|MadSci Network: Physics|
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 motion. 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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