22-March-2017: Lab 7: Physics 4A Lab -- Modeling Friction Forces

Eric Chong

Lab 7: Physics 4A Lab -- Modeling Friction Forces

Lab Partners: Lynel Ornedo and Harvey Thai

Purpose:  The goal of this lab is to model the trends of static friction and kinetic friction in five different experiments.

Introduction/Theory: Friction is a force that has two parts: static and kinetic. Static friction is friction that occurs when an object is stationary. It is harder to move an object when it is initially stationary than when it is already in motion. This is due to the irregularities between the surface the object is touching and the electrical charges on the surface. It takes more energy to move the object from rest than to keep it in motion. This is why static friction is a stronger force than its counterpart, kinetic friction. Kinetic friction is friction that occurs when an object is moving. Although its a weaker force, its effect is still visible. Many times objects slow down to a halt, and this is due to kinetic friction. When calculating for the friction force, there are friction coefficients that result from the surfaces of objects, and each one is different, depending of the type of surface. The equation for friction is this:

Ffriction = μN

(Note: N is the normal force applied to the object, and mu has different values depending on the type of friction and the type of surface)

Since the friction force relies on the normal force, it is expected for heavier objects on a surface to experience a greater amount of friction. By calculating the gravitational force on the object, and whether the object is on an inclined slope, we can find the normal force applied on the object. Furthermore, by calculating the amount of force needed to get an object to start moving, we can also deduce the static friction coefficient of that object and the surface. Lastly, once an object starts moving, we can apply a constant force on the object and have the object move at a constant speed in order to infer the coefficient of kinetic friction.

Procedure: 1) Static Friction

For this experiment, we set up a block, that has a mass of 175 g, connected to a string that is also connected to a hanging mass on the other end. The string is set upon on a pulley, with the felt side of the block on the table, and the hanging mass hanging in mid air from the edge of the table. This gives us the complete setup in order to find the force needed to get the block to move. Here is a picture of the setup:

After completing the setup, we then added some masses on the hanging mass until the hanging mass and the block started moving. The mass we measured on the hanging mass is a total of 90 g. Afterwards, we added more 200 g more on the block itself. This should increase the mass needed on the hanging mass in order to move the block. The measured mass for this trial is 200 g. We then did 2 more trials, each time adding 200 g more to the block. The results are 255 g and 420 g respectively. Here is a data table to summarize the results:



From here, we can find the force of static friction for each trial. To do this, we used a graph of the forces calculated from the mass of the hanging mass. We calculated the normal force by setting the normal force equal to the gravitational force. We also calculated the force of friction by calculating the gravitational force of the hanging mass. Here are the results and the graph:





From observation, the line is linear, and the reason this is so is that the equation of the force of friction resembles a line equation. Compare y=mx+b to Ffriction = μN. For this case, mu would be the slope of the graph, and the normal force is the x value inputted into the equation. From this logic, the slope of the line is the coefficient of static friction. Our value of the coefficient of static friction is 0.5105 +/- 0.02551. Now that we have the coefficient, we can find any value of the maximum force of static friction of the block by calculating its normal force.

2) Kinetic Friction
For this experiment, we attached a force sensor to the string that is connected to the block. We then connected the force sensor to LoggerPro and measured the force needed to keep the block moving at constant speed by pulling the block with the force sensor connected to the string. Here is a picture of the setup:



Once we measured the force, we then analyzed the average force on the graph and it looks like this:



From the data, the mean force is 0.4246 N. This gives us the force that will keep the block moving at constant speed.

We then added 200 g more to the block like we did in the first experiment, and pulled the string with the force sensor. We did this 3 times, for a total of 600 g in the third time, and measured the force each time. Here are the data from the trials:



From the data, we can see the averages of the force in each trial. The averages for each trial (4 in total) from least to greatest mass are 0.4246 N, 0.9319 N, 1.263 N, and 1.912 N. We then calculated the normal force on the block for each trial just like we did in the first experiment. Here is a data table to summarize the data:



We then graphed these results and looked at the slope of the line created. Here is the graph:



The slope of the graph, which is also the coefficient of kinetic friction in this case, is 0.2529 +/- 0.001228. From here, we can calculate any value of the maximum force of kinetic friction by calculating for the normal force applied to the block.

