MadSci Network: Physics

Re: Does a swinging bell have the same charecteristics as a pendulum

Date: Tue May 22 11:16:09 2001
Posted By: Juan Cabanela, Faculty, Physics, Astronomy, & Engineering Science, Saint Cloud State University
Area of science: Physics
ID: 989546835.Ph

You are correct that there is a similarity between how a pendulum is
treated and how a bell is treated when we are computing their 
physics.  In essense a bell is a pendulum.  That is, most bells have 
their mass off center from the axis of rotation.  Therefore, if we ignore 
friction for a moment, the stable equilibrium position of the bell is 
when it's mass is directly below its axis of rotation.  If we rotate the 
bell, this mass will be out of equilibrium and there will be a restoring 
force due to gravity of
        F = -m*g*sin(theta)
where m is the mass of the bell
      g is the local gravitational acceleration (about 9.8 m/s^2)
  theta is the angle the bell has rotated from it's equilibrium position
           (usually measured in radians).
The sin(theta) comes from the fact that we need only consider the 
component of gravity pointing in the direction of the pendulum's 

Now, when we deal with pendulums, we usually make a small angle 
assumption that theta doesn't exceed 10 degrees, an assumption 
that is not appropriate with bells swinging through 360 degrees of 
rotation.  Making this assumption allows us to quickly compute the 
period of the pendulum by noting that we for small angles
     theta ~ sin(theta)
that is, an angle measured in radians is almost identical to the sine 
of that angle for angles of less than 10 degrees (about 0.2 radians).  
Making that assumption, the restoring force becomes:
      F = -m*g*theta      
and again, assuming small angles, we can relate the length of the 
pendulum (or how far the center of mass of the bell is from the axis of 
rotation) to the distance it swings through as:
      s = theta*L
where s is the distance the end of the pendulum (or center of mass 
of the 
        pendulum) swings through...
      L is the length of the pendulum
and therefore the force equation describing this motion is:
      F = -m*g*(s/L) 
        = -k*s   where k = (m*g/L)
This is just a form of the equation describing the motion of a spring, 
which allows us to use all the equations developed for the motion of 
a spring to describe the motion of a pendulum.

However, in your case, you can't assume small angles, and therefore 
there is no simple way to describe the bell's motion.  The force 
equation is simple, just:
      F = -m*g*sin(theta)
but it's application is limited by the fact that theta can not be simply 
related to the displacement of the center of mass of the bell.  A 
solution to this problem is possible of course, but it requires partial 
differential equations and is usually reserved for beginning physics 
graduate students. :)

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