Determining Density of Cylinders/ Mass of an Unknown Object

Purpose: The purpose of this lab is to calculate the the density of three cylinders and the mass of an unknown object and calculate the percentage error of both.

We used the calipers shown in the picture below to find the diameter and the height of the cylinder in order to get the volume of the cylinder. We also used it in order to get uncertainty values for both of those measurements.

We also used the scale shown below in order to get the mass of the cylinder so that we could find the density of each cylinder. This also helped us acquire an uncertainty value for math.

Procedure: For the first part of the lab we used the scale and the calipers in order to get measurements of height, diameter and mass so that we could find the density of each cylinder. We also got the uncertainty of the measurements so that we could propagate the uncertainty of the density for each cylinder.

After that we went on to the second part of the lab which was the unknown mass hanging from force scales. We had to get the angle of the force scales measured and also the measurement that was on the force scales themselves. Uncertainty for each measurement was also needed so that we could propagate the uncertainty of the unknown mass.

Data: Below is the numbers we got for each of the cylinders.

We also got the measurements for the unknown mass which is also below.

Calculations: First, we found the density of each of the cylinders and then we propagated the uncertainty of the density for each cylinder by taking the partial derivative of each value that had uncertainty (with respect to density). The calculations are shown below.

We did the same thing for the unknown mass except we used forces instead of an equation. We then did the same thing for uncertainty below.

Summary: In this lab, we pretty much used the information given so that we could propagate for uncertainty and overall it was a success because I get it. We took measurements of a cylinder and the force scales on the unknown mass in order to find out what density and mass were respectively. We also came to the same values meaning that we all had the correct data and were able to get the purpose of the lab. 

Centripetal Acceleration as a Function of Angular Speed

Purpose: The purpose of the lab was to find a centripetal acceleration as a function of angular speed.

Procedure: First Professor Wolf set up the horizontal rotating disk and the accelerator by taping the accelerator to the edge of the rotating disk. Then he gave the whole class stop watches to record the period of the 4 rotations (which turned out to be 3 later). He gave us landmarks so that we could measure the period of the disk more accurately. He spun the disk at different speeds and we were supposed to record the period for each of them. There were 5 trials conducted.

Data/Calculations: Below is the data acquired of the different trials we conducted. They were then averaged and input into the data table to get a graph.

Below is the graph of the resulting data collected. We changed from 4 rotations to 3 rotations because some people did not get accurate results and we decided as a class that the information looked cleaner with a period of 3 rotations. We also got the radius of the spinning disk which was about 18 cm. The period was found by dividing the number rotations by 3 for each trial. We found omega (w) from the equation w = (2pi)/T (period). We then squared it in order for us to solve for the acceleration. We then used a = rw^2 to find acceleration. The slope of the graph, as a result, illustrates the radius of the spinning disk which was 18.31 cm which is really close to our value.

Summary: Overall, the lab was successful because we found that the radius is -1.7% off than what we experimented. The data was not perfect though because of human error in the fact that everyone did not contribute to the time and everyone got different times. Nevertheless, the data was really close with only a small percentage error which means that we did prove that there is a relationship between centripetal acceleration and omega (w) or angular speed. 

Modeling Friction Forces

Purpose: The purpose of the experiment was to utilize wooden blocks to see friction forces in action and to calculate the acceleration of the systems.

Procedure:
1) The first part of the experiment involved using a wooden block attached to a cup full of water with a string using a pulley system. The block was set on top of the table while the pulley was set at the edge. We let the cup full of water hang from the edge of the table by the string attached to the pulley system. This experiment was used to determine the coefficient of static friction between the block and the table. We measured it by adding water to the cup until the block just barely began to move. We did this for 1, 2, 3, and 4 blocks on top of each other. For some of these trials we had to use weights since the water in the was filled to the top and the blocks still would not move at all.

2) We then proceeded to use a Force Sensor to measure how much force did it take to pull the 1, 2, 3, 4 blocks respectively so that we could find the kinetic friction of the system respectively.