3) Static Friction from a Sloped Surface
For this experiment, we had an inclined slope and placed the block on the slope. We then lifted one end of the slope until the moment the block started to slip. At this moment, we measured the angle the slope makes with the table. The angle we got turned out to be 23.2 degrees. From here we can deduce the coefficient of static friction with calculations. Here are the necessary calculations:



By drawing a free body diagram, making the x and y-axis parallel and perpendicular to the direction of acceleration, and noticing that the Force of Static Friction is equal to the x-component of the gravitational force, we can solve for the coefficient of static friction, which turned out to be 0.43.

4) Kinetic Friction from sliding a block down an incline
For this experiment, we raised the incline slightly above the angle that we measured from the previous experiment. The angle we used was 25 degrees. We placed a motion detector at the top of the incline and connected it to LoggerPro. To ensure that the motion detector could see the block's movement, we taped an index card on one end of the block. Here is a picture of the setup:



After completing the necessary setup, we ran the experiment and measured the block's motion as it slid down the incline. Here is a picture of the data measured:



For this part, we looked at the slope of the velocity and took that as the acceleration of the block. The acceleration for our case is 1.519 m/s^2. The acceleration can be used to find the coefficient of kinetic friction through calculations, similar to the calculation of static friction. Here is a picture of the necessary calculations:



Through drawing the free body diagram and realizing that the summation of the x-components add up to the quantity of mass and acceleration, we can solve for the coefficient of kinetic friction, which turned out to be 0.30

5) Predicting the acceleration of a two-mass system
For this final part of the lab, we used the coefficient of kinetic friction we calculated from the previous part in order to derive an equation of acceleration. The derivation is this:



From here, we measured the mass of the hanging mass that moves the block. The mass of the hanging mass is 0.100 kg. Finally, we can use the measurements of the block mass, the hanging mass, and the kinetic coefficient in order to find the theoretical acceleration of the system. Here is the calculation:



The theoretical acceleration turns out to be 1.6945 m/s^2. Here is the experimental acceleration measured:



The experimental acceleration is actually 1.561 m/s^2. From observation, the experimental acceleration is less than the theoretical acceleration.

Conclusion: Through all five of these experiments, we were able to manipulate equations and derivations in order to give us the coefficients of static or kinetic friction for our models. From our models, we can find the maximum force of friction depending on the normal forces applied to the objects. Though this experiment was largely a learning purpose in order for us to see how friction works with the other forces, the lab was still prone to uncertainties, and this may extend to experiments in the future. Some sources of uncertainty were the surfaces. The surface of the block was very unsteady and could easily slip and slide from a single touch. By slightly shaking the table, or even due to the positioning of the block, the block was prone to slipping and ruin the calculated force needed to get it to start moving. Another source of uncertainty was the pulley itself. It is very possible that the pulley manipulated the tension and swayed the calculations. Since tension was very important in most of these calculations, this is the uncertainty that is crucial to our calculations. Perhaps using a more ideal pulley would make it better. And specifically in experiment 2, perhaps a source of uncertainty would be the pulling of the block, since it was largely unstable in the 4 trials we ran, which may have altered slightly our data, though we tried to combat this by taking the average. Overall, this lab was an endeavor to capture the mindset of how forces work with one another, specifically friction. Friction is reliant on the static or kinetic friction, depending on the type of surface and whether the object is moving, and the normal force applied to the object. The normal force is changed by whether the object is on an inclined slope at an angle. By drawing free body diagrams and looking at the x and y components, we can solve for the unknown variables and plug them into the other equations and find the value we are looking for. Friction is a force that occurs in everyday life, and by knowing how it functions, we can have a better understanding of how our lives work.

