Balance Torques and Center of Gravity

Our purpose in this lab is to study the rotational equilibrium of a meter stick and to also determine its center of gravity with a system of masses.

The materials needed for this lab are:

• Meter stick
• 3 mass holders
• masses for the holders
• knife edge clamp
• 3 mass clamps

INTRODUCTION:
When an object is not moving, it has Fnet=0. We are looking at balancing a meter stick on a knife edge clamp with masses hanging on each side. We aren't working in straight line motion when working with this particular lab, we are looking at rotational motion. The force in rotational motion is called torque.

Torque can be calculated with the following equation:

Since torque is a force and we are balancing the meter stick, the Tnet=0.

The Force is the weight force which is perpendicular to the lever arm. The angle should be 90 degrees. The lever arm would be the distance from the center of gravity to where the mass, or force, is at. Since our value for theta is 90, sin(90)=1 so we don't have to take consideration of the angle in this case.

PROCEDURE:

To begin the lab we first set the meter stick on the knife edge clamp and set it to where the meter stick was perfectly balanced. This will be where the center of mass is. We found that to be at 49 cm on the meter stick.

We measured out two different total masses of 100 g or more with the masses, mass hanger, and mass clamps because you must take into account the mass of the hanger and the clamps. They affect the torque as well since it is added to the meter stick. Our first mass was 220.6 g and our second was 171.5 g. We set our second mass 40 cm away from the center of gravity and tried to balance the meter stick with the first mass on the other end of the meter stick. That ended up being 31.16 cm away from the center of gravity. We calculated our values to see how close to zero for a Tnet we obtained. 

We calculated our Tnet value and compared it to how close to 0 we could get with our measurements we obtained.

Next we put the two masses in different locations but on the same side. We added the third mass clamp with hanger and masses that was 270.8 g on the opposite side and moved it until the meter stick was balanced again. We also found how close we got to a Tnet of 0 by solving.

We set the same weights from the first test on the same side and added one to the left and balanced it out. We calculated our Tnet and compared it to what it should have been.

For our next step, we replaced the third mass with an unknown mass. We left masses 1 and 2 on the right side at 30 cm and 15 cm respectively from the center of gravity. We adjusted the unknown mass to balance the meter stick. Our distance from the center of gravity was 30.2 cm. We solved for what the mass should be and then measured the mass to see how close to the actual we obtained. We calculated the percent error we had.

We changed our third mass on the left side with a new unknown mass. We found it's mass by using our equations. We compared the mass we obtained with the actual mass of the unknown mass and the mass clamp.

Next, we put 200 g (including mass clamps and hangers since they aren't part of the ruler and are causing the ruler to have torque) at the 90 cm mark and find a new balance point on the meter stick. We found that point to be 78 cm, which is 12 cm away from the mass hanging. The center of gravity of the meter stick is 29 cm away. We can say that the center of gravity of the meter stick is where the weight force is located because that's the point where the mass is concentrated. The new center of gravity will be at the 78 cm mark. We calculated what the mass should be and compared it to the actual mass of the meter stick. The mass of the clamp that holds the meter stick is irrelevant because it is keeping the meter stick in equilibrium.

We found the new center of gravity with a 200 g mass hanging at the 90 cm mark on the meter stick. We solved to find what the mass of the meter stick is and compared it to the actual mass.

We now had to picture a similar scenario with the 200 g weight still at 90 cm but now we add an additional 100 g at the 30 cm mark. We had to calculate to find it's center of gravity and where it should be on the meter stick. We found it to be at 65.5 cm once we did our calculations. I then went ahead and made it the center of gravity and checked to see if that center would give us a net torque of 0 or how close it would be.

We calculated to find the center of mass of the meter stick when a 200 g mass is hanging at the 90 cm mark and a 100 g mass is hanging at the 30 cm mark. We calculated our center of mass to be at 65.5 cm on the meter stick.

CONCLUSION:

In all of our calculations we had a very small percent error throughout the entire lab. This is because there is a very small window, as far as length is concerned, that we can obtain to keep our meter stick in perfect balance. Also when we were calculating our percent error, we weren't comparing the net torques because our theoretical value was 0 and we cannot have a 0 in a denominator. Instead we compared the theoretical length of where we should have put our mass and where our masses actually were on the meter stick. We weren't able to apply the last scenario of finding a center of mass to the actual meter stick to see if 65.5 cm was actually the center of mass when 200 g is hanging at 90 cm and 100 g is hanging at 30 cm. We did use calculations to see how close to 0 we got for a torque net but weren't able to compare to any tests.

