A uniform thin rod of length L oscillates as a pendulum about a pivot point that is a distancea from the center. For what value of a, expressed as a fraction of L, is the oscillation period aminimum?

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A uniform thin rod of length L oscillates as a pendulum about a pivot point that is a distance
a from the center. For what value of a, expressed as a fraction of L, is the oscillation period a
minimum?

Expert Solution

Step 1: Concept

A physical pendulum is made up of a solid object that swings back and forth on a pivot due to gravity.

The physical pendulum's time period is expressed as:

$T space equals space 2 pi square root of fraction numerator L over denominator m g l end fraction end root$

Where, I is the moment of inertia, m is the mass of the object, g is the acceleration due to gravity and l is the distance between pivot and center of mass.

The rod's length is given as

L is the length of the pivot, and the distance between the pivot and the center of mass is x.

The expression for an object's moment of inertia about any other point other than the moment of inertia is

$I subscript r o d space end subscript equals space I subscript C M end subscript space plus M x squared$

$I subscript C M end subscript$ is the moment of inertia at center of mass, M is the mass of rod and x is the distance between pivot and center of mass.

As a result, the moment of inertia of the rod about the pivot is:

$I subscript r o d space end subscript equals space 1 half M l squared space plus M x squared$

Subtituting the value of $I subscript r o d end subscript$ in terms of T

$T space equals 2 pi square root of fraction numerator I subscript r o d end subscript over denominator M g x end fraction end root T space equals 2 pi square root of fraction numerator bevelled fraction numerator 1 over denominator 2 M l squared space plus space M x squared end fraction over denominator M g x end fraction end root$

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A uniform thin rod of length L oscillates as a pendulum about a pivot point that is a distance a from the center. For what value of r, expressed as a fraction of L, is the oscillation period a minimum?