5/13 Finding the Moment of Inertia of a Uniform Triangle About it's Center of Mass

Purpose: To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

Procedure:

FIGURE (1)

We used the set-up pictured in FIGURE (1). Initially we did not have the triangular plate on the apparatus.We wrapped a string with a mass at the end around the pulley. Then we measured angular acceleration (FIGURE (2)) so we could calculate the inertia. We then added the triangular plate in the position pictured and measured the angular acceleration again (FIGURE (3)). Since we want to find the location of the center of mass of the plate, we needed to rotate the triangle to the longer side and measure angular acceleration once more (FIGURE (4)).

FIGURE (2)

FIGURE (3)



FIGURE (4)

FIGURE (5)
CORRECTION: the correct value for the 
third average acceleration is 0.932 not
1.864

A table of our data is in FIGURE (5). Once we had the angular acceleration for each trial, we were able to find the inertia of each triangle and therefore the inertia of just the triangle in each position using a formula we derived in a previous lab. (FIGURE (6)).

FIGURE (6)

A table of each value of inertia is shown in FIGURE (7). After we calculated the experimental values of inertia, we derived an expression to find the expected inertia of the triangle
 (FIGURE (8)). The equation for the inertia of each position turned out to be the same equation except the variables, b and h, were switched.

FIGURE (7)

FIGURE (8)

FIGURE (9)

After we found the inertia for the triangle around each edge, we were able to find the inertia of the triangle around the center of mass for each position. To do this, we used the parallel axis theorem (FIGURE (9)). Again, the equation was the same for both position, just the variables, b and h were switched. All that was left was to plug in the measured values of M, h and b. When we plugged those in, we calculated the values for inertia around the center of mass to be .000247 for the second trial and .000575 for the third trial.










Conclusion:

Our calculations seems close to the theoretical values but in fact, they were both more than 5% off. The inertia value .000247 was around 17% off and the inertia value of .000575 was around 9% off. Sources of error could come from not having the plate completely level, there may be systematic errors from when we connecting the apparatus to the computer, or there may be random errors. 

5/11 Moment of Inertia and Frictional Torque

Purpose: To find the moment of inertia and frictional torque of an apparatus with three disks.

Procedure:

FIGURE (1)

In this lab, we started with an apparatus as pictured in FIGURE (1). We took the measurements that are labeled in the the figure. Once we had the measurements for the each of the disks, we were able to calculate the mass of each of the disks using the total mass and the volume (FIGURE (2)). We needed the mass of each of the three disks so we could calculate the iof each of them and therefore the total inertia.

FIGURE(2)

In order to find the decceleration of the apparatus, we used a photogate. We stuck a piece of tape on the larger disk as shown in FIGURE (3). We then set the photo gate to measure in radians instead of meters. This allowed us to measure the theta that the disk (or tape) travelled through before it came to a stop t seconds later. 

Once we had that graph, we took the deriviative to get angular speed. Once we had the graph of angular speed we were able to do a linear fit and the acceleration was the slope. After all of this, we came up with an acceleration (due to friction) of -0.3236, and calculated the frictional torque to be -.00649. 

FIGURE (3)

Next we were able to set up the rest of the experiment. As shown in FIGURE (4), we tied a string to a cart and then wrapped the string around the smaller radius of the apparatus. We measured the angle that the track made with the horizontal to be 49 degrees. 

FIGURE (4)

Before we ran the experiment, we calculated the experimental value of time it should take the cart to go down 1 meter of the track (FIGURE (5)).

FIGURE (5)

Once we calculated the expected value, we were ready to carry out the experiment. We ran three trials and came up with the data in FIGURE (6). 

FIGURE (6)

Conclusion:

Overall, the experiment went well. We calculated the inertia of the disks to exactly what was expected (when rounded). We also came up with a way to calculate the frictional acceleration that apparently no other students had come up with yet in the course. There could have been error from our measurements, especially measuring the time for each trial. Our calculations were only around 2% off though, so overall, we were not too far off. 