3) Afterwards we used height to calculate static friction. By using a ramp and ring stand with a clamp we were able to set the ramp at an angle. We adjusted the height so that the block would barely move and then used that to find out the static friction of the system. This was done to only one block.

4) Next, we measured the kinetic friction of the system by doing a similar experiment as the height one except this time adjust the height so that the block would slide down. We put a motion sensor on the bottom of the ramp to record just how fast it was going and then use it to find the kinetic friction of the system.

5) After that we used the same set up as the previous one but instead we reinserted the pulley into the system and put a weight at the end of the pulley so it could hang. This was used to predict the acceleration of the system so that we could put the friction forces we acquired to the test and see how close we were to the results. The motion sensor was also placed at the bottom so we can get an experimental value of acceleration.

Data:
1) We calculated for the force of static friction for the first runs of the 1, 2, 3, 4 blocks respectively and also the Normal Force and put the in this chart below.

The graph below shows the resultant coefficient of maximum static friction and as you can see, the second and third data points were a little off.

2) After using the force sensors we came to the this graph below and using the cleaner data of when the force looks like it is almost constant, we were able to find the kinetic friction force.

4) Below is the graph of the resulting kinetic friction force used when using the motion detector when letting the block slide down the steep ramp.

5) Below is the data table that the motion sensor acquired for the the acceleration of the system when using the block-weight pulley.

The graph below is the resulting acceleration of the system that we found experimentally.

Calculations:
1) The calculations below were from the first part of the experiment were we used the block-water pulley system to find the coefficient of static friction. The coefficient of static friction was 0.406

3) Below is the calculation that we used to find the static friction of the block at an incline when we were adjusting height. The value is 0.364.

4) The calculations below were from the fourth part of the experiment and it shows how we got the kinetic friction of the experiment when on a steep inclined ramp. The value of the coefficient of static friction is 0.355.

5) The calculations below show how we came to the acceleration of the system of the block-weight system. The theoretical value we got is 0.565 m/s^2 and the experimental value was 0.6483 m/s^2.

Summary: We started by using the block-water pulley system to find the coefficient of static friction between the block and the table and graphing the force of static friction with the normal force of the block  to find a relationship. The relationship was the coefficient of static friction. For the experimental value we got 0.3285 and for the calculated value we got 0.406 which is +23.6% off. This number is pretty big but the reason for it was because we did not put the exact amount of water for maximum static friction since it requires for the object to remain still and in order for us to calculate it we have to let the block slide a little. It was also very difficult to get the block still having it moved for a little bit. 

Then we used the force sensor to calculate the kinetic friction of the experiment using the force graph of the recorded data. After that we found the coefficient of static friction of the wooden block on top of an inclined ramp. Sine it was a different surface we got 0.364. Next, we used the acceleration found from the block sliding down a steep ramp with the motion sensor to calculate the coefficient of static friction for the ramp. We got 0.355. Finally we predicted the acceleration of the system by using the previous coefficient of static friction. Our theoretical value was 0.565 m/s^2 while our experimental value was 0.6483 m/s^2. The percent error of the theoretical value is -12.8%. The reason being is because we can not get perfect reading from the machine and we also started putting stuff away before we knew we had to do this part of the experiment. We may also have gotten a different block which affected the result. Overall though, this lab was successful in showing us how the different friction forces affect objects on different surfaces. 

Projectile Motion Lab

Purpose: The purpose of the lab was to further use our understanding of projectile motion in order to predict the impact of a ball on an incline board.

Procedure: In order to carry out this experiment we needed a ruler to measure the distance the ball traveled, a small metallic ball as our projectile, a ramp with a ring stand so that the ball can have a certain initial velocity when launched, and carbon paper with a clear sheet of paper so that we can see where the ball has landed. First we set the ramp at an incline and the second one horizontal so that the speed of the ball can be measured. Then we did some tests to estimate how far the ball will launch and place the carbon paper accordingly. Next we measured the length of how far the ramp is and also how far the ball landed. We repeated this experiment 5 times to get the distance so we can get an initial velocity so that we could use it for the second part.