We weren't able to prove the last scenario to be true. I calculated what the Tnet would be with what we found if 65.5 cm on the meter stick is really the new center of mass with the given masses at the given lengths.

Human Power Lab

In this lab we are determining the power output of ourselves from walking up stairs.

In order to perform the lab we are going to need:

• two meter metersticks
• a stopwatch
• kilogram bathroom scale

INTRODUCTION

Power can be described as the rate at which work is done which can also be translated as the rate at which energy is converted from one from to another. To show this we look at the equation:

Change in PE = mgh

where:

• PE is the potential energy
• m is the mass of the object working
• g is the acceleration of gravity
• h is the vertical height gained

We can use this equation to find the change in potential energy which we need because the equation for  power output is:

Power = (change in PE) / (change in time)

 where:

• change in time is the time it takes to climb the vertical height

This is the unit analysis of the lab we are doing. The change in potential energy is read as (kgm^2/s^2) which is the same as Newton meters (Nm) which is equal to Joules.

PROCEDURE:

We first started this lab out by weighing ourselves on a kilogram bath scale. We measured our mass in kg. Once we weighed ourselves, we then went to the stairs that were down the hall from the lab and measured the height from the floor of the first floor to the floor of the second floor. We wanted the height in meters.

This is a sideview of the stairs to show the height we climbed.

We each started at the bottom of the stairs and were timed how long it took to reach the top stair. We performed two trials each for our data.

Once we all collected our two time trials we then calculated our value for our power output. We calculated our value in watts.

Once I found my average Power output in watts, I then solved for it in Horse Power.

DISCUSSION:

I calculated a % difference for what my values were compared to the rest of the class. 

The values I obtained for Power were greater in watts and in Horsepower by more than 10% for each. This meant that I output more power than the average of the class going up the stairs.

Answers to Questions:

1. It is OK to use your hands and arms on the handrailing to assist you in your climb because we are calculating the power our whole body outputs to get us up the stairs as fast as we can. If we use our hands and arms then we are outputing more power to get ourselves up the stairs.\

2. Some problems that can affect the accuracy of this experiment are human error in the stop watch because when told to go the watch isn't started when the person begins to move, it begins when someone says go. People's reaction times are sometimes slower than others. You can also obtain error from the way we measured the height of the stairs. We measured from the floor of the first floor to the half way point where the direction the stairs go is turned around, and then from there to the floor of the 2nd floor.

Conclusion:

I learned in this lab how to find the power output of a person. The process we learned is when something is changing in height. We were able to calculate our power going up the stairs. I output more power than the class average which means I was working harder since our height was all the same and Work can be expressed as m*g*h. I also learned how to apply the % difference when appropriate. We couldn't solve for a percent error because we weren't given an accepted value or a theoretical value to compare from. We only had others' values to compare to. This is why we used percent difference because we could only compare values obtained from fellow classmates.

Answers to Follow Up Questions:

1. They would both produce the same amount of work because they have the same mass, working against the same acceleration due to gravity, and traveling the same height. However, Hinrik would output more power than Valdis because Hinrik does the work faster and Power is a comparison of the amount of work done in a certain amount of time. The longer it takes a person to work, the less power output they will have.

2.

3.

4.

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.

Vector Addition of Forces Lab

In this lab we studied vector addition by graphical means and also by using vector components. The materials to replicate this lab are as follows:

• Circular Force Table
• 4 pulleys
• Masses
• Mass Holders
• string
• Protractor and ruler

Intro:
Vectors are arrows with a certain length, which describes the magnitude of an object, and direction. In order to add vectors you must have the vector components which describe how far along the x and y axes the magnitude of the vector travel which will give you <i, j> where i is the x direction and j is the y direction. To find the components of a vector that gives you the magnitude as well as the angle of direction you would use basic trigonometry identities.

To find the x component:

cosӨ = V_x /M

McosӨ = V_x 

To find the y component:

sinӨ = V_y/M

MsinӨ = V_y

Once you find your vector components the notation of your vector you will have <V_x ,V_y >. When you have at least two components you can now add them together by adding the x components and the y components to get the components of the two added vectors which will make up a new vector that you can graph.

Another method you can use to add vectors is graphically.

