September 21st, 2016: LAB 7: Modeling Friction Forces

Michell Kuang
Lab partners: Charles, Anthony
Date performed: 09/21/16

This experiment is essentially split into 5 parts. All of them experiment with static and kinetic friction but in slightly different ways. Using LabPro, we can record data and create graphs to find the coefficients of friction.

(1) We started with just one block that had a felt side (so as to make the movement as smooth as possible). This block has a mass of 190g. On the other end, we attached a string and put that over a pulley that was at the end of the table. At the end of that string, we attached a mass hanger (50g), and slowly added more mass to it, until the block started to slide. That mass at which the block started to slide will be the value we will use to calculate the max of static friction.

Setup of block with pulley on table edge and string with hanging mass

After that, we added another block on top of the first one. This second block has a mass of 135g. We did the same thing to test when the block would start to slide, and we repeated this with three and four blocks.

On the left: Data for first experiment

On the right: Data plotted onto Logger Pro to make a graph

(2) With this experiment, instead of a pulley and mass hanger, we attached to one end of the block a force sensor. This force sensor was connected to a laptop, which recorded data onto LabPro. This time, we determined kinetic friction by pulling the force sensor at a steady, constant speed, recording the speed onto Logger Pro with which we can find the kinetic friction. Like before, we did this 4 times, each time adding a block. The masses of the block are the same as in part 1.

Setup of two blocks with force sensor attached on wood cutout (smooth surface)



Logger Pro view: Measurement of force for each run (1, 2, 3, and 4 blocks)

Logger Pro view: Re-do of run 2 (2 blocks)

Logger Pro view: Data plotted to get coefficient of kinetic friction

(3) In this part,we used the block of mass 190g. Nothing is attached to the block. Instead, we rest the block on our horizontal surface, and slowly lift the wood cutout until the block starts to slip. We then recorded the angle at which the block began to slip, and used that to determine the coefficient of static friction. Our angle came out to be 22 degrees.

Setup of block on wood cutout being raised, with iPhone taped on to measure angle

Calculations for static friction

(4) In this fourth experiment, we kept the wood cutout inclined at an angle steep enough so that a block resting on it accelerated downwards. Again, we measured the angle of the incline to determine the coefficient of kinetic friction. In addition to that, we attached a motion detector to the ring stand holding up the wood cutout, and connected that to the computer so that we could use Logger Pro to measure the acceleration. We kept the board inclined at an angle of 32 degrees, used the 190g block, and our acceleration came out to be 2.729 m/s^2.

On the left: Logger Pro view: Measured velocity and acceleration of block sliding down incline

On the right: Setup of inclined board with motion detector

Calculations for kinetic friction

(5) Our last experiment for the lab is similar to the setup of the first part, except this time, instead of recording the added mass at which the block began to slip, we used a motion detector to record what the acceleration of the block was once the block started slipping from too much added mass. Then, using our results from part 4, we derived an expression for what the theoretical value of the acceleration should be. Once again, we used the mass of 190g.

Setup of block on horizontal surface with force sensor and pulley with hanging mass

September 14th, 2016: LAB 6: Propagated Uncertainty in Measurements

Michell Kuang

Lab partners: Charles, Anthony

Date performed: 09/14/16

In this lab, we learned how to calculate propagated error in density. To do this, we took measurements of three different cylinders that had different masses, and used this data to calculate the densities.

To measure the diameter and height of these cylinders, we used vernier calipers. To measure the masses, we used a scale.

Examples of measurements using vernier calipers

Data of measurements and calculation of densities

From this lab, we were able to learn how to use vernier calipers to take measurements. In addition to that, we were able to use our data to calculate various densities.

September 19th, 2016: LAB 5: Trajectories

Michell Kuang
Lab partners: Charles, Anthony
Date performed: 09/19/16

In this lab, we created a set up to experiment with an object in projectile motion. Using our knowledge of projectile motion and our experimental results, we were able to predict the position of the object when it hit an inclined board.

To obtain experimental data, we launched a steel ball from an inclined ramp that we created with an aluminum "v-channel" and ring stand. After trying that a couple times and examining the approximate location where it landed, we taped a piece of carbon paper onto the floor at that spot. Then, we launched the ball five times to verify that it landed at about the same spot every time. At the end of the table, we hung a plumb bob to mark the "initial position" to help us determine the distance from the table's edge at which the ball landed.

 

On the left: Five impact points marked by carbon paper

In the center: Inclined ramp setup

On the right: Plumb bob

Experimentally, we measured a horizontal distance of 48.7cm +/- 0.1cm. With the help of the plumb bob, we measured a vertical distance of 95.5cm +/- 0.1cm. Using this information, we found the time it took for the ball to hit the ground and the launch speed.

Calculations to find time and launch speed

Now that we had the launch speed, we were able to repeat the experiment, this time with an inclined board at the edge of the table. We measured the angle at which the board was inclined, and once again, taped a piece of carbon paper to the appropriate position at which we thought the ball would hit the board. We did this 5 times, and measured our experimental distance d along with the uncertainty at which the ball landed.

