section_path: Self-locking machines/Proof
section_name: Proof
paragraph_text: Proof
  
  When a machine moves in the forward direction from point 1 to point 2, with the input force doing work on a load force, from conservation of energy the input work W 1,2 {\displaystyle W_{\text{1,2}}} is equal to the sum of the work done on the load force W load {\displaystyle W_{\text{load}}} and the work lost to friction W fric {\displaystyle W_{\text{fric}}}
  
  If the efficiency is below 50% (η = W load / W 1,2 < 0.5 {\displaystyle \eta =W_{\text{load}}/W_{\text{1,2}}<0.5}):
  
  2 W load < W 1,2 {\displaystyle 2W_{\text{load}}<W_{\text{1,2}}\,}
  
  From Eq. 1 2 W load < W load + W fric W load < W fric {\displaystyle {\begin{aligned}2W_{\text{load}}&<W_{\text{load}}+W_{\text{fric}}\\W_{\text{load}}&<W_{\text{fric}}\end{aligned}}}
  
  When the machine moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to friction W fric {\displaystyle W_{\text{fric}}} is the same
  
  W load = W 2,1 + W fric {\displaystyle W_{\text{load}}=W_{\text{2,1}}+W_{\text{fric}}}
  
  So the output work is W 2,1 = W load − W fric < 0 {\displaystyle W_{\text{2,1}}=W_{\text{load}}-W_{\text{fric}}<0}
  
  Thus the machine self-locks, because the work dissipated in friction is greater than the work done by the load force moving it backwards even with no input force.