Lab 10 - Inelastic Collisions

Objective:  To analyze the motion of two low friction carts during an inelastic collision and verify that the law of conservation of linear momentum is obeyed.

Procedure:

1) Set up the apparatus as shown in fig 1. Use the bubble level to verify that track is level. Measure the mass of each object:

m₁ = 0.5024 kg

m₂ = 0.5134 kg

Connect the motion detector to the computer and open logger pro.

2) Check to make sure the motion detector is working properly by pressing collect then moving the cart nearest the detector back and forth. Does it provide a reasonable graph of position vs time?

Yes, as the cart is moved away from the detector, the functional value increases steadily, and as the cart is moved towards the detector, the functional value decreases.

Position the carts so that their Velcro pads are facing each other.

3) With the second cart (m₂) at rest, give the first cart (m₁) a gentle push away from the detector. Observe the position vs. time graph before and after the collision. Pictured in fig 2 is an educated guess as to what the position vs. time graph should look like.

Estimation of the position versus time graph

What should these graphs look like?

The slope of the graph after the collision should be less than the slope of the graph after the collision. Pictured in fig 2 is my prediction of what the graph should look like.

The slope of the position vs. time graphs will give us our velocities directly before and after the collision. To avoid dealing with friction, we will find velocities at the instant before and after the collision.

Is this a good approximation, why or why not?

Yes, because it will give us the instantaneous velocity directly before and after impact.

Select a very small range of data directly before the collision and apply a linear fit to the range. Record the slope (velocity) of this line. Repeat for a small range of data points directly after the collision.

v₁ = 0.4171 m/s

V = 0.2305 m/s

Pictured in fig 3 is the position vs. time graph of one trial run we completed.

Actual position versus time using the computer software

4) Repeat for two more collisions. Calculate the momentum of the system the instant before and after the collision for each trial and find percent difference.

The following calculations are for the first trial in this part of the experiment.

P_i = P_f

m₁v₁ + m₂v₂ = (m₁ + m₂)V

where:

m₁ = 0.5024 kg

m₂ = 0.5134 kg

v₁ = 0.4171 m/s

V = 0.2305 m/s

P_i = (0.5024 kg)( 0.4171 m/s) + (0.5134 kg)(0)

P_i = 0.21 kg m/s

P_f = (0.5024 kg + 0.5134 kg)(0.2305 m/s)

P_f = 0.23 kg m/s

Percent difference:

[(0.23 kg m/s - 0.21 kg m/s)/ 0.23 kg m/s]*100 = 10.5%

Pictured in fig 4 is data table of each trial and the momentum calculations

 

Data table.

 

5) Add 500 g of mass to cart number 2 and repeat steps 3 and 4.

m₂ = 1.01 kg

 

Pictured in fig 5 is the data table for the additional mass on cart two.



 

data table where weight is in cart 2

 

What do the graphs of velocity vs. time, and acceleration vs. time look like?

Pictured in fig 6 are my predictions of what velocity vs. time and acceleration vs. time should look like.



predicted graphs

 

6) remove the mass from cart two, and now add the mass to cart one.

m₁ = 1.00 kg

Pictured in fig 7 is the data collected for the trials with mass added to cart one.

data table where weight is in cart one

The average of all the percent differences that we had found was 10 % difference from the law of conservation of momentum.

How well is the law obeyed based on the results of your experiment?

The law of conservation of momentum was reasonably obeyed. Although we did encounter some larger percent differences, it may have been due to the range of data that was selected.

8) For each of the above nine trials, calculate the kinetic energy of the system before and after the collision. Find the percent kinetic energy lost during each collision. Show sample calculations here:

 

∆K/K_i * 100 = percent difference

[(K_f - K_i) /K_i ]*100

[(1/2*(m₁ + m₂)*V² – ½* m₁*v₁²)/ ½* m₁*v₁²]*100

[((m₁ + m₂)*V²)/ m₁*v₁²) – 1]*100

[((0.5024 kg + 0.5134 kg)* 0.2305² m/s)/ 0.5024 kg * 0.4171² m/s) – 1]*100 = 38 % difference

Pictured in fig 8 is the data showing the kinetic energy before and after collision, as well as the percent kinetic energy lost.

data for kinetic energy

9) Do theoretical calculations for ∆K/K_i * in a perfectly inelastic collision

1- a mass (m) colliding with an identical mass (m) initially at rest

 

