Purpose:
To find the period of multiple objects swinging back and forth on a pivot with a small change in theta.
We want to use the restoring force and small theta of the oscillation to manipulate our equations so that we can find the period of oscillation of a swinging object and then compare it to the experimental recording of the period.
Set Up:
For the experiment, we put the object onto a pivot and let it swing back and forth slightly.
A piece of tape was attached at the end of the object that would pass through a photogate to record the time of each period through LoggerPro.
Experiment:
1. Since we already had the steel ring for the first part of the experiment, we just needed to make a thin semi-circle and an isosceles triangle out of styrofoam. We did this using a saw in class!
2. Next we needed to measure the necessary elements of each object needed in later calculations (steel ring and styrofoam shapes):
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| Mass of Steel Ring = 396.8 grams |
 
Measured Data Recorded:
Solid Ring
Inner Radius = 0.1390 m
Outer Radius = 0.1154 m
Mass = 0.3968 kg
Isosceles Triangle
Base = 0.1931 m
Height = 0.1508 m
Semicircular Plate
Radius = 0.0983 m
3. Now for the next fun part, performing the experiment!
First we tapped the steel ring so that it oscillated back and forth (shown in the set-up) and found its period by looking at the mean of the data from photogate and LoggerPro:
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Period = 0.7181 seconds
Second we attached two aluminum loops on each side of each styrofoam shape to put a paper clip through, which that was taped to the stand. We tapped the styrofoam shape a small theta so that it swung back in forth in an oscillating motion, recording the period with photogate and LoggerPro and used it to find the average period by looking at the mean of the data:
Semi-circle concave up:
Period = 0.6838 seconds
Semi-circle concave down:
 
Period = 0.6789 seconds
Triangle right-side up:
  Period = 0.7843 seconds
Triangle up-side down:
 
Period = 0.6789 seconds
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Data Analysis:
Using our measured data, we now could calculate the theoretical period and compare it to the experimental period.
The Steel Ring
First we needed to find the new inertia of the ring. After finding the average radius we able to calculate the new inertia since we now knew the linear shift.
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| Inertia of the Steel Ring in this case = 3.23 * 10^-3 kg *m^s |
Next we used Torque = Inertia * Angular Acceleration so that we could manipulate it into giving us the period of the Steel Ring's swinging.
In this case Torque = mg * avg. radius * sin(theta) as well, where sin (theta) is so small it can written as just Torque = mg * avg. radius * theta.
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| Theoretical Period = 0.7175 seconds |
The semi-cirlce concave up
First we calculate the inertia and find the center of mass:
Next we calculate the theoretical period by applying Torque = Inertia * Alpha once again:
Period = 0.683
The semi-circle concave down
Now that we are rotating the semi-circle at the top of the round edge, we need to shift the moment of Inertia:
Now that we have the new moment of inertia that we found by using Euler's law (shifting the inertia of the center of mass linearly), we can find the theoretical period:
Period = 0.67 seconds
The triangle right-side up
First we calculated the inertia using methods of integration (pivot was at the tip of the triangle):
Next we found the period of the triangle using T=I*alpha:
Period = 0.783 seconds
The triangle up-side down
Last but not least, we found the inertia of the triangle pivoted at the center of its base, in this case we were able to use integration once again and substitute the y for H-y since the triangle was just being flipped:
Example:
Steel Ring
Last but not least, we found the inertia of the triangle pivoted at the center of its base, in this case we were able to use integration once again and substitute the y for H-y since the triangle was just being flipped:
We found the period for the last time using T=I*alpha:
Period = 0.6995 seconds
% Error:
Example:
Steel Ring
|(0.7181-0.7175)|/0.7175 x 100% = 0.0836% error
Overall:
Conclusion:
Overall:
Conclusion:
With an overall average percent error of less than 1% our experiment was successful is showing that due to the restoring force of the swinging and the small oscillations, we were able to make sin theta into theta, therefore simplifying our equation which allowed us to find the angular velocity using the equation angular acceleration = angular velocity^2 * theta. Therefore, we were able to derive and find the theoretical period successfully. Error may have come from friction on the pivot, making sin theta into theta (less accurate to the smaller decimal place), and pushing the object too much creating a theta that was larger and greater affected by changing sin theta into just theta.