15-March-2017: Lab 5 -- Trajectories

Eric Chong

Lab 5 -- Trajectories

Lab Partners: Lynel Ornedo and Harvey Thai

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

Introduction/Theory: Objects in a projectile motion are subject to changes in their x and y components of velocity and position, be it due to air resistance, gravity, inertia, etc. Each component is independent of each other and should be solved individually. The variable that connects the two components is time. Since the time it takes for both the moment when the y-component reaches the height of a certain position and when the x-component reaches the distance of the selected position from the point of origin are the same, the value for time can be applied to both x and y properties of the object. By solving for the time an object reaches a certain point using the kinematic formulas, we can use the time to find the velocity and position values of the object, and even predict the impact point of a ball on an inclined board.

Procedure: Professor gave us a page with all of the procedures we need. Here is a picture of the procedure and the pictures of the apparatus with two parts, one without the inclined slope and one with the inclined slope:

Part 1) For part 1 of the lab, we first found the initial velocity when the ball started shooting off of the edge of the table. To do this, we measured the height of the table to the ground and found it to be 0.947 m. We then identified approximately the distance at which the ball hits the ground from the foot of the table and taped a piece of carbon paper at the location so we could physically measure the distance that the ball marked. We launched the ball 5 times and measured the average of those 5 trials. We ended up with around 0.709 m for the distance traveled. In our calculations, we found the time it takes the ball to land, and then we used that time to find the initial velocity of the calculation. Here are the results and the calculations to find the initial velocity:

The results came out to be 1.612 +/- 0.002 m/s.

Part 2) For part 2 of the lab, we have to find the distance on the inclined board the ball would hit. To calculate this, we first derived an equation that would calculate the distance on the board the ball would hit based on the ball's initial velocity. Here is the derivation, where d is the distance on the inclined board:

Now that we have this derivation, we can measure the angle the incline and plug in the value of velocity we got from part 1 of the lab and the angle to get the distance on the ramp the ball would hit. We measured the angle of the incline to be 50 degrees. Here is the calculation:

Of course we must calculate the uncertainty in that measurement as well. The uncertainty in the angle is +/- 0.1 degree, and the uncertainty in the initial velocity is +/- 0.002 m/s. Here is the calculation:

Notice that I did not use the "square root form" when calculating the uncertainty, and both ways are fine. The result with the uncertainty turned out to be 0.982 +/- 0.319 m. When we actually launched the ball, the ball actually hit between 0.942 m and 1.003 m down the ramp. The theoretical result and the experimental result turned out to be accurate.

Conclusion: During this lab,  I mentioned that there may be uncertainties in the experiment. The biggest error came from possibly the inertia of the ball. The ball itself has some mass, and that mass might affect the inertia to the point of it being not negligible. Some other sources of uncertainty might be the placement of the ramp, as the angle measured might not be accurate. Our device for measuring the angle is also not appropriate for a lab setting, as we used a phone app to measure it, rather than using a protractor. And another uncertainty, though this may be negligible, is the air resistance. Another source of uncertainty that may affect the data could be the friction on the ramp. We were unsure whether the ramp was completely frictionless when the ball was rolling on it and off of the edge of the table. This could affect the initial speed and and ruin our calculations. These are some of the more obvious sources of uncertainty that we came up with.

Overall, this lab taught us the mechanics of how projectile motion problems work by having us calculate the motion of a ball in the air and where it lands.Gravity affects only the y-component of the ball, whereas the x-component is unaffected, unless stated otherwise. By calculating the time, we can slowly calculate each individual velocity and position components of the projectile.

13-March-2017: Lab 4: Modeling the fall of an object falling with air resistance

Eric Chong

Lab 4: Modeling the fall of an object falling with air resistance

Lab Partners: Lynel Ornedo and Harvey Thai

Part 1: Determining the relationship between air resistance force and speed.

Purpose: The goal of this part of the lab is to capture videos of coffee filters falling from the balcony with air resistance and model a position vs. time graph of the fall.