For the second part we pretty much did the same thing except we took a wooden plank and placed it at an incline. Using our data from before we predicted where the ball will land numerically and then used the experiment to see how far the ball actually landed. We used an angle measurement device to acquire the angle and assumed we knew final velocity. 

Data: The bottom diagrams show the measurements we acquired from the experiment.

Calculations; Below are the calculations made by our group. We found the initial velocity derived by using kinematics from the data from our first trial. Then we used that velocity to find the predicted distance that the ball will travel. Finally, we calculated the percent error of our values and got a 13.2% percent difference.

Summary: Overall, we used kinematics in order for us to predict the distance a ball will travel and then the distance a ball will travel on an incline. We set up ramps so we could give the ball an initial velocity and then calculate a value in order to predict where the ball will land on an incline. The value of our theoretical distance was 0.317 m and the experimental value was 0.28 +/- 0,01 m. The percent error between both values was 13.2%.
We were pretty close but some factors were present that made our data collection a little off. Sometime the ramp moved from the original spot and that might have changed the initial velocity which in turn affects the displacement. There was also human error in recording values because we can not read a ruler perfectly. We also shifted the experiment a little because we assumed we were done but had more data to acquire. The experiment was a success because we were able to closely predict the outcome with only a small window of error.

The Rocket-Powered Elephant

Purpose: Our goal was to solve a non-constant acceleration problem and to compare our calculations to the calculations made in excel.

There was no equipment used other than Excel in order to compare values to what was acquired via our own manual calculations.

Procedure: First we solve the problem by hand as shown in the picture below. The distance the elephant traveled before coming to rest was 248.7 m.

We then decided to find the distance by inputting information into Excel using the following information below. The whole class partook in this method.

Below is the graph with the information mentioned previously with time intervals of differing by 1 seconds each.

The graph below is the same one as the one on top except with the time intervals differing by 0.01 seconds.

The graph below is also the same one as the previous two except with time intervals differing 0.05 seconds each.

Summary: The results were fairly close to the ones found on our calculations. The only problem we as a class ran into was solving the problem by hand. We had to use the help of Wolfram-Alpha in order to finish our problem for we did not have sufficient knowledge to finish a problem that complex. Overall it was a fairly easy lab and the experiment was a success.

Questions
1) The results were fairly the same and not much difference could be found between the two.
2) In order for us to tell if the time interval is small enough is when we could see a really small difference between the change in distance and we could come to the conclusion that the answer becomes constant after the really small change. We knew if we made an error if we did not see the distance becoming constant or we went to  really far interval which did not physically make sense.

Coffee Filters and Air Resistance

Purpose: The purpose of the lab was to see a relationship between the air resistance force and speed of coffee dropping from a balcony.

For this experiment we used coffee filters, a 2 meter ruler, and video capture that was accessed through our lap top cameras. The coffee filters were used to measure the air resistance while the ruler was used as a scale when recording data from video capture. Video capture, overall, was used to get data from the coffee filters being dropped.

Procedure: First, we went over to the Design Technology Building in order to try and minimize the effect of drag when someone opens to door. One of us was then positioned atop of the balcony with the 2 meter ruler and the coffee filters while the other person was on the other side on top of the stairs recording the filter drop. We did this for 1, then 2 and all the way to 5 coffee filters on top of each other. We are doing this in order to find the terminal velocity which indicates that the gravitational force and the air resistance force equal each other.

The picture below shows the videos we captured during our data collection and the graph shows our data points acquired through the video capture. The slope near the end of the curve signifies the terminal velocity reached by each of the coffee filters. We need the terminal velocity in order to graph the Position vs Time. This process was done 5 times for each of the coffee filters used.

