03-June-2015: Physical Pendulum Lab

The purpose of this lab was to derive expression for the period of numerous physical pendulums, and then verify the predicted periods using LoggerPro.

Part 1: A Solid Ring
To find the period of a solid ring oscillating at small angles, we needed to find an equation for it's acceleration as a function of a negative constant and displacement. From there we would be able to find its period. Once we had this value, we would then use LoggerPro to find the actual period of the pendulum using a photogate. (see Fig. 1)

Fig. 1
Using LoggerPro and a photogate, we were able to find the period of the solid ring as it swung through small angles.

Since the ring was solid, we had to take its moment of inertia relative to both of its radii. When we measured the two radii, we got values of 0.05765 m and 0.06975 m respectively. The equation for the period of this pendulum would be given by 2*pi divided by it's angular frequency. After going through our calculations, we found the predicted period of the pendulum to be 0.718 seconds. When we ran the actual experiment, we found the period to be 0.720 seconds. Thus, we had a percent error of 0.376 %. (see Fig. 2 and Fig. 3)

Fig. 2
We were able to calculate the period of the solid ring be setting the angular acceleration into this form.

Fig. 3
Using LoggerPro, we were able to find the period of the solid and found that we were off by only 0.376%!

Part 2: An Isosceles Triangle about its Apex
To find the period of an isosceles triangle about its apex, we had to find two unknowns first. First, we had to find its moment of inertia. Next, we had to find its center of mass. Our calculations for finding the center of mass of the triangle can be seen in Fig. 4 and for the moment of inertia in Fig. 5

Fig. 4

Fig. 5
NOTE: There should be a "M" for mass next to the 12 in the final equation.

Once we had our equations we mad our prediction for the period, which turned out to be 0.703 seconds. (see Fig. 6) To test this, we cut out an isosceles triangle and measured its height and base to find a prediction for its period. (see Fig. 7)

Fig. 6
We calculated the period to be 0.703 seconds and we were off by only 0.92 %

Fig. 7
We cut out the isosceles triangle that we would use to test our theoretical value.

We found that we had a percent error of 0.92 %.

Part 3: An Isosceles Triangle about the Midpoint of its Base
To find the period of an isosceles triangle about the midpoint of its base, we again had to find an expression for its moment of inertia. Once we had this, we use Newton's Second Law to find an expression for the angular acceleration as a function of a negative constant and displacement. Once we had this, we were able to find its period. Our calculation for the moment of inertia can be found in Fig. 8.

Fig. 8

We plugged in the dimensions of the triangle as needed to find its period, which was 0.616 seconds. (see Fig. 9). Again, we used LoggerPro to find the actual period of the triangle. (see Fig. 10)

Fig. 9
Again, we used the formula for period being 2*pi/angular frequency to find our value for the period.

Fig. 10
We found that we had a mere 0.92 % error in our calculation from that of LoggerPro.

We found that we had a percent error of 0.92 %.

Part 4: A Semi-Circle about the Center of its Curved Edge AND the Center of its Base
To find the period of oscillation of a semi-circle about the center of its curved edge, we decided to first find its center of mass. This would be required to find the period of oscillation about the center of its curved edge and the center of its base. To do this, we decided to plot the semi circle and take its dm with respect to many little rectangles that would compose the semi circle. To see our full calculation, see Fig. 11. We found that the center of mass 4R/3pi above the center base of the semi circle.

Fig. 11
Finding the center of mass. 

Once we had the center of mass, we needed to consider its movement in small angle oscillations. This indicated that we would have to use Newton's Second Law, Torque= Moment of Inertia * Angular Acceleration. We would have to take the component of the disk's weight that was parallel to its motion (Perpendicular to the lever arm). However, to complete this equation, we needed the moment of inertia of the disks as they swung from different origins. Fig. 12 shows our calculations for both moments of inertia.

Fig. 12
In the red is our moment of inertia of the semicircle oscillating about the center edge of its curve. In the blue is
the moment of inertia of the semicircle oscillating about its center base.

Once we had the moments of inertia for the two differently orientated semicircles, we then proceeded to plug in our numbers and put angular acceleration into a SHM form. Once we did this, we took our omegas and calculated our periods. After running the experiments, we compared our expected values. (see Fig. 13 and Fig. 14)

Fig. 13-1
We found that our value for the period was only 0.6% of LoggerPro's value.

Fig. 13-2
Period LoggerPro found us for the semicircle oscillating about the center edge of its curve.

Fig. 14-1
We had to take into account the altered shape of the semi circle. It was not a completely circular curve, which may have correlated to our higher percent error.