Introduction/Theory: We modeled the fall of objects without air resistance before, but this time we want to take into account the air resistance that affects the fall. Supposedly, the equation of the force of air resistance changes with respect to the speed of the fall. We can model this in the form of a power law:

Fresistance = kv^n

where F is the force, k is the constant value that takes into account the shape and area of the object and n is the number that determines the shape of the graph. It is suspected that n should be 2, because the gravity force is still accelerating the speed of the coffee filter, until the force of air resistance brings the fall to constant velocity. It is at this point that the force of air resistance is equal to the force of gravity (mg), since the object falling is at equilibrium.

Procedure: First we went to building 13 and came across a balcony that is high enough to model the fall. Here is a picture of a picture of the building and balcony:

We have our laptops at hand and then the professor went up to the balcony and set up the coffee filters. Here is a picture of one of those coffee filters:

We used LoggerPro in our laptops and captured a video of the professor dropping a coffee filter from the balcony. Each time, professor would increase the number of coffee filters stacked on top of each other in order to increase its mass, and we would notice that the fall would start off faster each time. We recorded videos for 1, 2, 3, 4, and 5 coffee filters falling from the balcony. We also recorded one with 6 coffee filters just for good measure.

We then used LoggerPro and tried to plot the points on the video at which the filters' positions are each frame. By doing so, we are also graphing the position of the coffee filters over time. We did this for all of the videos and found the terminal velocity by taking some of the last few points where they are the straightest and used a linear fit and found the slope. Here are the results:

We then took the slopes of all of these graphs and used them in another graph. We made the y-axis of this new graph the force of air resistance (N), and the x-axis the terminal velocity (m/s). We know the mass of the coffee filters by measuring its mass and gravity is given, so we multiplied them by each other in order to get the force of gravity, which is also equal to the force of air resistance at terminal velocity. We graphed the values and did a power fit and here are the results:

On the graph, A is the value of k, x is the velocity, and B is the value of n. In this case, k is 0.006433 +/- 0.0005217, and n is 1.994 +/- 0.08755, really close to 2 as expected.  Now we can move on to part 2 of the lab.

Part 2: Modeling the fall of an object including air resistance

Purpose: The goal here is to apply the mathematical model we developed in part 1 to predict the terminal velocity of our various coffee filters.

Procedure: For this part of the lab, we used excel to set up a spreadsheet with this type of layout:

Note that delta x and x are not included in the layout, as the point of the spreadsheet is to find out the values up to acceleration. Like the previous lab, we made t depend on delta t and worked from left to right from there. Here is the general way to set up the cells of each column:

After we set everything up and we inputted the cells with the correct equations, we finally inputted the values and filled down the cells. We then examined the values of v, which is essentially our value of velocity at time t, and we searched for the point at which v is unchanging. This point is our terminal velocity. We did this for all 6 of our graphs from part 1. Here are the results for all 6 of the graphs:

We noticed that the values of v are slightly off from the values we measured by graphing, but they are still relatively close to one another. This concludes part 2 of the lab.

Conclusion: Overall, the lab documented the fall of an object that is subject to air resistance. We modeled the fall of that object and noticed that with an increased speed, there is an increase in air resistance. So, while stacking the coffee filters, we are also increasing the speed of the fall, since the force of gravity increases. This also increases the force of air resistance until it equals out the force of gravity and causes the filters to fall at constant velocity. We then plotted the position vs. time graph and found the terminal velocity. We can also calculate the force of air resistance by calculating the force of gravity. We then graphed the values in a force of air resistance vs. terminal velocity graph and found the values of k and n. This is how we found the equation of the force of air resistance.

One of the uncertainties in our data collection was the video recording. The video recording was poor quality and when we analyzed the position of the coffee filters, sometimes the filters would blur out as its speed increased, causing us to have difficulty pinpointing where its position was. This may have caused our our uncertainty in our points, as some of the points of the position vs. time graph are not linear at terminal velocity. This may have affected our slope, and thus affect the points on the force of air resistance vs. terminal velocity graph.

