# The oscillating meterstick

Remember this scenario from last week? We drilled a hole a distance x from the center of a meterstick,

suspended the meterstick from a pin through this hole, and then nudged the meterstick so that it rotated around the pin.

The meterstick has a mass M and length L (which should be 1 meter, of course). What's the moment of inertia of the meterstick around this hole?

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Right.

Okay, now, if we let the meterstick hang straight down for a moment, and then displace it by a small angle θ as shown, gravity will exert a torque on the stick. The magnitude of the torque will be

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Q:  What's the direction of the displacement θ?

Q:  What's the direction of the torque τ?

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Okay, so we know the torque and the moment of inertia. When we release the meterstick, it will start to rotate. How large will the angular acceleration be?

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Q:  What's the direction of the angular acceleration α?

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Now, as we saw last week, it turns out that if the angle θ is small -- say, 10 degrees or 15 degrees, then expressed in radians

Putting all this together, we can now say that the meterstick will accelerate like so:

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Q:  Hey!  Isn't the angular acceleration related
to the angular displacement?

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Why, yes, it is. In fact, it's the second derivative of angular displacement with respect to time,

So, putting everything together, this means that the motion of the meterstick -- as long as the angular displacements are small -- will follow the differential equation

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Q:  Can you write an equation which shows how
the angular displacement θ
will change with time?

Q:  Suppose that the offset between the
center of the meterstick and the hole is
x = 0.2 m.

What is the equation now, with numbers?

Q:  What will the period of the meterstick be?

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