Chopping Down A Tree

Classical Mechanics Level 3

A perfectly symmetrical tree has a trunk in the shape of a cylinder with diameter \(0.2~\mbox{m}\). Since the tree is symmetrical the center of mass is at some distance \(L\) above the ground along the axis of the cylinder. A rotten lumberjack with no concept that he's cutting down the most amazing tree ever grown uses a chain saw to cut the tree down. He saws horizontally right at the base of the tree (ground level).

What is the minimum value of \(L\) in meters such that the tree begins to topple over as soon once the chain saw reaches the position in the picture?

Details and assumptions

  • The chain saw blade is \(8~\mbox{cm}\) wide and \(0.5~\mbox{cm}\) high.
  • Treat the tree as a rigid body.

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