By Danielle Scheer
In June of 2000, my mom took away one of my coolest toys. She was a pink and purple “Sky Dancer Flying Doll.” Commercial here: http://www.youtube.com/watch?v=ETCLxxA8G_0
Why was my toy taken away, you ask? Well, the toy company Galoob recalled about 8.9 million of the dolls. The reason? The “The hard plastic Sky Dancers® dolls can fly rapidly in unpredictable directions, and can hit and injure both children and adults.” (CPSC.gov)
Over 150 children and adults had reported injuries due to the doll’s unpredictable flight path, their woes ranging from eye injuries (scratched corneas and temporary blindness), broken teeth, a mild concussion, a broken rib (a bit intense for a 12 inch doll, no?) and facial lacerations that required stitches.
Because so many of the injuries were reported to be so severe, I want to figure out with what force the Sky Dancer would hit a person in the rib, and if this would actually be enough force to break a human rib.
Sky Dancer Knowns:
Dimensions: .20754 m x .07874 m x .33528 m
M= 1 lb = .453592 kg
Treat the wings as a long uniform rod, and the torso as a uniform solid cylinder
Iwings= 1/12 mL2
Itorso= ½ MR2
Itotal= m(1/2R2 + 1/12 L2)
=.454 kg (1/2 (.03937)2 + 1/12 (.20574)2)
Estimate spins/second from video to determine rotational velocity:
If she spins in a full circle 9 times per second:
f= 9 rev/sec
ω = 2πf= 2π(9)= 56.55 rad/s
From video, we know it takes ~2 seconds for SkyDancer to reach that velocity from rest
ωf= ωi + αt2
56.55 rad/s = 0 rad/s + α(2s)
α = 28.28 rad/s2
KE=1/2 ΣI ω2 = ½ (.0019532945 kgm2)(56.55 rad/s)2= 3.123 J
Calculate velocity of Sky Dancer
V=rω= (.03937m)(56.55rad/s) = 2.226 m/s
How high does she go? (not really relevant, unless I calculated the change in energy if she hit a person’s face/different parts of the body...which i can do if I’m selected to present J)
mgh= ½ mv2 + ½ Iω2
(.4536kg)(9.8m/s2)h = ½ (.4536kg)(2.226m/s)2 + ½ (.0019532945 kgm2) (56.55 rad/s)
4.445kgm/s2 h = 4.247kgm2/s2
In the rib hit, assume all of the energy translates. The child stops the doll by doing work, and the body compresses by (assumed value) ~1mm= .001m
F= ΔKE/d = 4.247J/.001m = 4247 N
If the doll is released from a standing position, with the arms extended, and she comes back to hit the rib, the she would have travelled the full .9555 m up and down, so would height be ~2m, making KE= ~8.495 J, and F=8495N?
The ultimate force a bone can withstand:
The shear modulus for bone is 80x109 N/m2
Use the shear modulus because the bone is flexing/bending. When something bends the inside of the bend gets compressed while the outside gets stretched. The “layers” of bone have to move laterally to each other
Based on Abrams, Mohr, Engel, and Bottlang in “Cross Sectional Geometry of Human Ribs” (Portland, USA: 2003) the average cross-sectional area of a rib is ~20 mm2.
Kemper, McNally, Kennedy, Manoogian, Rath, Ng, Stitzel, Smith, Duma, and Matzouka, in “Material properties of human rib cortical bone from dynamic tension coupon testing” (VA: USA: 2005) found the ultimate stress of human ribs to be 124.2 MPa= 124,200,000 N/m2.
The rib can therefore endure a force of:
F=GA=1.242e8 N/m2 * 2.0e-5m2 = 2484 N
Therefore, it is possible that the SkyDancer doll did break a human rib.
I guess it’s a good thing my mom took my doll away from me...