Structural failure of a CD

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy.

Professor Starkman once asked us to use rotational mechanics to find out the properties of a CD spinning at 7200 RPM; this gave us some appreciation of the stress on a CD as it is spun. The Mythbusters once tackled the objective of trying to cause structural failure to a CD by creating unusually high rotational speeds to the CD to investigate the myth that a standard CD drive can under certain circumstances spin fast enough to cause a CD to break apart and turn into a lethal disc of shrapnel.

Let’s mesh these worlds, and see what it would physically take to destroy a standard CD.  The figure we were given in our physics class was 7200 RPM.  In certain disc drives, the regular speed may be more. 

Physical Information of CD-

Thickness (X) = 1.20 mm

Material = 100% Polycarbonate (tensile strength, σ_t, of polycarbonate is about 75 MPa)

Radius (R) = 12.0 cm

Density (d) = 1.20 g/cm^3

Let us make the assumption that since the hole is filled in, we have a complete volume of disc.  Plugging that in to our density:

m/V = d

m =dV

m = dπR²X

Mass (m) = 0.0650 kg

We have a radius and a thickness, which corresponds to a cross-sectional area of a CD on one side.  Recall that it only requires a break at one of these cross-sectional areas to fail.  This material is very brittle; do not expect much strain as a result of stress.  It ought to shatter.  This should simplify things.

I plan to evaluate the centripetal force caused by the spinning of the disc as a function of ω.  This disc must respond to a centripetal force with a normal force.  This normal force is dictated by its structural integrity.  Given that we have a specific area of interest, this force may be divided by area, leaving us with units of N/m^2… the same units as Pa, which is proportional to our tensile strength.  The units of tensile strength and pressure are identical.  Evaluate for the maximum possible ω which will cause a force that exceeds our tensile strength.

σ_t = F/A           (Force required to break divided by area equals tensile strength)

A = XR            (Cross-sectional area is equal to radius times height)

XRσ_t = F         (The force that is required)

F = mv^2 / r

v = Rω

F = mRω²

XRσ_t = mRω²

(XRσ_t / mR) ^½ = ω

(Xσ_t / m) ^½ = ω

This is the maximum possible angular velocity that we can achieve.  After plugging in the appropriate values:

((0.0012m)*(75E+6Pa)/(0.065kg))^ ^½ = 1177 rad/s

We now have a figure in radians per second, but disc drives are never advertised in such figures.  What does this translate to in terms of revolutions per minute, the preferred angular velocity measurement of the West?

1177 rad/s*(radian / second)*(1 revolution / 2π radian)*(60 second / 1 minute) =

11200 RPM

Our ceiling figure for angular speed of a CD is 11200! Um, wasn't it way faster on Mythbusters? =/

In all probability, as we estimate for error in this problem, our estimate is extremely liberal with its notion of structural failure.  In actuality, the polycarbonate material may be higher or lower than the one we listed; but every CD has a bottom and top layer which would likely enhance structural integrity.  Additionally, we did not account for the removed section of the disc (the hole in the center, into which an electric motor pushes a rotor that spins the disc.  

My point in this experiment is to reflect on the magnitude of stress on the CD in your disc drive as it whizzes around at 120 to 170 revolutions per second.  A modern engine will be on the redline when a CD drive is operating properly.