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Is there a relationship between these 2

As of now I am moving a piece blutac up and down a ruler to change my COG. I have connected the base of my ruler to a vibration generator and am measuring the highest amplitude to measure the resonance frequency.

The closer the COG is to to the base the higher my resonance frequency is.

enter image description here

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  • $\begingroup$ Do you have data from your experiment? Please can you post them into your question as you did with your earlier question? $\endgroup$ – sammy gerbil Feb 12 at 20:02
  • $\begingroup$ yup I have added the data i collected! $\endgroup$ – ritzrori Feb 13 at 7:53
  • $\begingroup$ Thanks. I guess you have calulated the position of the COG. Do you have the raw data - viz. mass of blu-tak and ruler and position of blu-tak? As BillN shows, resonance frequency depends on the moment of inertia as well as position of COG. $\endgroup$ – sammy gerbil Feb 13 at 17:38
  • $\begingroup$ the ruler was 11.8 g and the blu tac os 6.7g $\endgroup$ – ritzrori Feb 13 at 18:11
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If you are talking about a pendulum in a gravitational field, yes there is a relationship between the two. If you do a free-body diagram and analyze the situation, you will get a differential equation: $$\frac{\mathrm{d^2}\theta}{\mathrm{dt^2}}=\frac{-mgr}{\mathcal{I}}\sin\theta,$$ where

  • List item $\theta$ is the angular position relative to vertical (direction of the gravitational field of magnitude $g$,
  • $t$ is time,
  • $m$ is the mass of the pendulum,
  • $r$ is the distance of the center of mass from the pivot point, and
  • $\mathcal{I}$ is the moment of inertial about the pivot point in the plane of oscillation.

Based on this equation the angular frequency of (small) oscillations will be $$\Omega = \sqrt{\frac{mgr}{\mathcal{I}}},$$ which will be in radians per second if all the other quantities are in SI units.

So, a change in the position of CoM due to a moveable mass could change the frequency as long as the moment of inertia changes at a different rate. For a moveable mass on a stick, that is the case.

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  • $\begingroup$ The OP is vibrating a ruler which is positioned vertically and clamped at the lower end in a variable frequency oscillator. $\endgroup$ – sammy gerbil Feb 12 at 21:17
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    $\begingroup$ @sammygerbil It's an analogous situation with a torque-driven restoring action and the CM and MoI changing at different rates. It's still a type of pendulum and not a linear spring-mass system. Linear spring-mass doesn't have torque if you ignore the rotation of the spring. $\endgroup$ – Bill N Feb 13 at 15:28
  • $\begingroup$ im assuming m in the formula is center of mass? $\endgroup$ – ritzrori Mar 15 at 22:57

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