In this picture you can see that in order to find V₃ you would draw it from the tail of the first vector which is V₁ to the head of the last vector which is V₂.

Procedure:
In this lab we are given 3 masses and an angle by our teacher. The 3 masses and angles we were given were:

• 150 g @ 0°
• 110 g @ 70°
• 250 g @ 135°

We were given a conversion to convert the masses into lengths for the magnitude of our given vectors. The conversion was 1cm = 20 g. We used the conversion and found the magnitudes of our vectors:

• 7.5 cm
• 5.5 cm
• 12.5 cm

We graphed the 3 vectors using the head to tail method first to find the magnitude and the direction of the angle.

We measured the length of the resulting vector with a ruler to be 14.0 cm and 87.75° with a protractor. 

Once we found the resulting vector using the graphic method we then solved for the resulting vector by finding the components of the given vectors.

With these components we were able to find the magnitude and angle of the resultant vector.

Data Analysis:

Now that we have a magnitude and direction for our new vector we now convert it to grams by using the conversion 1 cm = 20 g which gives us 280.4 grams for our resultant vector. We then set up our circular force table with the four pulleys. At the center there is a ring with 4 strings tied from it that hung off of the pulleys. The first 3 pulleys were set at the given angles, 0°, 70°, and 135°. The fourth was set 180° opposite the angle we found which was 87.8. This gave us a new angle 267.8° which now makes our vector negative. After having all of the pulleys set we then hooked the massholders to the ends of all of the strings and added the masses onto the massholders:

The center ring was in equilibrium on the circular force table which means the masses hanging at the end of the strings which hung from the pulleys at the correct angles suspended the ring in the center without touching anything.

We showed the direction of all the vectors to understand the direction they are going on the table.

Conclusion:

We calculated our percent error to be 0.14%. It was a low because we were able to control most of our experiment since we were only trying to find one vector from 3 vectors. The one that was more off was when we solved for it graphically only because we were physically measuring rather than working with the components. In this lab we learned how to add vectors by their components and also graphically with the head to tail method.



Acceleration of Gravity on an Inclined Plane Lab

The purpose of this lab is to find the acceleration of gravity by observing the motion of a cart on an inclined plane. In this lab we will be using:

• Logger Pro Software
• motion detector
• ballistic cart
• aluminum track
• woodblocks
• meterstick
• small carpenter level

Intro:
We will be using Logger Pro to collect the position vs. time data for the ballistic cart as it accelerates along the aluminum track. We are not going to include the effect of friction because friction will act with the force of gravity as the cart moves up the track and against gravity while it travels down the track. With this, we can use the average acceleration of the cart moving up and down the track with:

(a1+a2) / 2

to determine the average acceleration of gravity as the friction will slightly increase acceleration of gravity while the cart moves up the track and decrease slightly as the cart moves down the track. The average acceleration of the cart is equal to the acceleration due to gravity, g, times the sine of the angle of the track.

We can use this picture to find the acceleration of an object on an incline plane. We have θ because that is equal to the angle of the plane with the table. We also have the acceleration of the freefall and the acceleration of the object parallel with the inclined plane. If we look at them as vector components we see that we can add | a_parallel |+| a_{perpendicular} | = a_freefall

We see that g is the same as the a_freefall and can come up with the equation:

sinθ = opposite / hypotenuse

sinθ = | a_parallel | / g

Knowing this we can substitute in the average acceleration of the cart going up the track and going down the track:

| a_parallel | = (a1+a2) / 2

and we can arrange the previous equation:

sinθ = [(a1+a2)/2] / g

g sinθ = (a1+a2) / 2

and in order to find g, the acceleration due to gravity, we divide both sides by sinθ:

g = [(a1+a2)/2] / sinθ

where a1 and a2 are the accelerations of the cart, θ is the angle of the track and the table, and g is acceleration due to gravity.

Procedure:

First, we hooked up the Logger Pro software and opened up the graphlab file which is where we recorded all of our data. We then set up the aluminum track with one side having 1 wood block underneath its feet which raised up the track to give it an incline. Using the bubble level, we made sure the width of the track was level to make sure there wouldn't be any other effects of friction that could taint our results. After that we set the motion detector at the top of the track having it face the bottom part of track where the cart will be coming from that way we could measure its position, velocity, and acceleration.



