Final Lab

Thanks for a great semester!

2 December 2014: Period and Moment of Inertia Lab 21

Objective:
The purpose of this lab is to be able to find the moment of inertia of a real life ordinary semicircle and use it to its period.

Procedure and Results:

The first thing to do is to cut out a semi circle out of a piece of cardboard. Some semicircles were drawn using a compass but ours was made using a protractor. It is important to note that you can not cut out a circle and then cut it in half; it has to be a semicircle.

The next thing to do was to to tape little hangers to each side of the semicircle. It is important to make sure the holes of the hangers are as close to the edge as possible as the later calculated moment of inertia is along the edge and not further out.

Next, we predicted the period we believed the semicircle would have. We did this by first calculating the moment of inertia of the semicircle and then relating it to period. Our calculations can be seen below:

As you can see from the picture, we calculated our period to be 0.636 s.

Next, we wanted to test the validity of our prediction so we set out to measure the semicircle's period.

We hung the semicircle we cut out by the hangers we had attached to a rod secured to a ring stand by clamps. On the same ring stand was attached a photogate with the use of more rods and clamps.The photogate recorded the oscillations of the semicircle and the data was saved to Logger Pro.








The recorded period was:

Our predicted period was .636 and its actual period was .611. Clearly they were not the same value so to find out by how much we were off we found percent error. Percent error can be found by:

So, our percent error was 3.93%

Conclusions:
We do not really know exactly why our calculations did not fall within the expected one or two percent error. We believe our calculations and our approach is correct and yet we still did not get the correct value. We believe that the source of error lies in the handling of the semicircle. We cut the semicircle a few days before we actually measured its period. We think during that time, the semicircle was bent and damaged causing its period to be distorted. Even though it did not meet the expected percent error, it is still amazing that we are able to find the period of a real life object cut by our own design with a percent error of <5%.

20 November 2014: Mass Spring Oscillations Lab Lab 20

Objective:
The purpose of this experiment is to find the spring constant of a spring using our own methods and compare it to the spring constants of other springs.

Procedure and Results:

The set up for this lab required the hanging of a spring by a rods and clamps, This this spring, some masses could be added.

The procedure to find the spring constant of our spring started with placing a motion sensor below the spring and recording its height without any weight attached to it. Its height was .21 m We then added 500 g to the spring and again recorded its height. Its height with the mass was 4.9 m. We then divided both these heights as they were the amount that the spring was stretched. This gave us our value for our spring constant (k):

k= 4.9/.21= 23.3 N/m

After we had found our k, we went on to measure the mass of our spring. Our spring had a mass of 27.7 g.

We then had to find the period of our spring. We did this by adding many different masses to the spring and recording the spring's oscillations with the motion detector used earlier. We then analyzed the motion of the spring and its resultant graphs. The graphs can be seen below:

From this data we were able to determine using the position part that the spring had a period of 0.42 s. We then compared our data to other people's data and placed their values into this organized table:

From this table we, were able to construct a graph using all the periods of all the springs and all the spring constants of all the springs:

As you can see the fit for this graph was a power fit.

From the table above we could also create using all the masses of all the springs and all the periods of all the springs:

The curve fit for this graph was a power fit.

Conclusions:

From this lab, we were able to learn how to find the spring constant of a real spring. Also, we were able to find its period. From the graphs of everyone's data, we can learn that as mass of the spring increases so does its period. This makes sense because if the spring has more mass, it will be able to oscillate more as the 1/3 mass of itself contributes oscillations; if there is more mass, there is more to contribute. Also, as spring constant increases, period decreases . This makes sense because a more firm spring will resist oscillations more than a weak spring.

20 November 2014: Conservation of Linear and Angular Momentum Lab 19

Objective:
The purpose of this lab is to investigate the conservation of angular momentum about a point that is external to a rolling ball. This differs from other labs since other labs looked at the angular motion of an object rotating about some internal axis.

This lab had three parts so it will be divided accordingly.
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Part 1:

The first set up was set up like this. A ball is slid down a ramp some height off the ground. It will then hit the ground at some point L away from the end of the ramp. A piece of carbon copy paper was placed on the floor to find exactly where the ball hit. This is a picture of the actual set up:




We measured that the ball landed .522 m away from the edge of the ramp. After we knew this distance, we were asked to find the launch speed of the ball.

Our calculations showed that the launch speed was 1.18 m/s.

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Part 2:

This part of the lab required the use of a rotational apparatus. This apparatus allowed tow disks to rotate freely. A ball would be launched from the ramp shown and caught by a little ball catcher attached to the top of a torque pulley on top of one of the rotating disks. Once the ball hit the ball catcher, the disk would begin to rotate. The point is to find out at what rotational speed the disk would rotate. But first, moment of inertia of the system must be found.