5/4 Angular Acceleration

Part One

Purpose: To note the difference in acceleration of a system when we change the hanging mass, radius of the torque pulley, and the rotating mass. 

Procedure:
We started with the setup in FIGURE (1) and took the measurements of the masses and diameters of the disks, pulleys, and hanging mass.

FIGURE (1)

FIGURE (2)

The values we measured are listed in FIGURE (2). Once we set up the apparatus, connected it to the computer, and set up the sensor settings, we turned on the compressed air so that the disks can rotate separately. We started with just the hanging mass attached to a string that was wrapped around the pulley. In the next trial, we added twice the hanging mass and then three times the hanging mass. In the fourth through sixth experiment we changed the size of the pulley to a larger one. In the last two experiments we also changed the disk. So in the fifth experiment we had only the top aluminum disk rotating and in the sixth experiment we has the top steel disk and the bottom steel disk both rotating. The data we recorded was the angular velocity. We took the derivative of that velocity to get the angular acceleration. (FIGURE (3)-(8)).

FIGURE (3)

FIGURE (4)

FIGURE (5)

FIGURE (6)

FIGURE (7)

FIGURE (8)

Once we recorded all the data, we were able to find the average acceleration for each trial, simply by adding the up and down acceleration and dividing by two (FIGURE (9)).

FIGURE (9)

We can notice some patterns in FIGURE (9). When we increased the mass of the hanging mass, the acceleration increased as well. It increased about one radian per second when we doubled the mass, and another radian per second when we tripled the original mass. The acceleration increased with a larger hanging mass because the torque on the pulley increased from the initial trial. In trail four, the first trial with the larger pulley, we noticed that the acceleration increased. While keeping the larger pulley and changing the top steel disk to an aluminum one we can notice that the acceleration drastically increases. This is because the inertia of the lighter disk is much smaller than the steel disk, since the aluminum disk is lighter. In the last trial, when we added the rotating bottom steel disk, is slightly slower than the initial acceleration. This could also be due to the amount of inertia that is needed to move two disks instead of just one. 

PART TWO

In class, we derived an equation to solve for the inertia of the system for each trial. We plugged in the measurements we took for each trial to the equation and came up with each inertia (FIGURES (10) & (11)).

FIGURE (1)

FIGURE (11)

We also calculated the expected value of inertia for each system using the equation: I = 1/2*m*r^2

For the first four trials, the expected value of inertia was .00276, for trial five the expected value was .000929, and for the last trial it was .1162. 

Conclusion:

The first five trials were very close to the expected value of inertia. They were all within 4% of the expected value (most fell under 2%). There is an error in the sixth one though. I think our group might have measured the value of acceleration wrong- maybe we did not have the lab pro calibrated correctly. Other sources of error in this lab would come from measuring incorrectly. The relationships between hanging mass, radius, and rotating mass are evident in our data though. When the hanging mass was increased to twice as much the acceleration also doubled. And when we tripled the hanging mass we saw the same pattern. This does not affect the inertia though because the mass and radius of the pulley and rotating disk stayed the same. In the fourth trial, it is shown that the size of the torque pulley does not affect inertia, because inertia is dependent on the size and mass of the rotating disk. In trial five we changed the rotating disk to a lighter, aluminum one and, as expected, the acceleration increased dramatically. The inertia decreased in this trial since the mass was lighter. In the last trial, we expected to see that the acceleration decreased from the original acceleration. We saw a slight decrease but not as big as we expected. That is why I believe this is where our error was.

4/27 Pendulum Lab

Purpose: To find the launch speed of a ball using a ballistic pendulum, with conservation of momentum and energy.

Procedure:

FIGURE (1)

We used the ballistic pendulum shown in FIGURE (1). We took measurements of the mass of the ball, holder, and length of the wire, which were 7.67 g, 80.9 g, and 20.2cm respectively. Since we wanted to find the speed of the launch, we needed to find theta. With this apparatus all we had to was launch the ball and the holder pushed a wire to a certain angle measure, which we were able to just read. We calculated this to be 17 degrees. Once we had all the measurements, we were ready to calculate velocity (FIGURE (2)).