    

On the left: Setup with inclined board placed at edge of table

In the center: Five impact points marked by carbon paper

On the right: Angle of inclined board measured with iPhone 

Our measured value of d came out to be 0.44m +/- 0.05m. We then did the calculations to see how our theoretical values compared to our measured value.

Calculations to find theoretical value of d

Our values turned out to be accurate, and in comparison with our measured value, our theoretical value was identical. By repeating the experiment several times, we were able to decrease our chance of error and thus got this result.

September 12th, 2016: LAB 4: Object in Free Fall with Force of Air Resistance

Michell Kuang
Lab partners: Charles, Anthony
Date performed: 09/12/16

PART 1:
In the first part of this lab, we experimented with an actual falling object, using logger pro and video capture to help us get our data. We do this to determine the relationship between the air resistance force and speed.

Using the video capture feature in logger pro, we recorded 6 runs of a falling object (we used coffee filters). The first run had 1 coffee filter, the second had 2, and so on. There is a feature on logger pro that allows us to track the movement of the coffee filter on video by marking its position frame by frame. We set the 1 meter mark, and then did this with every run. Then, with our recorded data, we again used logger pro, this time utilizing its graph and equation fit functions, and applying a linear fit for the position vs. time graph. 

Logger Pro view: Run with 1 coffee filter; data of time and position and graph with linear fit

The slopes given in the statistics boxes of our graphs from each run are our estimated terminal velocities. We took these values of each run and plotted them onto a new logger pro file. Then, we formatted it so that our variables fit appropriately with the function we want, and we got our values of k and n from the fit for the graph. The uncertainties are also given.

Logger Pro view: Plotted data with graph; Value of A is k and value of B is n

PART 2:

In the second part of the lab, we charted our data on the lab following the model of a power law: 

Fresistance = kv^n

Following this model, we are able to predict the terminal velocity of each run. Along with some constants, we had columns for various variables including time, velocity, position, and more.

Excel sheet of plotted data to find terminal velocities

A lot of the data on my lab is not consistent with the data shown in certain pictures because we messed up and didn't correctly label our videos. To produce data with runs of varying number of coffee filters, we used some of another group's videos. The data consequently became inconsistent and incorrect.

September 12th, 2016: LAB 3: Non-constant Acceleration

Michell Kuang
Lab partners: Charles, Anthony
Date performed: 09/12/16

In this lab, we are given a non-constant acceleration problem to solve. The analytical, step-by-step process and solution are also given in the manual, but our goal is to solve the problem with the help of excel.

By using excel, we essentially split the problem into its variables by column. Then, all we had to do was write out functions for excel to solve, and get the answer. The analytical approach gets x=248.7.

We first set up a few constants, and below those, we have our variables that we need to fill. We have a column for time, acceleration, average acceleration, change in velocity, velocity, velocity average, change in position, and position. We start with ∆t=1. The time at which we want the position is approximately 19.69 seconds (solved with analytical approach).

Excel sheet with ∆t=1

As one can see, our value of x with this numerical approach is fairly accurate. However, we will vary ∆t to see if it will make a difference. First, we changed it to 0.1 seconds, and then to 0.05 seconds.

On the left: First half of excel sheet with  ∆t=0.1

On the right: Second half of excel sheet with  ∆t=0.1

Again, the value of x that we get with this variation is still fairly accurate compared to the answer from the analytical approach. In comparison to our values obtained with ∆t=1, the smaller variation of ∆t gives us a more precise value. Having looked over the steps of the analytical approach in the manual, one can tell that it can be very tedious to solve, and there are a lot more chances for errors in the problem solving. With excel, we save a lot of time. 

If you wanted to know if your value of ∆t was small enough without knowing the analytical approach's result, you can just look at your values of x. Looking at the excel sheet with ∆t=0.1, you can see that when the x values reach about 248, there are a lot of times at which the answer is that. So to make sure that your time interval is small enough, check to see if you can spot out the value of x at which it repeats (so it stands out). The values of x will begin to decrease after that point. Finally, we will practice one more time using excel to solve a similar problem with different given values.

On the left: First half of excel sheet with M0=7000, b=40, F=-13000

On the right: Second half of excel sheet with M0=7000, b=40, F=-13000

September 7th, 2016: LAB 2: Free fall and learning about excel; Error and Uncertainty (Make-up, joined class late)

Michell Kuang
Lab partners:
Lab performed: 09/07/16

PART 1:
In this lab, we test to see if the acceleration of an object in free fall will equal the acceleration of gravity (9.8 m/s^2) if no other forces acted on it. To do this, we used an apparatus from which the object was released and a strip of spark paper that recorded the data.

Spark generator apparatus with tripod base and spark paper strip

Every 1/60th of a second, the spark generator left a dot on the spark paper. After recording a good amount of data points, we lined the spark paper up with a meter stick, and marked the position of each dot.