So first we put the rotating system in motion and to recorded data from its motion. From this data we were able to construct graphs that produced angular acceleration. The graphs can be seen below:

There is frictional torque acting on the system so the average of both angular accelerations will be taken: (5.934+5.339)/2= 5.637 rad/(s^2).

Now that we know angular acceleration we could find the moment of Inertia of the disk by these equations:

We then simply plugged in the known values of mass, radius of the torque pulley, and the above value of alpha to determine that the moment of inertia = .001057 kg*m^2.


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Part 3:
Now that the moment of inertia of the disk is known and the launch speed of the ball, we can find the rotational speed at which the disk will spin once the ball is caught. The calculations for this mainly used conservation of angular momentum and can be seen below:

So the angular momentum of the disk was 1.95 rad/sec.
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Conclusions:
The two parts of lab helped us find the value for the last part of the lab. From this lab we found that from launch speed kinematics and moment of inertia, we can find the rotational velocity of a real life disk in motion.

18 November 2014: Angular Collisions Lab 18

Objective:
The purpose of this lab is to use knowledge about angular momentum and conservation of energy to find the height to which a ruler will rise when it collides with a stationary lump of clay.

Set up:

This is the set up for our lab. We used rods and clamps to be able to hang a ruler so that it rotates freely around a screw. The ruler would later be raised and it would hit the stationary lump of clay that was placed atop of rod.












Procedure and Results:
The first thing to do was to gather data about the setup and system. Firstly, it was important to note that the screw is not at the end but at the 0.02 mark. Also, the mass of the ruler 0.087 kg and the mass of the clay 0.028 kg. Being that we're using a meter stick, the length of the stick 1 m.

Next we decided to run first predict the height to which the meter stick would rise. Calculations can be seen below:

As you can see from our calculations, we predicted that the ruler would rise 0.303 m.

We then decided to test the validity of our prediction. We mounted a camera in front of the meter stick and a dark board behind the meter stick so that the clay would be more visible. We then raised the meter stick and let it go. It hit the clay and rose to some height. We captured the whole event on video and later analyzed the video.

We set coordinates on the video so that we would be able to find distances and velocities from the video. From the video, a table of values was created from which we made a graph:

As the table shows, the highest the meter stick ever rose was 0.233 m and from the graph we can see that as the ruler gained distance in the negative x direction, it also gained height, which can be expected from its rotational motion.

Conclusion:

Our prediction for the height that the ruler would reach is much higher than the height it actually reached. When we calculate percent error, we find that we have 23% error. Unfortunately we think that most of error came from the video analyzing. As you can see from the picture, the video came out blurry which made it hard to pin point exactly where the clay was at every frame. Also, we mounted the camera at the bottom which distorted our view of the system and hence the coordinates we placed upon our video. Had we mounted the camera higher perhaps our coordinates would have been more accurate.

13 November 2014: Triangle Moment of Inertia Lab 17

Objective:
The purpose of this lab is to determine the moment of inertia of a uniform triangle about its center of mass. We must also determine the angular acceleration of a hanging mass when the triangle is at different orientations:

Setup:

A triangle was given to each group. First the moment of inertia of the triangle must be found. But then, a the triangle must be placed onto a disk that is spinning freely without friction. A string is wrapped around a pulley and attached to a disk. This string then goes over another pulley and is attached to a hanging mass. There apparatus comes with a built in rotational sensor that can record angular velocity and position.









Procedure and Results:
The first thing to do was to gather data on the everything we were to use including measurements of the triangle and measurements on the disks upon which it would be placed:

Next, we had to calculate the moment of inertia of the triangle. A picture of our calculations can be seen below:

Since we had found the moment of inertia, it was time to test it using the apparatus shown above. First we found the angular acceleration of the system without a triangle. The graph can be seen below:

We must take the average of both the acceleration as the mass is going down and the acceleration as the mass is going up. This because there is some frictional torque in the system. So the average of both these values is (2.993+1.946)/2 = 2.469 rad/(s^2)

Next we placed the triangle on the disk pulley holder system and again found angular acceleration. We placed the triangle on the disk with the long side on the bottom.

Again we took the average of both values and found the average to be (2.564 + 1.470)/2= 2.017 rad/(s^2)

Then we rotated the triangle on the disk pulley holder system so that that the short side was on the bottom. We again found angular acceleration:

We took the average again and found it to be (3.013-1.670)/2 = 2.34 rad/(s^2).

