September 29 2014: Angular speed for a particular rotating apparatus

Purpose: To find a relationship between angular speed(w) and angle created by the string relative to the vertical on a particular rotating apparatus.

This is the rotating apparatus that the class used to calculate the rotations and the time for this lab. There is a string attached to one of the ending of this apparatus, where a weight is located, and it was our reference.

First, we took some time to calculate the measurements of this apparatus such as height, length of each arm, length of the string, etc... Then we calculated the time of 10 rotation at 8 different speeds and we are going to attempt to find the angular speed for each trial by utilizing the given equations and using a computer to calculate the values for us.

This is the model that our professor drew on the board for us to use as reference of what measurements to take from the apparatus.

We drew our own model of the apparatus, and as possible to see, we have the measurements of each relevant part of the apparatus, and we are going to solve for h2, and theta according to each trial.

This is the picture of the values that we got for each of the eight different trials that the class did. Each time was taken for 10 rotations of the object.

 These equations were the ones that we found relevant to finding the angular speed for each trial. From those equations, we expected to get numbers to create a graph for the lab.

In order for us to perform the calculations easily, instead of finding each angular speed for each different trial by hand, we plugged the numbers in excel and created formulas to calculate the values faster.

This is a graph of w(rad/s) vs. f(theta). Our expectation was to get the slope close enough to 1, which would show that our calculations for the angular speed would be accurate, we got 0.9987, a number really close to 1.

Summary: First, we got the measurements of each relevant part of the rotating apparatus. Then the professor turned on the apparatus and we timed 8 different trials at different speeds. We drew a model of the apparatus with our measurements and tried to find a equation where the angular speed(w) would be a function of theta. We got w^2=g*tan(theta)/d+l*sin(theta) to be our equation. We used each different trial to get a different value of w, and we plugged those numbers in the computer. After that we used the equation w=2*pi/time to calculate the angular speed(w(rad/s)) relative to the time for each rotation. We also plugged this numbers in the computer and created a graph w(rad/s) vs. w=function of theta. The slope of that graph had to be close to 1 to indicate that our calculations were accurate, and we got 0.9987 which is considered close. This showed that our calculations/measurements were good.

September 24 2014: Centripetal acceleration as a function of angular speed

Purpose: To find the centripetal acceleration as a function of the angular speed of the system.

This is a picture of the apparatus, which is an accelerator with the disk, the picture is not clear because our group was not close to the professor's desk, and we forgot to take a closer picture.

The whole class did this lab together, each group got a stop watch, and the professor set up an accelerator with a disk that rotates. Each group was supposed to find the time in which the disk did four rotations. We did this five times, with different speed in each trial. With the average of times, we attempted to reach to a centripetal acceleration by plugging our data into the computer and creating a graph.

This is a graph that the professor obtained when plugging the data in his computer, we got this graph because he posted in moodlerooms. 

This is the average of time that the class had for the different speeds when executing the lab.

This lab was more of a consensus of data information between the groups. Each group had to time four rotations, and of course, this lead to human errors plus uncertainty. Another factor that contributed to errors was that some groups did not calculate the time for the rotations. With the average from the groups who timed the lab, we created a graph and from it we were able to calculate the centripetal acceleration of the system, showing that exists a relationship from the angular speed and the centripetal acceleration.

Sep 17 2014 - Modeling friction forces

Purpose: Utilize wooden blocks to find the friction forces acting on the blocks in various situations, and also calculate the acceleration of the system.

This is a motion sensor, which is the apparatus that we used in order to get our data into our computer, such as position, time, velocity, etc.

This is a force sensor, which is the apparatus that we used to calculate the kinetic friction acting on the wooden blocks as we dragged them.

