Centripetal Force

In this lab we are verifying Newton's Second law of Motion for the case of uniform circular motion. To better understand this, we are going to try and show that the force of circular motion to cause a spring to stretch is equal to the force of a mass hanging to stretch the spring the same amount.

The materials needed for this lab are:

• Centripetal force apparatus
• Metric scale
• Verneir Caliper
• Stop Watch
• Slotted weight set
• Weight hanger
• Triple beam balance

INTRODUCTION
Before starting this lab we need to understand a couple concepts first. The centripetal force apparatus rotates a known mass in a circular path with a known radius. When we time the motion for a number of revolutions we can find the distance traveled and calculate the velocity. We can use Newton's Second Law to determine the velocity with the equation:

F = (mv^2)/r

 m: mass of the object

v: velocity

r: radius

F: centripetal force

This is derived from the equation F = ma and in uniform circular motion the value for acceleration, a, is given by:

a = (v^2)/r

PROCEDURE

To set up this lab we first measured the mass of the weight. We then put a centripetal force apparatus on the table and leveled it by adjusting the legs appropriately. Next, attached the weight to the end of arm of the apparatus on a string and let it hang until it stopped moving. We adjusted the post to where the weight was directly above the post. Once the post was tightened down, we attached the spring to it.

Once everything was set up, we then measured the radius of the apparatus from the center of the rotating pole to the string where the weight hung from. This is going to be our radius, r, for the equations we use.

After having the measurements we spun the apparatus until the spring stretched far enough to where the tip of the weight reached the post. We timed how long it took the apparatus to go around 50 complete revolutions. We took 3 trials of this and put our measurements on to a table.

The next part of the lab is to now find the Force that is required to stretch the spring the same distance that spinning the apparatus did. In order to do that you must attach a string to the weight opposite of the spring and hang a mass hanger over the pulley of the apparatus. Once doing that, begin adding slotted weights to the hanger until the weight is over the post just as it was when we were spinning the apparatus. Record the mass and calculate the force that was required to stretch the spring. 

Repeat this experiment except this time around, add a 100 g slotted weight to the hanging weight. 

DATA ANALYSIS

All the data that was collected we put into tables. In order to do so we first had to do some calculations. We had to find the linear speed, the centripetal force, the force of the hanging mass, and the percent difference.

First we calculated the linear speed for each trial and found an average.

Once we had and average velocity, we could then calculate for our centripetal force with the given equation F = (mv^2)/r. We also solved for the force of the hanging mass

These were the calculations for the mass of 0.4492 kg. We also did the calculations for the mass of 0.5492 kg. The table represents all of our calculations.

DISCUSSION

In this lab I learned how to calculate the centripetal force of an object in a circular motion. I also learned that the centripetal force to stretch the spring with a weight attached moving in a circular path is equal to the force required to pull the string. This makes sense because the faster you spin something, the force on that object that is pulling it away is greater. 

Some sources of error that could have occurred in this lab is that we may not have given a completely constant velocity on while rotating the apparatus. Also there were a couple of times where the weight wasn't completely over post; sometimes the spring was not stretched enough and sometimes it was stretched too much.

Drag Force on a Coffee Filter Lab

The purpose of this lab is to observe and study the relationship between air drag forces and the velocity of a falling coffee filter. In order to do this lab the materials needed are:

• Computer with Logger Pro
• Lab Pro
• Motion Detector
• 9 Coffee filters
• Meter Stick

INTRODUCTION

When an object falls, it experiences a drag force that points in the opposite direction of which way the object, such as a ball, is moving. When this drag force reaches its maximum force, which is equal to the Gravitational Force pointing in the opposite direction, it has reached it's terminal velocity. This is a constant velocity because when added, the Drag Force and Gravitational Force leave an Fnet = 0. Even though the Fnet is 0 doesn't mean that there isn't any movement. This just means that there is no acceleration.

When calculating drag for this lab we are going to look at the equation:

F(drag) = k|v|^n

and when you compare this equation to drag equation which is

F(drag) = (1/4)Av^2

where A is the surface area of the object and v is the velocity you can see similarities. In the equation we are going to use in the lab k is equal to (1/4)A because it is a constant that will work for all the trials we are going to do even though we are doing trials of different numbers of filters falling. They all have the same surface area.

PROCEDURE

In order to do this experiment, we set up Logger Pro on the computer and labeled our graph on the x axis as Time (s) and the y axis as Position (m). We set our motion detector on the floor facing up and set the data collection rate on Logger Pro to 30 Hz. We also opened an EXCEL spreadsheet and made column titles for velocity of each trial and row titles for how many filters were used for the experiment. We grabbed 9 filters and stacked them into a packet. You must be careful of not damaging any of the filters because it is very important that we keep the surface area the same because since our k value is constant for all trials our A needs to stay the same to keep this true. 

Once we had our graph and table ready, we clicked the "Collect" button and dropped the 9 filters above the motion detector and recorded the data. We did a linear fit to the part of the Position Vs. Time graph where the coffee filters decreased in position at a constant velocity (when the graph made a straight line) just before they hit the ground. We copied the slope which is the velocity into trial 1 for 9 filters into the spreadsheet and repeated this for a total of 6 times. We then calculated the average velocity of the 6 trials.

This graph is one of the trials that shows the constant slope, or terminal velocity, of the falling filters.

Once doing 6 trials for 9 coffee filters and finding the average, we repeated the experiment for 8 filters instead, then 7 filters until we had 6 trials for each amount of filters until we reached 0 filters.

This is the table of the terminal velocities found for all of our trials that we did.