Once having the lab set up we then solved to find the angle that θ would be by doing some simple trigonometry. We measured out 50 cm on the length of track and used the level to make marks on the table vertically below where the 0 cm were on the track and where 50 cm were on the track. We found that length to be 49.95 cm. Now, in order to find the measure of angle θ, we use the equation:

cosθ = adjacent side of the angle / hypotenuse

which, when using our measurments, comes to be:

cosθ = (49.95cm/50cm)

(arccos(cosθ)) = arccos(49.95/50)

θ = 2.56°

Before starting we opened up two graphs in Logger Pro, one of the position vs. time and one of the velocity vs. time to compare the two as they are occuring. We did a few practice trials to get ourselves comfortable trying the system out and making sure everything was going to work smooth for the lab. Once doing that, we did 3 trials of pushing the cart up the track and letting it come back down with the effects of gravity.

Results:

When doing this experiment and collecting data graphs, we expect to see the Position vs. Time graph to be a positive parabola because as time goes from 0 to

We did two linear fits on the velocity vs. time graph to obtain our a1 and a2. When finding a1, we did a linear fit when the velocity was moving at a constant rate less than 0 and for a2, we did a linear fit when the velocity was moving at a constant rate greater than 0. Those are going to make up our total average velocity that we used in the equation.

Trial 2:
We kept the set up the same but then we raised the incline to become steeper by adding another wood block underneath it. We measured out 50 cm on the track again and also used the level to make marks on the table vertically below the 0 cm and 50 cm mark to measure out the the length of the base of the "triangle" that the track and table make and that came to be 49.75 cm.

cosθ = (50cm/49.75cm)

arccos(cosθ) = arccos(50cm/49.75cm)

θ = 5.73°

Results:

Again, we took a linear fit of the negative velocity (a1) and the positive velocity (a2) that way we could take an average of the two for one average acceleration.

Conclusion:

In the first set of trials when we set the angle of the incline, θ, at 2.56°, our g-experimental value was 7.09 m/s^2. Our accepted value of g, g-accepted, is 9.80 m/s^2. With these numbers we can calculate a percent error that occurred  with the equation:

% error = |[(measured-actual)/actual]| X 100

=|[(7.09m/s^2 - 9.80m/s^2) / 9.8m/s^2]| X 100

=27.6%



Our second set of trials which ran on the elevated track were set at an angle θ = 5.73°. The average g-experimental value was 7.24 m/s^2. We can calculate the percent error:

|[(7.24m/s^2 - 9.8m/s^2) / 9.80m/s^2]| X 100

=26.1%

When we look at the position vs. time graphs of the cart in motion, we expect to see a positive parabola because as it starts traveling up the track it moves closer to the motion detector at a decreasing velocity which gives it a curve and not a line because the velocity is changing at a rate and not a constant. Once it gets to its highest point on the track, which is the closest point to the motion detector, the acceleration due to gravity takes over the cart and cart's position slowly starts to increase at a rate away from the motion detector which gives it an increasing curve. The graph would never have a negative position either because the cart approaches the motion detector which would have 0 position and once gravity takes over the carts velocity, the cart's position increases.

The velocity vs. time graph would be different though. It would be a line going in the positive direction that starts negative, crosses the x-axis which means it has velocity of 0 m/s at one point, and then becomes positive. This happens because the carts position is decreasing until it reaches its highest point as time increases which gives the cart negative velocity. The graph for velocity increases though since g is moving in the direction opposite direction until velocity reaches 0 m/s, which is the carts closest position to the motion detector. Then the velocity becomes positive because it starts increasing a changing rate as it travels in the same direction as acceleration and moves away from the motion detector increasing its position.

In this lab we observed the acceleration due to gravity of a cart as it travels up and inclined aluminum track as the velocity decreased and reached 0 m/s once the cart made it to its closest point to the motion detector and increased as the position increased. I learned how to calculate the acceleration due to gravity on an incline plane and showed the proof on how to find your experimental g. I also learned how to read and interpret the graphs such as the describing why the object has a parabolic position graph while it moves on the inclined plane and why the velocity graph is linear since the acceleration is constant. We had a large percent error when working with this lab but one major thing that could have contributed to the large error was that we could have taken a larger measurement when we were trying to find our angle of θ. We should have taken a larger measurerment of the track to find our angle because we could have gotten a more accurate measurement because there would have been a larger difference in our hypotenuse and the base leg of the triangle. I also think we had a lower percent error when we raised the angle of θ to be a bit higher also because the acceleration due to gravity could have more effect on the object since the plane is steeper.