Pendulums

Summary

• A simple pendulum is…
  • a small, compact mass
  • at the end of a light, inextensible string
  • suspended from a pivot
  • that is free to swing back and forth
  • under the influence of gravity alone.
• The period (T) of a simple pendulum is…
  • independent of its mass (m);
  • proportional to the square root of the length of the string (T ∝ √ℓ);
  • inversely proportional to the square root of the local freefall acceleration (T ∝ 1/√g); and
  • roughly independent of the amplitude angle (θ) for small angles.
  
  

  T = 2π√	ℓ
  	g

  
  
• A physical pendulum is…
  • a rigid body of any size or shape
  • suspended from or attached to a pivot
  • that is free to swing back and forth
  • under the influence of gravity alone.
• The quantity analagous to the length of a simple pendulum is the equivalent length of a physical pendulum…
  
  

  ℓeq =	I
  	Md

  
  where…
  
  

  ℓeq =	equivalent length of a physical pendulum [m]
  I =	its moment of inertia about the pivot (not about the center of mass) [kg m2]
  M =	its mass [kg]
  d =	the distance from the pivot to the center of mass [m]

• The period (T) of a physical pendulum is then…
  
  

  T = 2π√	ℓeq	= 2π√	I
  	g		Mdg