (technically, I marked a corner of the piece).
Notice that it isn’t quite a constant velocity (that would be a straight line in an x-t graph). Instead, this piece seems to be slowing down (appears—but maybe not actually slowing since I am just looking at the corner of a spinning piece). From the quadratic fit, I can get the velocity at t = 0 seconds at around 453.2 m/s (1014 mph). Is this what it should be? For an object moving in a circle (like it was at the beginning of the motion), the following shows the relationship between the angular velocity, the radius of the circle and the linear speed.
Using an R of 0.12 m and an ω of 3.914 x 10 3 rad/s, I get a linear speed of 469.68 m/s. BOOM. Almost exactly the same as from the video analysis. But what about the acceleration? From the quadratic fit, the CD piece has an acceleration of -2.138 x 10 -5 m/s 2. Why would it have an acceleration like that? I would guess it’s due to air resistance. I think there is something cool here, but I am going to leave it as a homework question (see below).
What about the rotation of the piece of the CD that flies off? If I measure the position of two ends of the piece, I can use those to get an angular orientation of the piece. Here is a plot of the angle of that piece as a function of time.
This gives it an angular velocity of 6.05 x 10 3 rad/s. But why should this piece be spinning at all? I have two guesses to start with. First, if you watch the way the CD breaks one end of the piece becomes free before the other end. That means that the rest of the CD is still pulling on one side of the piece which causes it to spin on release. My second guess has to do with angular momentum. If you consider the piece while it is still connected to the CD, that piece is rotating (it goes from oriented up to down and back to up as the CD spins). If the piece has angular momentum before the break, it should still have angular momentum after the break.
Summary and Homework
First, here is a diagram showing the path of a piece of the CD.
Before the CD breaks, the rest of the CD exerts a force on the CD piece causing it to move in a circular path. After the break, this force is no longer there. What do objects without forces on them do? The correct answer is that they don’t change their momentum (velocity would be acceptable). Moving in a straight line means the object has constant velocity.
But wait! The piece does accelerate after the break. Yes, I think this is due to air resistance (just a guess).
Now for some homework. There is much to be done.
If a piece broke of in such a manner that it shot straight up, would you be able to see a change in speed due to the gravitational force? If so, could you use this acceleration to check that the frame rate is correct? If not, why not?
If there is indeed a significant air resistance on the piece of CD, can you detect this force on other pieces too? Should a smaller piece have a higher or lower acceleration due to air resistance? Estimate the mass of a piece (based on the size) to get a ball park figure for the acceleration due to air resistance.
For a piece of the CD on the outside, estimate the forces needed to keep the CD together for rotation rate of 3,900 rad/s. What is the breaking strength of a CD? You might want to assume the CD is polycarbonate plastic to get a value of the tensile strength. Is this why the CD broke? I suspect that the break was caused by high spin rate along with wobbling of the disc.
Why does a broken CD piece spin (or why do some spin)? Try looking at other pieces and measure their post-break spin rate. Do other pieces spin the same?
Big project: track all the pieces of the broken CD for a short time after the CD breaks. Estimate the mass of each piece based on the area density of a CD and the area of a piece. Is kinetic energy conserved (should it be)? Is angular momentum conserved? Make a plot of linear speed as a function of starting distance from the center of the CD. Should this plot be linear?
I really want to see the answer to some of these homework questions. If you don’t answer them, I might.