Explanation: First we are going to find the static friction force acting on the wooden blocks by attaching a string and a pulley connecting the blocks and a cup filled with water. After that, we are going to connect a force sensor to the wooden blocks to calculate the kinetic friction that acts on the blocks. Then we are going to build a ramp and find at which angle the blocks start to slide so that we can calculate the static friction of that system. After finding the static friction, we are going to increase the angle of the ramp so that the block slide with constant speed so that we calculate its kinetic friction. The finalize, we are going to create a system using weights, and a pulley that will pull the wooden block from the bottom of the ramp to the top of it, and by doing this we are going to calculate the acceleration of the system.

This is the system that we created using the wooden block and the cup of water to find the static friction.

 This data table shows the friction force obtained from, 1,2,3,4 wooden blocks respectively. We used those numbers to create a graph where we would find the coefficient of the static friction.

This is the graph from the data table above, as we can see, points 2 and 3 are a little off, probably due to uncertainties and irregular surface.

This is a graph obtained from using the force sensor to calculate the kinetic friction of the wooden blocks, we did the same calculations for all 4 blocks.

This is the ramp that we created to find the angle which the block starts to move and find the static friction. We also used this ramp to calculate the kinetic friction by increasing the angle of the ramp.

This is the graph that we used in calculating the kinetic friction for the angled ramp.

 This is how we calculated the acceleration of the system, as possible to see, the weight on the right will pull the wooden block if I stop holding it.

 This is the data table from the system that we run to calculate the acceleration.

This is the graph of the system used to find the acceleration obtained by the wooden block being dragged.

 These calculations were used in the first part of the lab, where we utilized the cup of water to find the static friction of the system. Notice that part two did not require calculations, because all we did was use a force sensor to find the kinetic friction.

 These calculations were used in the third part, where we found the the static friction of the system by creating a ramp inclined with certain angle.

 These calculations were used in the fourth part, where the increased the angle of the ramp in order to find the kinetic friction of the system.

The handwritten part of top of this picture are the calculations used for us to find the acceleration in the last part of the lab, where weights pulled to wooden block from an inclined ramp.

Summary: We started this lab by using a cup of water attached to 1,2,3,4 wooden blocks to find the static friction of them. By using those numbers in a data table, we could find the coefficient of static friction in the system. We then, attached the wooden blocks to a force sensor and by using those numbers in the computer, just like the first part, we were able to find the coefficient of kinetic friction. After this, we created a ramp with certain angle, where the block would start to move, and that would give us the maximum static friction in the ramp system. When the angle and friction were found, we increased the angle significantly, such that the wooden block would slide with certain speed, and from that we were able to calculate the kinetic friction in the ramp system. To finish the lab, we attached a string connecting some weights and the wooden block in order for us to find the acceleration in which the block was pulled by the weights.

Sep 15 2014- Measuring the Density of Metal Cylinders and Mass of Unknown Obejct

Purpose: Calculate the density of 3 different metals and the mass of an unknown object, and calculate the uncertainty of those measurements.

This is a caliper, which we used to find the diameter and the height of the three metals. We also used a regular scale to find the mass of those three metals.

With those calculations, we plan to find the density of each metal from the equation density=mass/volume, and by using a given equation dP/P = dm/m + dh/h + 2dd/d where dm, dh, and dd are the uncertainty of each measurement, we are going to find the uncertainty of the density, which is dP. The scale has uncertainty of 0.1g and the caliper has uncertainty of 0.01cm. After that, we are going to analyze an unknown object hanging by two springs and by calculating the tension and angle of each spring we are going to determine the object's mass.

This picture shows the calculations of the three metals: aluminum, bronze and copper. We found the uncertainty of the density for each one of them.

This picture is a continuation of the calculations for the three metals. The lower part of the picture shows the calculations made in order for us to find the mass of the unknown object.

This is a picture of how the unknown object was hanging. Unfortunately, I don't have a picture of the actual object, so I used the drawing from the lab sheet to demonstrate the scenario that we analyzed.

Summary: First, we used a caliper and a scale to find the height, diameter and mass of the three metals: aluminum, bronze and copper. Because we were given the uncertainty of the caliper and the scale, which were cited above in the brief explanation, we were able to plug those numbers in the equation which was also cited in the explanation. From that, we were able to find the density of those metals and the uncertainty of that density value. After that, we analyzed a hanging object and by calculating the tension and angle of each spring holding that object, we could find an estimate of the object's mass.

