Friday, December 6, 2013

The Rockefeller Christmas Tree

Rotational Dynamics:





What would be the moment of inertia of this pulley as it raises the star for the Rockefeller Christmas tree? (Pulley rotates about its center, initially at rest)

Assumptions:
            Pulley radius = .381 meters
            Frictional Torque = tfr = 2.50 mN
            Force Applied to cord = 2000 N
            Angular speed = 15.0 rad/s2

St = RFT - tfr
      = (.381 m)(200.0 N) – (5.50 mN) = 756.5 mN

I = St/a
  = (73.7 mN) / (15.0 rad/s2) = 50.43 kg m2

Using the moment of inertia we can now determine the angular acceleration of the pulley, and finally the linear acceleration of the star

Assumptions:
            Mass of star: 250 kg
a = (mgR - tfr) / (I + mR2)
  = (2000 N)(.381 m) – (5.50 mN)
      (50.43 kg m2) + (250 kg)(.381m2)
 = 5.19 rad/s2

Linear Acceleration of star:
 a = Ra
    = (0.381 m)(5.19 rad/s2)
    =  1.98 m/s2
            This answer makes sense, the set up crew would want to move this star very slowly as to prevent any swaying or damages to the crystal or its surroundings!


Stress and Fracture



This year’s tree is from Connecticut, with a trunk about 4 feet in diameter and 75 feet high. How much would the tree shorten once the 200 N compressive force of the star is exerted on the tree? (Assuming long uniform rod)

Length of tree = 75 feet = 22.9 meters
Cross sectional area = p(.61m)2 = 1.17 m2
Compressive Strength of wood parallel to grain = 35 x 106 N/m2


F/A = (2000N)/(1.17) = 1709 N/m2  
(which is less than the compressive strength so it will not break)
L = Lo  F
         E    A
 = [(22.9m) / (10 x 109N/m2)] x 1709 N/m2

=3.91 x 10-6 m

Therefore the tree will shrink negligibly when the star is mounted.


Sources: 
http://www.nydailynews.com/new-york/rockefeller-center-christmas-tree-lights-city-article-1.1537918

http://www.swarovski.com/Web_US/en/crystal_society?contentid=10007.229549

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