FIGURE (2)

Using momentum and energy, we were able to find the speed of the launch to be 4.80 m/s

Conclusion: 

We also calculated uncertainty for the speed since we had the uncertainty of all of our measurements. (FIGURE (3) & (4) )

FIGURE (3)

FIGURE (4)

Errors in this lab would have come from our measurements. 

4/22 Collisions in Two Dimensions

Purpose: Look at two-dimensional collision and determine if momentum and energy are conserved.

Procedure:

FIGURE (1)

We started with the set up shown in FIGURE(1), and we connected the computer to the camera and adjusted the settings accordingly. We then set a steel ball in the center of the table and collided an aluminium ball with the one in the middle while capturing video (shown to the right) for one trial and another steel ball for the next trial. We used LoggerPro to track the movement and collision of the balls (FIGURE (2)).
Once we had the graphs of the collision we could figure out whether or not momentum and energy were conserved.



FIGURE (2)


Conclusion:
Sources of error in this lab could come from a few places. We were not one hundred percent sure that the table was exactly level, so if it was unleveled that could affect out measures. The camera that we used also distorts the image, so when the balls got to the edge of the frame, it was a less accurate reading of position. A third source of error is how well we tracked the balls' position. Since we were tracking the position by adding individual dots for each frame, it is likely that the dot was not in the exact center of the ball.



4/15 Impulse-Momentum

Purpose: To prove the Impulse-momentum theorem in elastic and inelastic collisions.

Procedure:

Part One 

FIGURE (1)

For the first part of the lab we were observing collision forces that change with time. We connected a spring plunger to a rod (FIGURE (1)), which we were then able to clamp to our table. Next, we mounted a force sensor on a cart at the same level as the spring plunger. Once we had things all set up we collided the cart into the plunger several times and observed what happened.
QUESTIONS:
1. The net force exerted on the cart just before the collision is zero.
2. The magnitude of the force on the cart is maximum when the plunger is at its maximum compression.
3. The net force exerted on the cart after the collision is also zero.
4. The collision takes about a second.

We added a motion sensor to the opposite end of the track that the spring plunger was on. We also made sure our ramp was level. We opened the correct file from the computer and made sure all of our devices were calibrated. Then we started recording our data as we gave the cart a push toward the spring plunger and let them collide. (FIGURE (2)). We then took the integral during the time the collision took place and got a value of -0.9211. 

FIGURE (2)

The change in momentum of the cart was close to the impulse applied by the spring plunger. We calculated a value of -0.8516 (FIGURE (2.5)), which means the calculation was off by about 8%.

Part Two

In the third part of the lab we changed that mass of the cart by adding 200 g. Then we repeated the same steps as part two and took new data. (FIGURE (3)) In this trial we took the integral of the force of the collision and came up with -1.060.

FIGURE (3)

When we calculated the change in momentum (FIGURE (3.5)) we got a value of -1.006. Again, these are close- they are off by about 5%. With this outcome though, we are able to say that the impulse is equal to the change in momentum, even with a larger cart.

FIGURE (3.5)

Part Three

For the last part of the lab, we changed the set up slightly. Instead of using a spring plunger to create and elastic collision, we used a piece of clay to create an inelastic collision (FIGURE (4)). Our prediction was that the impulse would be smaller than the elastic collision and the impulse and change in momentum would still be equal to each other. 

Again we pushed the cart toward our "clay man" and collected data. (FIGURE (5)) We took the integral of the force and came up with -0.3392. This confirms one of our predictions: the impulse of the inelastic collision was smaller than the elastic collision.

We calculated the change in momentum (FIGURE (6)) and came up with a value of -.275. While this still seems close to the impulse value, it is around 23% off.

FIGURE (6)

Conclusion:
Errors in this lab could have come from having the clay spread too thin. This would cause the nail to hit the wood instead of the clay stopping it. The nail probably wouldn't have enough force to go into the wood so it would bounce back slightly.