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STATICS - CASE STUDY

    Introduction


Problem Description
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The defending soldiers of a castle are launching a watermelon over the castle wall at an advancing army. To do this, they have designed a catapult with a heavy spring.

What is known:

  • The catapult and spring have the dimensions shown.
  • The spring constant k is 160 lb/ft, and the spring is in a relaxed position when the catapult is vertical (i.e. the relaxed length is 5 ft).
  • The collar of the spring is attached to a frictionless bearing.
  • The center of mass of the catapult (without the watermelon) is located at 2L/3.
     
    Questions


Dimensions
  After the watermelon has been launched, at what angles θ can the catapult come to a rest? Are these equilibrium positions stable?

     

    Approach

   
  • If the system is conservative, we can find an equation for the potential energy of the catapult in terms of θ.
  • The values of θ that yield zero for the derivative of the potential energy indicate the equilibrium positions for the catapult.
  • Use the second derivative of the potential energy to determine which positions are stable, and which are unstable.