Below is the graph of Speed vs. Force that is used to determine k and n in the equation Air Resistance Force = kv^n. This graph was not used though because the third data point was really off.

Below is the new graph used for finding out the actual values of k and n. The third data point was not included because we thought we would get a better correlation if we excluded it. We got 0.007 for k and 1.793 for n.

Calculations: In order to verify that we were able to get the correct values, we plotted the values of k and n into excel in order to find the final velocity. The results proved to be the same so it was accurate.

Below are our calculations that we did for the experiments.

 Summary: First, we dropped the the coffee filter from the top of the balcony so that we could gather the data in order to find the terminal velocity. This was done by capturing videos of the filters falling. Then we found the final velocity by graphing the Position vs. Time for each of the coffee filters. Then we used the terminal velocities to graph the Speed vs. Force which gave us the k and n for the equation Air Resistance Force = kv^n. We then used Excel in order to verify our k and n values which proved to be correct. Overall, this lab proved to be successful since we came to the same results and there could be some uncertainty because interpreting the data can not be 100% accurate. All in all, the air resistance force is directly related to the speed as proved by the lab results.

Sparker Free Fall Lab and Standard Deviation of Data

Purpose: The purpose of the lab was to see if gravity was actually 9.8 m/s^2 ignoring all other external forces. It also served as a way to use Excel as a means of recording and displaying data effectively.

We used a machine called a sparker generator which records the location of an object in free fall on very thin sparker paper in order for us to observe the location of where the object is at specific time intervals. An object is first held on top of the electromagent and then released and at that moment the sparker generator creates an imprint on the sparker paper with accuracy in order for us to calculate acceleration.

Purpose: In order for us to get the data in order to get gravity as close to as 9.8 m/s^2, we have to use the sparker generator with sparker tape to measure the distance between the markings it will generate for us. After having dropped the object, the sparker tape will have imprints . These imprints are used to measure the distance the object dropped. We then grabbed a ruler in order to measure how far apart the imprints were from each other. This gave us the displacement of the object which will be used to calculate the velocity of the object as well as the acceleration which turns out to be gravity. 

Data & Calculations: After having found the values, we put them into an excel spreadsheet (all calculations were done in Excel). Knowing that the sparker generator creates imprints every 1/60 of a second, the distances between points are going to gradually increase since the acceleration of gravity is taking effect. We then calculated the mid-interval time between each imprint or point using the formula (time) + 1/20 s, the mid-interval speed between each point by using the formula (distance between each point) / (1/60 s).

This table represents the relationship between distance and time using a quadratic fit because in order for there to be acceleration, there needs to be an exponential change in distance. We were able to get such a result because as the object drops, the distance changes due to the acceleration of gravity.

This other data table shows the relationship between the mid-interval speed and the mid-interval time. It allows us to see that the speed of the falling object has a slope which means that there is an existing acceleration. We used a linear fit because change in speed is gradually increasing which shows us that there is an existing acceleration.

This other data table is the standard deviation of the whole class' results in order to compare how close we came to the constant of gravity as a whole. The result was about 955.3 cm/s^2 which is pretty close. Our result was 949.3 cm/s^2 which is number 7 on the graph.

Summary: This lab was successful because even though we did not get the actual 9.8 m/s^2 we were really close. We did find a pattern within our data points and they do make sense because in order for there to be acceleration, the correlation of distance has to quadratic and the correlation of speed has to be linear. Our result was 9.49 m/s^2 which was -0.31 m/s^2. The error percentage was approximately -3.16 % which in fact is relatively not too far off. We did not get these results because we have to consider the fact that we are humans so our data collection is not the most accurate. Another factor that prevented us from getting that result was air resistance and the machine that recorded the object falling is not going to be 100% accurate because no machine is perfect either. Overall, after collecting the data from the sparker generator and using it with Excel to find the acceleration of gravity we can say that this lab allowed us to see measure how fast objects fall by finding the constant of gravity.