2- a mass (2m) colliding with a mass (m) initially at rest

3- a mass (m) colliding with a mass (2m) initially at rest

 

 

Conclusion:

During  this lab, we were able to analyze the motion of two low friction carts during an inelastic collision and verify that momentum is conserved. We learned that if we can take a system where there are no outside forces acting within the system, the momentum will be conserved. Possible sources of error in this lab would include the following:
1) Selecting a bad range of data from the position vs. time function, resulting in an inaccurate velocity value.
2) The carts may have taken a few milliseconds to actually stick together.
3) Friction was present in the system, resulting in a slight loss of momentum.
Further investigations that could be followed after this lab would be to determine the effect of the impulse on the momentum, or maybe even taking friction into account.

Lab 8 - Human Power

Purpose:  To determine the power output of the students in the class, and to create an average for them.

Procedure:
Step 1:      Determine your mass by weighing yourself on the bathroom scale. The scale read in Newtons and read 636 N, so the mass had to be solved for:

F_w = 636 N

636 N = ma à where a = g

636 N/ 9.8 m/s² = m

m = 64.9 Kg

Step 2:  Next, the height of the stairwell that we were to climb needed to be measured. Two measuring sticks (each 2 m in length) were taken to the stairwell and placed directly on top of each other. The stairwell was measured to be 4.29 m.

Step 3: At the command of the timer, one person waits to begin their short trip up the stairs, and then the timer stops the watch once they reach the top of the stairs.

 

Step 4:  The trial runs for every person are repeated, as we can have two separate times to do calculations with.

 

Step 5:   From the two trials runs up the stairs, the average time was t = 4.62 s. Calculate the personal power output.

Taking the time, the change in potential energy is evaluated as follows:

∆PE = mgh

∆PE = (105.0 kg)*(9.8 m/s²)*(4.29 m )

∆PE = 44414.41 Nm

Power (W) = (∆PE)/( ∆t)

Power (W) = 4414.41 Nm / 4.62 s

Power (W) = 955.5 W

hp = 955.5W (0.00134102209 hp / 1 W)

= 1.28 hp

 

 Step 6: The average power output of the entire class was then evaluated in one excel spreadsheet. Pictured in fig 3 is the data collected from the entire class.

Data Sheet for the entire class with average watts and HP.

Possible Sources of Error:  There are several factors for the errors in this lab.  The largest one is the time keeping.  If the timer and the person going up the steps were not in sync, then the time would be off a great deal.  Other factors might include the health of the person.  It is possible that there might have been a physical defect that would prevent them from doing this lab to their full potential.

 

Questions:

1) Is it okay to use your hands and arms on the handrail to assist you in your climb up the stairs?

Yes, although you are using multiple limbs in order to make your way up the stairs, there is still work being expended to get to the top. More energy is exerted in pulling yourself up for a faster time.

2) Discuss some of the problems with the accuracy of this experiement.

This was covered in the PSOE section above.

Human Power Follow-Up Questions:
1) Since the change in potential energy is the same for both people, the person who completes the journey in the fastest time will expend the most energy. Since power output is change in potential energy over change in time, we can see the smaller the time, the greater the power output.

2) mg = 1000 N

h = 20 m

t = 10 s

Power (W) = (∆PE)/( ∆t)

Power(W) = (1000 N * 20 m ) / (10 s)

Power (W) = 2000W, or 2 KW

3)Brynhildur climbs up a ladder to a height of 5.0 m, if she is 64 kg:

a) What work does she do?
The work that Brynhildur does climbing up the stairs is lifting her 64 kg mass up to a height of 5 meters.

b) What is the increase in gravitational potential energy of the person at this height?

∆PE = mgh

∆PE = 64 kg * 9.8 m/s² * 5.0 m

∆PE = 3136 N m

c) Where does the energy come from to cause this increase in PE?

The energy required to lift her up the ladder comes from her muscles both pulling and pushing her way up the ladder.

4) Which requires more work: lifting a 50 kg box vertically for 2 m, or lifting a 25 kg box 4 m?

      They require the same amount of work, although the 25 kg mass is being lifted to twice the height, the 50 kg mass is being lifted to a height half the amount, meaning it takes the same amount of work.

Lab 7 - Centripical Force

Objective: 
To verify Newton's second law of motion for the case of uniform circular motion.

