
STATICS  CASE STUDY



Introduction

Problem Graphic


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.