Fig. 14-2
Period LoggerPro found for us for the semicircle oscillating about the center of its base.

We found that our first prediction for the semicircle oscillating about the center edge of its curve was good, in that we yielded only a 0.6% error. However, our second run for the semicircle oscillating about the center of its base had a larger 1.3% error. This may have been due to the fact that our semicircle was not completely round, thus throwing of its actual moment of inertia.

Error and Conclusion:
Although the lab represented little error, we must still account for some issues. In cutting the shapes, we were unable to make perfect and exact shapes, which in turn may have altered the actual period of oscillation of the system. Another issue was finding the perfect point to have the triangle oscillating. Using cable rings, we tried to get the edge of the triangle and semicircle as close as possible to being at the pivot.

In all, however, the lab was successful and allowed us to verify the period of multiple oscillating pendulums.

20-May-2015: Conservation of Energy and Angular Momentum Lab

The goal of this lab was to predict the height at which a simple pendulum would rise after colliding with a stationary piece of clay.

Setup:
To set up this lab, we used a rotary motion sensor as a pivot for a meter stick to act as a pendulum on. Once the meter stick was attached to this pivot, we setup a small piece of clay on the ground directly in path the meter stick would be falling. To ensure the collision would be inelastic, we wrapped the clay and meter stick with tape, sticky side out. Ideally, they would stick together after the collision (see Fig. 1). We then measure the mass of the meter stick as well as the clay.

Fig. 1Setting up the experiment.

Once we had the masses, we found the position of the pivot point for the simple pendulum. This was located roughly 0.0085 m away from the top end of the meter stick. Once we had all this basic information, we were ready to begin finding our prediction.

Calculating the Height:
To find the height at which the clay and meter stick system would rise, we split the collision into three components. First was the initial fall of the meter stick after being held horizontally. Second was the change in angular velocity after the collision. Third was the final height the system would reach after the collision.

PART 1: Initial Angular Velocity
To find the angular velocity of the meter stick directly before the collision, we used conservation of energy. Initially, as the meter stick was position in a completely horizontal manner, we only had gravitational potential energy. If we called the center of mass of the meter stick in the vertical direction our zero point for GPE, then our final energy would be rotational kinetic energy. (see Fig. 2)

Fig. 2
To do this calculation, we set our zero at the vertical center of mass of the meter stick.

Directly before the collision, we found our angular velocity to be 5.445 rad/s.

PART 2: Final Angular Velocity
To find the angular velocity of the system directly after the collision, we used conservation of angular momentum. However, to find the moment of inertia in the final stage of the collision, we had to include the moment of inertia of the clay as a point mass. (see Fig. 3)

Fig. 3
We used conservation of angular momentum in this section and included the moment of inertia of the clay.

Directly after the collision, we fond our angular velocity to be 3.858 rad/s.

PART 3: Finding the Height
To find the final height of the clay, we again used conservation of energy. To make the problem simpler, we called the pivot point of the meter stick our zero for GPE. This in turn would make our GPE values negative. In the final energy section of the equation, we multiplied our heights relative to the cosine of the angle at which the meter stick would make with the vertical. (see Fig. 4)

Fig. 4
To make this problem do-able, we tried to find the angle at which the pendulum swung rather than the actual height of the clay.

We found the theoretical angle of the pendulum to be 63.6 degrees. To find the height, we assumed that the clay was directly at the end of the meter stick. Thus, the height it rose would be the difference between the length of the clay to the pivot point, and the cosine of the length of the clay to the pivot point. In final, we predicted the height of the meter stick to be 0.5514 meters.

PART 4: Comparing to the Actual Height:
Using video analysis, we recorded the collision of the meter stick and clay. Setting up a horizontal and vertical axis, we placed a point directly at the maximum height of the clay. (see Fig. 5). We found this to be 0.4842 m.

Fig. 5
Using LoggerPro, we were able to find the actual height the clay rose.

We found our percent error to be 13.87 % at the conclusion of the lab. (see Fig. 6)

Fig. 6
Percent Error Calculation.

Error and Conclusion:
There were a few sources of error in this lab. For one, we assumed that the meter stick was completely uniform and it's center of mass was located directly at the 50 cm mark. Having degraded over time, this may not have been entirely true. Also, we chose to ignore the fact that energy was lost in other forms during the collision, such as in sound or friction. Thus, it makes sense that our predicted value for height is a little bit over the actual value.

Overall, the lab was a success and we were able to prove that energy and angular moment was conserved in this inelastic rotational collision.