Procedure: 
The equipment used for this lab are as follows:
1. Centripetal force apparatus
2. Metric scale
3. Vernier Caliper
4. Stop Watch
5. Slotted weight set
6. Weight hanger
7. Triple beam balance


The idea of this lab was to manually spin the centripetal force apparatus so the weight is close to the marker, time 50 rotations for each weight to figure the velocity using Newton's second law.   After a few practice runs with the apparatus, we were ready to begin.  While one of the lab partners counted, one manually spun the apparatus another used the stop watch to track the time it took for 50 rotations.  The hard part of spinning the apparatus is that the weight had to match with the indicator post on the apparatus as it spun.  Spinning it too fast would make the weight pass the post, while spinning too slow would not reach the post.  This was repeated 4 more times for a total of 5 runs.  With all of this data, we were able to calculate a relatively accurate average of the velocity.

The cetripetal force apparatus also had another post that would let us calculate the cetripetal force of the bulb by attaching weight to a string until the bulb was even with the indicator post.  The amount of weight that was attached to the bulb to get it to center over the indicator post was .6kg.  This is illustrated below.



Apparatus with the weight to measure the Centripetal force needed to center the bulb over the indicator post

After the first set of runs, we added another .1kg of weight to the bulb and started the experiment over.  In the end, we had data to calculate the cetripetal force for a .5 kg and .6 kg bulbs, along with the measured centripetal force that it took to center the bulb over the indicator post.

Data: For both of the experiments, the information was entered into Excel to do the calculations.

Data captured from the experiments and information derived from formulas used

Trial - The number of the trial out of the 5
Time - The amount of seconds that it took for 50 revolutions
LDT - 2PIr - 2*pi*.1754 measured in meters
Mass - the mass of the spinning bulb
Radius - The measurement from the center of the spinning pole to the center of the indicator post
Average Velocity - LDT / Time measured in m/s
Calculated Centripetal Force - (mass - Velocity ^2) / Radius measured in Newtons
Measured Centripetal Force - the mass of the weight it took to hold the bulb over the indicator post * 9.8(gravitational measurement) measured in Newtons
Percent Error = Absolute value of ((Measured C force - Calculated C force) / Measured C Force )* 100

Possible Sources Of Error:  Timing was the key with this experiment.  If the person spinning the bulb could not keep the bulb over the indicator post, the timing would be off.  If the bulb spun too fast, the time would be less, and if the bulb spun too slow the time would be too much.  In addition, the timer had to be paying attention to the person counting the rotations to ensure that the stop watch was started and stopped at the correct times.  This was the cause for most of the trials that were done a second time.  We tried to ensure that the count was correct by having 2 of the lab partners counting at the some time.  This prevented us from starting over at least 2 times.

Questions:
1. Calculate this force and compare with the centripetal force obtained in part 3 by finding the percent difference.  This was completed with the excel data.  The percent difference at .5 kg was 1.42, and at .6 kg it was 1.95.
2. Draw a force diagram for the hanging weight and draw a force diagram for the spring attached to the hanging mass.

Conclusion:  I feel that the lab overall was successful.  We were very close to the measured centripetal force with very little difficulty.
The purpose of this lab was to verify Newton's second law of motion in a uniform circular motion using the formula F = (mv^2) / r.  Using this formula I was able to calculate the centripetal force of the bulb as it spun on the apparatus.  This was a great way to see the calculations that we do in the problems in action with the spinning bulb.
If the force and radius are the same, and only the weight changed, then the only other variable that effect the net force is the velocity of the spinning bulb.

Lab 5 - Working with Spreadsheets

Objective:  To become familiar with electronic spreadsheets by using them in some simple applications.

Procedure: Starting with a clear Excel spreadsheet, we followed the instructions and entered 5 in column A, and a 3 in column B and PI/3.  The columns were labeled Amplitude, Frequency and Phase, to look like this:


Step 1 - adding headers and data

 

The next set of instructions were to add another column, label it X and put a 0 in the first field of the column.  In the next column, label it f(x) and enter the formula "A2*sin((B2*D2)+C1)".  This is what the spreadsheet now looks like:

Step 2 - adding more columns and the formula

 

The next step was to add a series of values in the X column ranging from 0 to 10 Radian, step by .1.  The idea was to use Excel to do this, so after the first .1 was entered, the rest of the column could be added from the previous field.  When this was completed, it looked like:



Step 3 - Getting familiar with Excel

Using Excel, the X column was extended to 100 radians.  The f(x) formula was extended to cover the range from 0 to 100.  This information was cut from Excel and inserted into the Graphical Analysis, where a graph of the data was created.  Using the program to analyse the data that was entered with the data set function, the following data was created:

 

Step 4 - importing the data to the graphical analysis program





Using another equation [f(t) = r0 + v0(t1-t0) + 1/2a(t1-t0)^2], setting up the series in Excel:





Step 5 - Creating the new data sheet

What the data looked like with the curve fit when it was imported into the Graphical Analysis software:

Step 6 - Adding new data to the computer software

Data:  All of the data was provided in the procedure.  No additional data needed.