10 September 2014- Trajectories

Purpose: To use our understanding of projectile motion to predict the impact point of a ball on an inclined board.

 This is the set up of order equipment to perform the experiment, where the metal ball land somewhere in the carbon paper on the ground.

This is our second set up, where the ball will land somewhere on the inclined board and we will predict the distance which the ball traveled.

We are going to build a trajectory where the ball will land somewhere on the ground, where a carbon paper will be taped for us to know where it landed. By calculating the height of the table and the distance where it landed, we are going to find the ball's velocity when it is released from the trajectory. After this, we will attach an inclined board and estimate where the ball will land on the board.

These are the calculations our group did in order to find the ball's velocity when it is released from the trajectory, and the estimated distance where it will land on the board.

First, we built the trajectory in which the metal ball would travel. We calculated the height of the table to the ground, as well as the distance that the ball landed from the table on the carbon paper. We did this experiment about five times, to have a good estimate of the average distance. Using this information, we used kinematics to calculate the speed of the ball at the moment when it goes off the table. We then attached the inclined board to our equipment and calculated the degree which the board was inclined. From that, we used the equations: dcos(theta)=Vot and dsin(theta)=gt^2/2 to find the distance that the ball would land on the board.

Our experimental value gave us a distance of 0.28 meters on the board.

The theoretical value for d which we found was 0.317 meters on the board. Our theoretical value was .037 meters off. The relative difference was -11%. The uncertainty or error between the numbers could be human errors, natural errors, or air resistance acting on the ball during the period which it is in free fall.

September 10 2014 - Modeling the fall of an object falling with air resistance

Purpose: Determine the relationship between air resistance force and speed with air resistance by modeling the fall of coffee filters from a balcony.

The picture shows a 2 m ruler to set the height/distance for our experiment; and a coffee filter, which will be our object thrown from the balcony.

First, we are going to record our experiments dropping 1,2,3,4,5 coffee filters from a balcony in order to get our data with LoggerPro. As we drop the coffee filters, we are going to have the 2m ruler on the top of the balcony to set our parameters of the distance which we are from the ground. With the data collected, we will determine the final velocity of the object by using LoggerPro to set a graph of position vs. time. With the final velocity, of each experiment, we are going to create a graph of speed vs. Force of each coffee filter to determine k and n from the equation F=kv^n. by finding those values, we should be able to determine the relationship of air resistance with speed.

 This picture shows the videos of our experiments on the left, and on the right is possible to see a graph of position and time, the slope determined the velocity of the object falling. We did this 5 times, each one representing the quantity of coffee filters that were dropped.

 This picture shows the Speed vs. Force graph of all five experiments. As shown, the experiment using 3 coffee filters was off the expectations when compared to the other 4 experiments.

In order for us to have a more reasonable value of k and n, we didn't include the experiment using 3 coffee filters in our final graph of Speed vs. Force as seen in the picture above.

 To make sure that our result of final velocity was accurate, we did the solution numerically by using the values of k and n obtained in the graphs above. The results were accurate, meaning that our calculations were good.

This picture shows the calculation of the force in each of the five experiments that we did.


Summary: To start our lab, we practiced dropping the coffee filters in class, just to get an idea of how LoggerPro works. After that, we went to the Design Technology building and from its balcony, we did the execution of our lab by dropping 1,2,3,4,5 coffee filters and recording with the computer. After recorded, we used LoggerPro to plot a graph of distance vs. time for us to find the final velocity of each experiment with the coffee filters, and we did calculations to find the force of air resistance in each of coffee filters. We used the equation F=kv^n as standard, and after making a graph with speed vs. force, we were able to find values for k and n. With that equation, we found the relationship between air resistance force and speed. To make sure that our results were accurate, we found a solution numerically, which was close enough to our solutions analytically, showing that our calculations were accurate.

