A physics student measures the period of a physical pendulum about one pivot point to be T. Then he finds another pivot point on the opposite side of the center of mass that gives the same period. The two points are seperated by a distance L. Can he find the acceleration due to gravity, g, without measuring the moment of inertia of the pendulum? Why?(adsbygoogle = window.adsbygoogle || []).push({});

My answer:

For a physical pendulum, the angular speed is

[tex]\omega=\sqrt{\frac{mgd}{I}}[/tex]

Where m is the mass of the pendulum, I is the moment of inertia at the axis of rotation and d is the distance to the center of gravity of the pendulum. So, the period of the pendulum is

[tex]T=2\pi\sqrt{\frac{I}{mgd}}[/tex]

Since the two points have the same period, then the ratio of the moment of inertia about those points to the distance between that point and the cg should be the same. ie

[tex]\frac{I_1}{d_1}=\frac{I_2}{d_2}[/tex]

Using the parallel axis theorem,

[tex]

\frac{I_{CM}+md_1^2}{d_1}=\frac{I_{CM}+md_2^2}{d_2}[/tex]

simplifying,

[tex]I_{CM}=md_1d_2[/tex]

Therefore, the student could find g without measuring the moment of inertia of the pendulum. He just needs to find the location of the center of gravity.

But how do I relate L to my answer????

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# Physical Pendulum

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