Once having this information we created a graph in Graphical Analysis of the Number of Filters vs. Average Terminal Velocity. When doing so we got the graph of the function we looked at in the introduction:

F(drag) = k|v|^n

We took a power law fit to show the equation of the graph where the value of k = A and the value of n = B.

CONCLUSION

When interpreting this information we first look at the Position vs. Time graph. There is a time right before it hits the ground when there is a constant change in the position with respect to time. This means that it is in a straight line and we have a constant velocity. This is what we call our terminal velocity or terminal speed.

When we performed a power law fit to the Number of Filters vs. Average Terminal Speed graph we were given an A value and a B value. The A value is equal to the value for k in our equation F(drag) = k|v|^n and the B value is equal to n.

The value for k when we go back to the equation F(drag) = (1/4)Av^2 is the constant (1/4)A. It is a constant because our value for A (surface area) is constant since all of trials we did were equal to eachother. This is because when the filters are stacked the surface area never changes since there are none being added around the bottom one. This means we can find the surface area of the coffee filters since the k value is constant.

k = A

A = 1.39

k = 1.39

k = (1/4)A

1.39 = (1/4)A

A = 5.56

Our B value is equal to n which should be equal to 2 since the original equation shows v^2. Our value for B was 2.21. We can calculate our percent error with the equation:

% error = |(accepted-experimental)/accepted| X 100

% error = |(2-2.21)/2| X 100

% error = 10.5%

We had a fairly high percent error but we can make that number smaller because some things we can take into affect would be was there air currents that were going on in the room. Another would be the filters because since they were stacked some could have caught more air resistance since they could've been hanging off the edge of the bottom one a little bit. 

In this lab I learned how to calculate values of terminal speed of a falling object. We used coffee filters and found their terminal speeds when being dropped from 1.5 m.

Working with Spreadsheets Lab

In this lab we are going to become familiar with spreadsheets by using them in  some applications. We are going to need:

• A computer with the EXCEL software
• Graphical Analysis software

In labs, we are always collecting data and spreadsheets are a great way to present your results in a lab report. They make the tables you would need for someone that is reading the report to easily understand what quantities were obtained for a certain part of the lab. They keep your data neat and organized as well.

Procedure:

We began by opening up Microsoft EXCEL and saved the spreadsheet as Practice Spreadsheet 1. Our first table we created was one that would calculate the values of:

f(x) = A sin(Bx + C)

Our initial values we are going to use are:

• A = 5
• B = 3
• C = (pi)/3

We put these values on the right side of the spreadsheet. We put each value in a column and labeled them. In the Amplitude column we put 5, which was cell R2, the Phase column we put (pi)/3 in cell T2, and the Frequency column had 3 in cell S2.

We then made column headings labeled "f(x)" which we put in K1 and "x" which was in cell J1. In the x column, we put the first value as 0 and below that we put 0.1. We highlighted both cells and in the bottom right hand corner of the highlighted cells is a square. We clicked on the box and dragged it down until we reached the value of 10. Doing this creates all of the cells needed for the column because it recognizes the change in values and creates them all without having to type them in one by one.
In the In the "f(x)" column, we are going to create a formula to calculate the function. To create a formulat, you must put an = to let the program know you want it to calculate something. In  cell K2 we created the function by inputing =H2sin(J2D2+I2). It will calculate the value for f(x) for us so we don't have to. Again, we highlighted the cell and in the bottom right hand corner we dragged that to fill the column as far as the x column went. This calculates everything so we don't have to. In order to see the equation that creates the value in the "f(x)" column, you can press "Ctrl~".

We made a copy of the table with the values and then one with the equations of the first 20 rows.

We copied the table by highlighting the whole thing and dragging into the program Graphical Analysis. In Graphical Analysis it created a graph of the table we made with the equation. We highlighted a part of the graph and performed a curve fit to show the equation with the same values we gave it. On the graph, we titled it "Graph of Excel Spreadsheet", labeled the y-axis as "f(x)" and the x-axis as "x". We printed out the graph.

Once we finished that part up, we then repeated a similar process to calculate the position of a freefalling object. The next function we are going to look at is the kinematic equation:

x₁ = x₀+v₀(t_f –t_i)+.5g(t_f-t_i)²

for the values we used: 

g = -9.8 m/s²

v₀= 50 m/s

x₀= 1000 m

(t_f-t_i)= 0.2 s

These values were also put in celss with column headings. g was in column N, v was in column O, and x was in the P column. In column F, we labeled it as time, and we put the time value. In F2 was 0 and F3 was 0.2. We highlighted the cells and dragged the box down until the time reached 20 s. In the G column with the header Position. Here we typed the equation =P2+O2*F2+0.5*N2*F2^2. We highlighted that cell and dragged it to fill all the time values.

We copied a table of the first 20 values calculated for the position. We also showed a table of the equations by pushing "Ctrl~" and copied that table as well.

We dragged the table of values into the Graphical Analysis as we did before and created a graph of the table. In the graph, we titled it "Free Falling Particle", labeled the y-axis as "position (m)" and the x-axis "time". We fit the data with a curve fit using a quadratic type equation (y = A+Bx+Cx^2) to get values for A, B, and C.



The values of A represented a similar value to our initial position, the B value represented the velocity, and C was 1/2 of the value of g.

Conclusion:
In this lab we learned how EXCEL can be used to create a spreadsheet of values that we find during a lab. We learned that you can copy that table into Graphical Analysis that way you can create the most accurate graph of the data that you find. We can use this skill in just about any lab because we are always using experimental data in equations to find values of the lab. Take the free fall part of this lab for instance. We performed this lab at the beginning of the semester and could've used a spreadsheet with a given equation to create the graph of what occurred.