Possible Sources of Error:  Unless the numbers were keyed incorrectly into Excel, or the graphical analysis program, there were not other sources of error.

Questions:
1. How do these compare with the values that you started with in your spreadsheet? The data from the curve fit matched the data that was used to create the series of numbers that were pasted into the graphical analysis program.

Conclusion: I can see how using MS Excel when working with the lab experiments can be a huge advantage.  Excel can be used to calculate data like series or formulas to verify or use the numbers with the lab software that is used for the lab experiments.  In future experiments, I will be using this software to assist with gathering and analyzing the data.



Lab 6 - Drag force on a Coffee Filter

Objective:  To study the relationship between air drag forces and the velocity of a falling object.

Procedure: Using coffee filters as the falling object, we were to track the velocity using the logger pro software and the motion detector.  To do this, we started with 9 of the coffee filters about a meter and a half above the motion detector and let them fall freely until they hit the floor.  The motion detector and the logger pro software would track the position over time.  Using the curve fit for a sample in the graph that was created, we were able to determine the velocity of the falling filters.  Then with excel, we tracked this motion for 5 trials and determined an average velocity of the 9 filters.  One of the filters was removed from the stack and the procedure started all over.  After the five trials were captured, another filter was removed and the same procedure followed. This continued until there was only 1 filter, and five trails were completed.  A copy of the excel sheet is posted below.

The information that was used to populate this table was pulled from the position versus time graphs that were created with the logger pro software, and the curve fit that was used to get the

Questions:
1) In the formula F-drag = K V^n, what should the n be?  The n should be a 2.  The velocity would be parabolic in nature since it is being increased by the weight (f-gravity * mass) exponentially.  This  would be true until the object reaches terminal velocity where f-drag = weight.
2) Why is it important that the shapes stay the same? Since we are experimenting with the drag force, and this is directly effected by the cross section of the object, any change in the shape of the object is going to change the cross section and therefore have an effect on the drag of that object.
3) What should the position versus time graph look like? Explain. Since the object is falling over time, the graph should be a straight line representing the distance from the sensor as time goes by.  Therefore, the shape of the graph would be a slope.
4) What should this slope represent? Explain.  Once the object can no longer accelerate, where f-drag = weight, then the object has reached terminal velocity.  Terminal Velocity is where a falling object can no longer accelerate because the drag force and the gravitational force plus mass are in an equilibrium.
5)

PSOE:
1) As stated in question 2, the shape of the coffee filters should be relatively the same no matter the count used for the experiment.  Our set of filters were slightly worn, and no matter how much we tried to keep the shape consistent, they always seemed to lay out as flat again.  This lessened somewhat as the experiment went on, but it did seem to have an adverse effect and this was really seen in the average velocity over time chart that was published in the procedure.
2) As always, the tools and equipment that were used are not the most precise ones available.  This does not prevent us from doing the experiments, just makes our results a little fuzzy.
3) Try as much as I did, but I am sure that I was not able to drop the filters from the same position every single time.  For the trials where the numbers seemed extremely different from the other trials, the trial was repeated to try to minimize this as a source.
4) No matter how many times I explained to one of the lab partners that the motion detector picks up movement in a cone, he still insisted on waving the measuring stick in front of, and around the testing area.  This also lead to several of the trials being run again in an effort to remove this as a source.

Conclusion:
The experiment was a very time consuming one.  Between the number of trials that had to be ran again due to the sources of error that were explained, and the fact that there were 45 base trial runs, this turned into a lengthy experiment.  That is not to say it was a boring or tedious experiment, just a long one.  This ended up being one of the reason that some of the trials that needed to be redone were not.  We simply ran out of time and barely got the lab completed as class was ending.
One thing that should have been done was to control the area around the experiment a lot better.  I think that this might have helped tremendously with the numbers that were calculated from our trial runs.

