2014年11月26日星期三

Finding the moment of inertia of uniform triangle about its center

The purpose of this lab is determining the moment of inertia of a uniform triangle about its center of mass. Therefore, we will use computer to determine the real inertia of the object, use formula to figure out the inertia, and compare them.
As the picture shows below, we set the triangle on a holder and disk. The upper disk floats on a cushion of air, so there is not fiction when the disk rotates.A string is wrapped around a pulley on top of and attached to the disk, and go over a freely-rotating pulley to a hanging mass.
We measured data:
hanging mass: 0.025 kg
radius of top pulley: 0.025 m
However, first of all, we have to find the inertia 1 of disk without the object and the inertia 2 of the disk with object, and use inertia 2 subtract the inertia  1 in order to get the inertia of the object.


From this function, we can substitute numbers to get the inertia. And then, the only number we need to know is angular accelerations.
angular acceleration of empty disk:
 average angular acceleration: 2.191 rad/s^2
After substitute data:
Inertia of empty disk is 2.78*10^-3 kg*m^2

angular acceleration of the entire disk with triangle:
 average angular acceleration: 2.0225 rad/s^2
After substitute, we get the inertia is 3.012*10^-3 kg*m^2
Therefore, we were able to get the triangle inertia:
I1= 3.012*10^-3 kg*m^2 - 2.78*10^-3 kg*m^2= 2.328*10^-4 kg*m^2

And we also try to measure another inertia of triangle in different direction, so we turned the triangle 90 degrees:
the acceleration of triangle after turning: 1.8365 rad/s^2

After substitution, we get the inertia I2 is 5.3952*10^-4 kg*m^2
Thus, for different direction of the same object, their inertia are different.
And then, we need to calculate the inertia with formula. We know the inertia of triangle that around one side of it: I=bh^3/12
We also get the b=0.098 m and h=0.149 m, and the distance d from side to center of mass is 0.149/3 m.
We apply parallel-axis theorem to this question: Icm=I-md^2.
However, for the both of experiments, the inertia is not quite equate to what we got from experiment.

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