September 8th 2014: Non-constant acceleration problem

Purpose: Solve the elephant problem, given in class, numerically, as well, as analytically.

We did not use any apparatus, the only equipment used was Microsoft Excel, for us to compute the data information.

The class solved this problem with the professor, as a whole group. We followed the steps of the professor to reach to an answer analytically, then we solved the problem numerically by using Microsoft Excel to perform the calculations. This allowed us to know at what time and what distance the elephant reached a complete stop.

This table shows the time interval of 1 second between each distance taken.

This table shows the time interval of 0.1 seconds between each distance taken.

This table shows the time interval of 0.05 seconds between each distance taken.

The problem analytically solved in class.

The time interval and distance that the class decided to use for parameters in Excel.

Summary: We had a problem where an elephant in movement had to stop before it reached a cliff. With the numbers given in the problems, the class as a whole was able to reach to a conclusion analytically. To make sure that our results were accurate, we used Microsoft Excel to perform numerical calculation and see if our results were reasonable.

Questions

1) The results were close to being the same.

2) The time interval would be small enough when the distance was close to being the same, or when the distance became to show up negative. This would mean that the elephant reached a complete stop.

3 Sep 2014: Determination of gravity and learning some features of Excel

Purpose: The purpose of this lab is to get as close as possible of the real value of gravity "g" as well as leaning a little more of the capabilities of Excel when computing data.

This apparatus is called spark generator, which is a electromagnet with power supply that give us some data to conclude our project. It provides us marks on a spark-sensitive tape at specific intervals so that we have records of those points, which are the distances of the object dropped as it reaches the ground.

With the data of the distances given by the apparatus, we are going to use a ruler to calculate each distance from the origin. Each mark has a time interval of 1/60 of a second, and as we all known, those distances are going to increase as it reaches the ground because gravity is acting on it. Then we calculated the distance between each point, the mid-interval time for each point with the formula: (time)+1/120, and the mid-interval speed with the formula (distance between each point)/(1/60). We compute all this data in Microsoft Excel, creating a table, and with that we created two graphs: one for Mid-interval speed vs. mid-interval time and one for distance vs. time. The slope of the graph Mid-interval speed vs. mid-interval time gave us an approximation of what is gravity.

 This table is our collection of data, each column has a title of what those numbers mean.

 This graph is the distance vs. time, its was originated with the data from columns A and B from the first picture.

This is the graph of Mid-Interval Speed vs Mid-Interval time, originated from columns C and D from the first picture. The slope of the graph is the approximated acceleration of gravity.

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This table is the collection of data from all groups in our class, our group number is #7. From this table, we could compare our results to the other groups.

Summary

First, we used the spark generator apparatus to collect the data needed for us to calculate each distance point. From that, we went to the computer and typed a table with our data plus the formulas given in the lab sheet. We were able to generate a two graph with our data information, one containing distance vs. time and another one containing mid-interval speed vs. mid-interval time. From those graphs, we found their slopes to reach an approximation of acceleration due to gravity. Then, we compared our results with other groups to see if our data was accurate.

Questions

1) For constant acceleration, the velocity in the middle of a time interval is the same as the average velocity for that interval because the both will become a function of time, and the time will be the same.

2) The acceleration due to gravity can be obtained from our velocity/time graph because it will be the slope of that graph.

3)To find the acceleration due to gravity of our position/time graph, we find the slope and multiply by 2.

Our results were a little off when compared to the real value of gravity = 9.81 m/s^2. We got 9.49 m/s^2 which was not far from it, but not so close. Some reasons for that could be systematic errors that interfered in our conclusions and also human errors, which are errors that happen from us being human beings; that is, when we calculated each distance point, they might be a little off. Another reason could be air resistance when the experiment was executed, leading to errors that affected our final result in finding the acceleration of gravity.

Difference between expected and experimental values

Relative difference:  (9.49-9.81)*100/9.81 =  -3.27%

Absolute difference: 9.49-9.81= -.32 m/s^2