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A pendulum is a physical object thatundergoes small angular oscillations under the restoring force of gravity. |
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{excerpt} A pendulum is a physical object thatundergoes small angular oscillations under the restoring force of gravity. {excerpt} In its simplest form, a pendulum consists of a point mass at the end of a massless string attached to a suspending point, or a point mass attached to a massless rod that is set to pivot about a fixed point. if allowed to swing freely, the mass exhibits [simple harmonic motion] about its equilibrium position, with [natural frequency] of {*}ω{*} \\ {latex}\begin{large} \[ \omega = \sqrt{\frac{g}{L}} \]\end{large}{latex} \\ Here [g|gee] is the acceleration due to gravity of |
Here g is the acceleration due to gravity of 9.8
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m/sec
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and
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L
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is
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the
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length
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of
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the
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string
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or
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rod
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between
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the
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pivot
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point
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and
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the
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mass.
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A
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more
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nuanced
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model
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is
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to
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view
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the
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pendulum
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as
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an
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extended
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mass
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having moment of inertia I pivoting
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about
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a
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fixed
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point
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that
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is
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not
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the
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,
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and which
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hangs
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downward
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from
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the
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point
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of
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suspension
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under
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the
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influence
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of
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gravity. The distance between the pivot point and the center of mass is L , and the mass of the object is m. Small excursions from the equilibrium position feel a restoring force due to torque from the force of gravity.In this case the natural frequency ω is given by
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\begin{large} \[ \omega = \sqrt{\frac{mLg}{I}} \]\end{large} |
The restoring force is not truly linear in displacement, since the sine of the angle of displacement enters in, but if the displacements are small the small angle approximation makes the restoring force very nearly linear in angle, and the equations of motion for the pendulum become identical to those of a mass on a spring.