Calculate Pendulum Frequency: Length, Weight & Height

[aletof]

Hi!

I want to build a machine, but I need some precise calculations to complete it. I basically need to have various pendulums of the same length but different frequency

I was thinking on using some weights that I could set higher or lower on the pendulum to adjust the frequency.

I want to calculate the frequency given the following information:

-Length of the pendulum (will be constant)
-Weight of the material used throughout the pendulum (will be constant)
-Additional adjustable weight
-Height of the additional weight

I hope it is not to difficult to come up with a formula that considers all of this.

Thanks in advance!

 

[russ_watters]

Welcome to PF. The wiki is always a good place to start: http://en.wikipedia.org/wiki/Pendulum#Period_of_oscillation

 

[aletof]

Thanks, I was reading this article: http://en.wikipedia.org/wiki/Center_of_percussion
which describes somewhat my problem, unfortunately I haven't been able to deduce a formula that would enable me to precisely calculate the real (I'm actually going to build it) frequency of a pendulum, given the above described criteria.

I reckon I need something like this:

http://www.wolframalpha.com/input/?...scillations.I-.*PendulumSmallOscillations.m--


Looking at the schematic I want to keep the length static, and move the red ball up and down along the black line (it will be rigid, not a string), thus modifying the period/frequency.

 

[Doc Al]

You'll want to treat it as a physical pendulum. See: http://hyperphysics.phy-astr.gsu.edu/Hbase/pendp.html" .

You'll need to calculate the rotational inertia about the pivot and the distance of the center of mass from the pivot.

 

[aletof]

What I understand is that this is the formula that's going to help me:

Where:
-"m" is the total mass of my pendulum
-"g" is gravity
-"L" is the length from the pivot to the end of the pendulum
-"Isupport" is the moment of Inertia which I have to calculate with:

Where:
-"r" is the distance from the pivot to the center of mass?

I'm not sure if I'm interpreting things correctly on this one, I need to perform the calculation for about 20 different pendulums, so deducing a formula where I can input my data and get the result is my priority.

Now, it troubles me that this will work only for "small displacements", what exactly is a small displacement on this case?

Thanks for your help so far!

 

[Doc Al]

I'm not sure if I'm interpreting things correctly on this one, I need to perform the calculation for about 20 different pendulums, so deducing a formula where I can input my data and get the result is my priority.

Depending on how you are distributing the mass along the pendulum, you can simplify the calculations for the center of mass and rotational inertia.

Now, it troubles me that this will work only for "small displacements", what exactly is a small displacement on this case?

The link I gave describes where the 'small angle' approximation comes in. You can always extend it to greater accuracy, if you like.

 

[aletof]

Depending on how you are distributing the mass along the pendulum, you can simplify the calculations for the center of mass and rotational inertia.

The pendulum will be a straight wooden stick and the weight will be a lead or steel piece that I can move along the length of the stick, thus changing the center of mass.

Ideally all the pendulums will have the same weight, but just by offsetting the steel weight I will be able to change the frequency significantly.

I guess I need to figure out how to calculate the center of mass, based on the position and mass of the steel weight as well as the rotational inertia, I could then input those on the above mentioned formula and it should work?

 

[Doc Al]

The pendulum will be a straight wooden stick and the weight will be a lead or steel piece that I can move along the length of the stick, thus changing the center of mass.

If you can neglect the weight of the stick compared to the steel weight, you can treat it as a simple pendulum. Otherwise, you'll need to treat it as a physical pendulum as we've been discussing.

I guess I need to figure out how to calculate the center of mass, based on the position and mass of the steel weight as well as the rotational inertia, I could then input those on the above mentioned formula and it should work?

I suspect you'll be fine by approximating the steel weight as a point mass. Then you can easily calculate the combined center of mass and combined rotational inertia. The rotational inertia of a point mass is mr²; that of a stick rotating about one end is 1/3mL². (All of this can be found on the hyperphysics site.)