# Find the period of a small oscillation

## Homework Statement

A rod of lenght ##2L## is bent at point of its middle so that the rods now created are in a upside V shape and the angle between them is ##120°##. The system oscilates. Find the expression of the period of oscilation.

## Homework Equations

3. The Attempt at a Solution [/B]
I thought of one thing. Since the angle is ##120°## the system oscilates the same way as a normal pendulum rod of lenght ##L/2##. If that is true all that follows is the center of mass which for a rod would be ##L/2## i think. Therefore we have imagined a pendulum of lenght ##L/6##. Its period is ##T=2*\pi * \sqrt{\frac{L}{6g}}##. Is this correct?

gneill
Mentor
A pendulum that is not a simple bob on a massless string does not behave in the same way. You need to look at the moment of inertia around the pivot point.

Look up: "Physical Pendulum". The Hyperphysics web site would be a good place to start.

Well it seems i need to find the center of mass. But that means that the center of mass for a poll of lenght ##L## is ##L/2##. I can think of the center of mass being located at point ##L/4## below the point ##O##. Is that correct?

gneill
Mentor
Well it seems i need to find the center of mass. But that means that the center of mass for a poll of lenght ##L## is ##L/2##. I can think of the center of mass being located at point ##L/4## below the point ##O##. Is that correct?
Could be. Show how you worked it out.

The poll of lenght ##L## has a center of mass located at ##L/2##. ##(L/2)*cos60=L/4## but that gives the period of ##\pi*\sqrt{\frac{L}{g}}##. The solution in the book read ##T=4*\pi* \sqrt{\frac{L}{3g}}## How can this be? That means that the center of mass is at ##4L/3## No idea how...

gneill
Mentor
What did you use for the moment of inertia of the pendulum about its pivot point?

What did you use for the moment of inertia of the pendulum about its pivot point?
I didnt. I should have used ##I=mL^2/3## but i dont know how to use it to get the period.

haruspex
Homework Helper
Gold Member
2020 Award
I didnt. I should have used ##I=mL^2/3## but i dont know how to use it to get the period.
Do you understand how to write down and solve the differential equation for a pendulum? It involves angular acceleration and moment of inertia.
The formula for period is obtained from that.

gneill
Mentor
I didnt. I should have used ##I=mL^2/3## but i dont know how to use it to get the period.
Did you investigate the term "physical pendulum" as I had suggested previously? The period of such a pendulum depends upon the moment of inertia about the pivot point and the distance of the pivot point from the center of mass.

Hint: when you formulate your moment of inertia, remember that your pendulum was originally a single rod of mass m.

I see now that the period of a physical pendulum is ##2*\pi*\sqrt{\frac{I}{mgL_{cm}}}##
I know that the moment of inertia of a rod is ##mL^2/3## but that inputed doesnt give correct answer. I need the center of mass. Im not sure what to do

I came up with something. The center of mass of these two rods is a rod of lenght ##L/2## staying vertical and oscilating. With above equation and inputing values of ##I=mL^2/3## ##L_{cm}= L/4## i get ##T= 4*\pi*\sqrt{\frac{L}{3g}}## correct?

haruspex
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