Overall, the lab teaches the concept of air resistance and how speed affects the magnitude of it. We now know that air resistance can equate to the force of gravity while objects are falling, and we can use this in our future calculations.

8-March-17: Lab 3 Non-Constant acceleration problem/Activity

Eric Chong

Lab 3 Non-Constant acceleration problem/Activity

The Problem: A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on a level ground. At that point a 1500-kg rocket moounted on the elephant's back generates a constant 8000 N thrust opposite the elephant's direction of motion. The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the

m(t) = 1500 kg - 20 kg/s*t. Find how far the elephant goes before coming to rest.

Introduction: For this activity, we are given a problem about an elephant with a rocket that is thrusting backwards while losing mass. There are two ways to approach this problem. The first way is the analytical approach, which essentially relies on the use of calculus, physics laws, and other equations in order to find the distance the elephant traveled. We did not do this ourselves, for our professor showed us how to do it in the handout because this approach is very time-consuming. Here is how he did it:

The other way to approach this is the numerical approach, where we use an excel spreadsheet in order to link values of certain variables with other variables in order to help us find the distance at the elephant travels. Here is the initial layout of the spreadsheet:

(Note: M0 is the initial mass, V0 is the initial velocity, b is the burn rate, F is the force of the thrust, t is the time, x is the position, and a is acceleration. The rest of the variables, such as v_avg and delta x should be self-explanatory.)

We first filled out the given values in the spreadsheet (M0, v0, b, F, and delta t). Then, we manipulated the columns at the bottom portion of the spreadsheet by filling out formulas that correspond to them.

For t (time), the formula is equal to the previous time plus delta t, with the initial time being 0. This gives the next subsequent values of time based on the change in time.

For a (acceleration), the formula is the force divided by the quantity of initial mass minus b (burn rate) times t (time).

For a_avg (average acceleration), the formula is the final acceleration plus the initial acceleration of the time interval, all over delta t.

For delta v (change in velocity), the formula is the average acceleration at that time interval multiplied by the change in time.

For v (velocity), the formula is the previous velocity plus delta v.

For v_avg (average velocity), the formula is the final velocity plus the initial velocity of the time interval, all over change in time.

For delta x (change in position), the formula is the average velocity at that time interval times the change in time.

And lastly for x (position), the formula is the previous position plus the change in position.

After we filled out the formulas and inputted the values and used the fill down function, the resulted spreadsheet looks something like this:

We first set the delta t as 1 second and recorded the values. We then set delta t as 0.1 seconds and 0.05 seconds and recorded their data. Here are the values in the form of a table:

We observed that the smaller the delta t, the more accurate the position is, since essentially we are making thinner rectangles under the position vs. time curve and minimizing the area left out by the corners of the rectangles, the same idea as taking the integral, when the rectangles are infinitely thin under the curve. As expected, through this method, the answer is around 248.7 meters, the same distance as when solved through the analytical approach.

Conclusion: Overall, this little exercise is a means to show how each value relates to one another in the form of an excel spreadsheet. It helps us practice the fundamentals of deriving equations and thinking how we can obtain the values of each variable.

1.) The answers are relatively close, as through the analytical approach the answer is 248 m, and through the numerical approach the answer is around 248.7 m.

2.) We can figure out that the time interval is small enough and find the answer, even without the reference of the analytical result, by looking at the digits of the results based on the value of the time interval (1 s, 0.1 s, or 0.05 s), and seeing at which point does the value stop changing at a significant rate. Once we notice that the results stops changing significantly for a specific value of the time interval, we can estimate to the decimal point that matters to us, and in this case is to the tenths place (248.7 m), and we can then claim that the time interval is small enough. For example, in the previous picture, we see that the time interval of 1 second is too large, as the values for the distance varies too much within that time interval. Thus, by reducing the time interval to 0.1 second and then 0.05 second, we found the result to be more accurate. That is how we knew when the time interval is too large, and when it is small enough.