Lab 3 - Acceleration of Gravity on an Inclined Plane

Objective: To find the acceleration of gravity by studying the motion of a cart on an incline, while gaining more experience with the tools that will be used during lab.

Procedure: Using the motion detector, aluminum track, and cart we tested the acceleration of gravity.  The steps were pretty simple in this process.  First we leveled the track, with one end elevated.  The motion detector was located at that elevated end to track the cart.  Measurements were taken to calculate the angle of the track, which would be needed later to calculate the gravity.  A few test runs were done to make sure that the motion detector was working, and to give us an idea of how hard to push the cart.  As the cart traveled up the track it would slow to a stop and come back to the bottom of the track.  The motion detector would capture the movement of the cart and the computer would graph it.

Data: For the first set of test runs, block of wood was turned on its side to elevate one side of the track.  The measurements were taken to determine the angle of the track:

 Using the calculator, sin theta = 9.55/228 = 2.40 degrees.

The motion graph for the first set of tests we all similar to this one:

Motion graph at 2.40 degrees

Once these were understood, the testing could begin.  For this height, 3 tests were completed.  The graphs from these tests are shown below:

Test 1

Test 2

Test 3

Once these were completed and everyone felt comfortable with the results, The track was raised and 3 more tests were performed.  Here are the corresponding graphs:

Using the calculator, sin theta = 18.65/228 = 4.69 degrees.

Motion Diagram at 4.69 degrees

Test 4

Test 5

Test 6

Possible sources of error: This lab seemed to be stricken with error.  The first go around one of the linear fits was not completed, and this was not noticed until after the equipment was put away.  This was partially due to the fact that most of the team members were not reading the documentation on how the lab was to be done.  As a result, after half of the team had left, the second half stayed to completely redo the lab.  Most of the team just had to take his word that he did the lab correctly.

Common sources of error are the measurements.  We have measured the best we can with the yardstick that was provided, and the motion detector gets distracted easily and picks up various other movements.  This was the cause for several runs to be completed a second and third time to be able to get data that we could work with, and was closer to the expectations.
 

Questions: For each run, we were expected to calculate the force of gravity using the formula that was provided in the write up.  For each test run, the calculations are illustrated below.

Conclusions:

Lab 2 - Acceleration of Gravity

Objective:  To determine the acceleration of gravity for a free falling object, and to continue to gain experience with the data collecting hardware and computer programs.

Procedure:  Once the hardware is set up and tested, toss a ball into the air and track the position versus time and velocity versus time using the graphing software.  Track this for at least 5 repetitions and average out the data.

Data:  Below are the 5 data captures that were used for this experiment.  Curve fit and line fit were used to determine the best tests to use



Ball Toss Test 1

Ball Toss Test 2

Ball Toss Test 3

Ball Toss Test 4

Ball Toss Test 5

Possible sources of Error:
1. The easy answer to this is that we had different students tossing the ball in different ways.  One student would throw the ball higher than another, there was hesitation and the way the ball was held seem to make a difference as well.

2.  Since the tracking sensor is just that, and sensor, it would sense other movement with the movement of the ball.  For example, several of the test that were not used were due to the movement of the ball thrower, people walking by and other movements that occurred during the testing.

3.  Our own interpretation as a group.  What I mean by this is that if one of us thought that it was a good result, someone else may have wanted to throw it out.  This may have caused us to settle for a less accurate test, or may have given us an improvement.  Hard to say really.

Questions:
1. Why should it be a parabola?  The position versus time graph should be parabolic because gravity = 9.81 m/s^2.  Acceleration is constantly slowing down on the way up, and constantly speeding up on the way down at the same rate.  In relation to this experiment, gravity slowed down the ball on the way up, and sped it up on the way down.  When interpreting to velocity versus time, the line is linear.

2. Why does the slope have a negative slope, and what does the slope of this graph represent? The graph represents gravity, and since gravity is -9.81 the linear graph will slope in the negative direction.

3.  Results from falling body experiment.

Percent error = (measured - actual)/actual * 100

Actual = 9.81

Average from the 5 Tosses

Conclusion: The experiment produced results that were close to what was expected.  The percent error could have been better with a little more time. and more tests.  The results were pretty close to the given force of gravity that was used in the percent error calculation.

In the end, we were able to determine a close proximity to the force of gravity with the experiment that was performed.  Given additional time and practice, I feel confident that we could have come very close to 9.81.