We then took the time find the actual moments of inertia of the triangle:

Moment of Inertia of a Short Triangle

Moment of Inertia of a Tall Triangle

Next, using our average angular accelerations found above we were able to calculate the moment of inertia of each system:

With this equation for I, we simply plugged in values and found the moments of inertia of each system organized in the table below. It is important to note that after we found the moment of inertia of a system and the moment of inertia of the system with a triangle, we subtracted the value of the moment of inertia of the system from the value of the moment of inertia of the system and triangle to find the moment of inertia of just the triangle. In other words:

I(triangle+system)-I(system)=I(triangle)

Conclusion:
Our results prove that we were closer to calculating the moment of inertia of a short triangle than a tall triangle. Nevertheless, both percent errors fall within an acceptable range proving that we were correct in our calculations of moment of inertia of a triangle along its center of mass.

4 November 2014: Moment of Inertia Lab 16

Objective:
The purpose of this lab is to find the moment of inertia  of a disk. We must also find the deceleration of the disk due to friction. The validity of these two values will later be tested by making a prediction.
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There are three parts to this lab and so this blog will be divided accordingly

Part 1:
This part required that we find the moment of inertia of a spinning disk. This are pictures of the disk:

side view of disk

top view of disk

 As can be seen from the picture there are not just one object to calculate the moments of inertia but three: two small cylinders and one big disk.

The first thing to do was to gather the appropriate data needed to find moment of inertia. This included measurements and masses. The mass of the whole disk was given but the masses of the individual cylinders and disk were calculated. The measurements were gathered with the use of the caliper.







This is a picture of our calculated masses and the moment of inertia:

As you can see from the picture we round the radius of the disk was 0.0997 m and the height of it was 0.0155 m. The radius of the small cylinders was 0.0155 m and the length of each cylinder was 0.51 m. 

We then used these values to calculate the volume of each individual piece by the equation V= pi* (radius ^2)* height. We found the volume of the each small cylinder to 3.87 *10^-5. And the volume of the big cylinder was found to be 4.84*10^-4. Lastly we found the total volume to be 5.615*10^-4. 

With these volumes, we were able to calculate the masses of each individual piece by looking how much percentage of the total mass is made up of each individual piece. We knew that the whole apparatus 4.825 kg. We found that the percentage of the each small cylinder was 6.89% and thus the mass of each cylinder was 0.3324 kg. Lastly we found that percentage of the disk was 86.20% and thus the mass of the disk was 4.159 kg.

With these masses, we could find the moment of Inertia contributed by each individual piece. It was then found that each cylinder did not contribute to the moment of inertia. Therefore, the moment of inertia is just the moment of inertia of the disk which is defined by the equation I= mass*(radius^2). Therefore the moment of inertia of the disk was found to be 0.0207.

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Part 2:

This part of the experiment required that we find the angular deceleration of the disk as it slows down due to frictional torque acting on the apparatus. We did this by placing a piece of tape on the disk. We then set up a camera to record the disks' spinning, gave the disk a spin, and later analyzed the video.

We saved the video to our computer for further analysis and from the video, we were able to construct a graph. The data from the video gave us the tape's position and velocity in the x and y direction. Using the velocity in the x and y direction we were able to graph speed of the tape by the formula speed = sqr root((xvelocity^2) + (yvelocity^2)). Here are picture of the data tables and graphs:

As you can see by the linear fit we applied to our graph, the deceleration (a) is -0.04607. We then know that angular deceleration can be known by the equation: angular deceleration(alpha) = a/ radius. so then: 

our angular acceleration is -0.462 rad/(s^2).

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Part 3:

The last part of this experiment is to test the validity of our calculations. To do this, we connected this disk apparatus to a dynamics cart. This cart was to roll down an inclined track for a distance of one meter. But first we needed to calculate and predict how long the cart it should take for the cart to travel 1 meter from rest.

Here is a picture of the setup:

As you can see, a string attaches the cart to the disk apparatus, Also, the cart is on a track placed at some angle. The cart will be released at the top from rest.

Here is a picture of our calculations and prediction:

So as you can see from the calculations, our track was angled at 40 degrees. And we predicted our cart to take 8.27 s to travel on meter.

After this we actually let the cart go and timed the amount of time it took for it to go 1 meter. This was done using a stop watch. The time the cart took was 10 s.

We then used our predicted time and our actual time to find percent error:

So our percent error was 17.3 percent.

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Conclusion:

Our percent error was way too high to be valid. In truth we do not know exactly what went wrong. Our calculations are correct. We suspect that the problem lies our manner in getting angular deceleration and the way we analyzed the video. If our a is wrong then our alpha is wrong and thus this whole calculations are wrong. Although we do not know exactly what went wrong at least we know our math and approach is correct. This was an interesting lab as it showed us that calculations can predict real life events.