Dotted spark paper aligned with meter stick (positions recorded in cm)

We took this data along with some other information and entered it onto an excel sheet. Because we want to record the time in intervals of 1/60th of a second, we enter the time in cell A3 as "=A2+1/60", A2 being the initial time (0 seconds). The equal sign makes it so that what we enter into the cell becomes a function. Then, instead of doing this with every cell going down column A, we can use Excel's fill function. This will automatically fill the column with the corresponding cells (so in cell A4, "=A3+1/60" will be entered, and so on).

On the left: Excel spreadsheet with data for time, distance, delta x, mid-interval time, and mid-interval speed

To continue, we made two graphs: one of column D versus column E, and another of column A versus column B. For both graphs we chose a corresponding graph fit and displayed the equations and R-squared values.

On the top: Graph of mid-interval time v. mid-interval speed with linear fit

On the bottom: Graph of time v. distance with polynomial fit of order 2

PART 2:

In the second part of the lab, we learned how to calculate the uncertainty in our data whilst implementing our learned knowledge of excel through analysis of the class' data (from part 1). The lab manual goes over in detail how to find uncertainty, but there are two ways.

The first way is called taking the average deviation of the mean. With this method, you take the average of the absolute value of all the deviations from the mean to make the values positive. The second way, which we used for our data in this part of the lab, is to square them, average them, and then finally, take the square root.

Formula used to analyze class data

In excel, we create three columns. The first will have the class' values of g and the average of those values, and the second will be the deviations from the mean. The third we had to construct ourselves to find the uncertainty, so we finish the formula by squaring column two's values, and finally taking the square root of the average of those values.

Excel sheet with class' g values and the final calculation of the standard deviation of the mean

September 7th, 2016: LAB 1: Finding an unknown mass by deriving a power-law equation for an inertial balance (Make-up, joined class late)

Michell Kuang
Lab partners:
Lab performed: 09/07/16

With this experiment, we try to understand the relationship between mass and periods of oscillations by varying the masses with which we use to create a rate of oscillation. The periods are recorded with an inertial balance and photogate.

Instead of using the conventional balances, we used an inertial balance to measure masses of various objects. In doing so, we can find the inertial mass of an object, instead of the gravitational mass, and notice the type of relationship that is formed between mass and period. Additionally, through this method, we can record oscillation periods with which we can form a power-law equation. With that equation and our known variables, we can solve for an unknown mass.

Inertial balance setup; photogate apparatus on left to record data onto computer

We start by varying the masses we set on the tray (0g, 100g, etc.) and recording each of the oscillation periods with Logger Pro. This proves that there is, in fact, some kind of relationship between mass and period.

On the left: Recorded data of all our runs with varied masses

On the right: Logger Pro view- run with 800g of mass on tray

Given in the lab manual is the equation: T= A(m+ Mtray)^n. By taking the natural log of each side, we get: lnT= nln(m+ Mtray). We plug in our known data into Logger Pro (mass and period), and create equations for the unknowns. In the end, our only unknown variable is Mtray, whose parameter we vary in attempt to find a correlation coefficient as close to 1 as possible (this will result in a nice straight line for the linear fit of the corresponding graph). Since it would be very tedious and difficult to find an exact value of Mtray, we instead find a range for what it could be. We find that our Mtray must be between 325g and 375g. The corresponding values of y-intercept and slope are recorded for later use of the power-law equation.

On the left: Logger Pro view- Data of several variables and corresponding graph with linear fit

On the right: Recorded values of Mtray min and max, along with corresponding A and n values

With our newly found data, we are able to move on to the extension of the lab in which we must find the masses of two unknown objects (we chose a partially empty water bottle and a cellular phone). In this part of the experiment, we record the oscillation periods for the two objects in order to calculate the inertial mass. Using our equations, we find a range of values for the inertial masses of the objects. We then record the gravitational masses of the objects (obtained by measurement on a scale), and compare those values with the ones we calculated.

On the left: Unknown object #1 (water bottle) on tray of inertial balance apparatus

In the center: Logger Pro view- run with unknown object #2 (cellular phone) on tray

On the right: Recorded data of both unknown objects

On the top: Calculations of min and max inertial masses of unknown object #2 (cellular phone)

On the bottom: Calculations of min and max inertial masses of unknown object #1 (water bottle)

The comparison of our results with the measured gravitational mass show that there was some error produced during the process of this lab. I believe this could have been an error in data collection or calculation. The calculated values of mass for the unknown objects are negative, which is immediately incorrect because our objects can't have negative masses. I initially thought that it was simply a syntax error because my results for the masses of the water bottle created a range that the value of the measured gravitational mass would have fit if the numbers were just positive. However, after completing the calculations for the masses of the cellular phone, I saw that there must have been a greater error somewhere else because my values for this object were almost identical to the values calculated for the water bottle. By looking at the numbers, it seems as though this may have been avoided with different values of Mtray.