3.) For this problem, we would do the same numerical approach with the spreadsheet. After setting everything up and getting the values, here are the results:

According to the data table from the numerical approach, the distance the elephant would travel is around 164.04 m (as seen in the highlighted cells).

6-March-17: Lab 6 Propagated uncertainty in measurements

Eric Chong

Lab 6: Propagated Uncertainty in Measurements

Lab Partners: Lynel Ornedo and Harvey Thai

Purpose: The goal of this lab is to practice and get acquainted the use of calipers and calculated uncertainty. Calipers are different than what we are used to, so this is our introduction to such an equipment.

Introduction: Up until now, we have never used calipers. This short lab is a means to help us know how to read this precise instrument. Calipers have two scales built into it. The first scale is the main scale, and is read normally like any other scale such as rulers. There is also a smaller second scale, which is slightly more complicated to read, but is not too hard once we knew how to read it. This smaller scale is a means to get precise measurements in the smaller decimal places such as 0.02 cm. The lab contains two different cylinders of different types of material, and we must measure the density of both of them with calculated uncertainty. Calculated uncertainty is essentially carrying over the uncertainty we gained from raw measurements into our results, since whatever we measure cannot be better or worse than what we get out of it.

Procedure: First off, we prepared the calipers and the aluminum and zinc cylinders.





We started with the aluminum cylinder and measured the diameter of its base with the calipers. To do so, we placed the base of the cylinder in the clamp and tightened it until it held the cylinder in place. Then, we read the main measurement normally by looking at where the first of  the bottom markings lined up with the main markings. Then, we looked at where the subsequent small markings first lines up with one of the markings. This is important because you may notice that some of the markings do not line up with the main markings as if the calipers were built wrong, and this is intentional. The first instance the small markings line up with the main markings, aside from the very first marking, indicates the hundredths place of the measurement for the diameter. The measurement we got for the diameter of the aluminum cylinder is 1.26 cm. We did the same for its height and got 5.09 cm. Finally, in order to get its density, we need its mass, so we put it on a balance and measured 18.32 g. I will talk more in-depth about uncertainty when I get to the calculations.

The process is the same with the  zinc cylinder, and we got 1.32 cm. for its diameter, 3.28 cm. for its height, and 28.90 g for its mass. Here is a table with all of the measurements:



We put plus or minus 0.01 next to our numbers because that is the decimal place that we are least certain of. Since we are pretty much eyeing up the small secondary markings of the calipers, we are uncertain of whether that is the exact measurement. Thus, we have to put the 0.01 in order to indicate that that is where our uncertainty lies.

Next, we manipulated the density equation in order to give us something that is easier to use with our measurements. In order to calculate the uncertainty, we have to do partial derivatives for each variable and put it in its "square root form" in order to give us the uncertainty of the result. Here is what the calculations look like.



Finally, this becomes a matter of just plugging in numbers for each cylinder. Here is the calculation for the aluminum cylinder:



And here is the calculation for the zinc cylinder:



As such, the density of the aluminum and zinc cylinder are 2.887 +/- 0.046 g/cm^3, and 6.439 +/- 0.0995 g/cm^3, respectively. We left the uncertainty of the zinc calculation as 0.0995 and not 0.1 because we wanted to indicate where the uncertainty is. Writing the rounded 0.1 as 0.100 does not help as well, so we made a decision to leave the uncertainty as 0.0995, though that means we probably should have increased the decimal place of the zinc density by one in order to match the uncertainty. Nonetheless, the process is what matters and we now know how to calculate the uncertainty while learning how to use calipers.

Conclusion: Overall, this short lab about calculating the density of the cylinders demonstrates the use of calipers and how we deal with uncertainty in our measurements. We first take the measurement, look at the decimal place that is most uncertain and we calculate the uncertainty based on that using partial derivatives in the square root form. This way, we can look upon measurements and see how precise we are.