APR-01-2015 Centripetal Force with a Motor

Centripetal Force with a Motor

Purpose:  To determine the relationship between an angle theta with the vertical and omega on a motor-driven apparatus swinging a mass connected to the apparatus via a string.

Procedure:  The apparatus was set up previously by our instructor.  It is depicted below.

(Motor-driven apparatus swinging a mass connected by a string)

A stopwatch was used in order to determine the time required for the swinging mass to make 10 complete revolutions.  This is used to determine the period T of the circular motion.  The period can then be used to calculate what omega is, this gives us our measured value of omega.

We can also calculate what omega SHOULD be by taking additional measurements, including the radius of the motion, the height of the object from the floor, and the length of the string connecting the object and the apparatus.  We can use these values, along with the application of Newton's Laws, in order to come up with a theoretical value of omega.  Firstly, in order to accomplish this, we need to express omega in terms of an angle theta.  This derivation is depicted below (along with a diagram of the apparatus and relevant measurements).

(Derivation of omega in terms of an angle theta)

We took measurements in six different instances.  In each run, we recorded the time it took the object to make 10 revolutions about it's circular path, the objects total distance (radius) from the center of the circular path it's traveling, as well as the height at which the object was relative to the ground.  Having previously measured the total height of the apparatus to be 200 cm, we can utilize the object's height relative to the ground in order to determine the angle theta being made with the vertical.

Shown below is the gathered data; values in highlighted in green are our sources of uncertainty which propagate our uncertainty in our calculated value of omega, the values highlighted in orange should be nearly equal (except they are not, due to propagated uncertainty).  The measured omega value was attained using the formula omega = (2*PI) / T     where T is the period of circular motion.

(Excel spreadsheet organizing data)

If we plot our measured value of omega versus our calculated value of omega on an xy graph, the slope SHOULD be 1.  Displayed below is my attempt to use LoggerPro on my Windows machine...

(LoggerPro graph failed attempt at linear fit)

For some reason, after naming my axis and entering values, I'm unable to have the program proceed with a linear fit.  HOWEVER, I am able to input values on a plain xy graph and the linear fit works....  This is depicted below.

(LoggerPro graph displaying slope)

We came out with a correlation of 0.9994 and slope of 1.063.  These small discrepancies are due to the propagated uncertainty in our measurements of time (from stopwatch) and our height measurements (which was taken from a 2 - meter stick).

MAR-25-2015 Centripetal Acceleration vs. Angular Frequency

Centripetal Acceleration vs. Angular Frequency

Purpose:  To determine the relationship between centripetal acceleration and angular speed.

Procedure:  For this lab, the apparatus depicted below was previously set up by the instructor.

 (Full apparatus setup)

A photogate was used to measure the length of time it takes for the disk to complete a rotation.  The apparatus was powered by electricity at varying voltages.  The acceleration was recorded via an accelerometer hooked up to LoggerPro software.  We gathered data for 10 rotations with an initial start time to to the completion of the 10th rotation t10, increasing the voltage to produce smaller rotation times and greater centripetal acceleration.  The radius of the disk was measured at approximately 13.8 cm.   

Data Gathered:  Shown below is an Excel spreadsheet containing all the data which was gathered, as well as calculated values of angular speed and values the radius should be.

(Data gathered and organized in an Excel spreadsheet)

When the measured acceleration is plotted against our calculated value of omega^2, the slope of said graph yields the radius of the disk!

(LoggerPro graph of acceleration vs. (angular speed)^2)

The slope of this line is 13.71 cm, this value for the radius of the disk matches well with our average calculated value, as well as the physically measured value.  The uncertainty in this exercise is mostly in the physically measured value of the radius of the disk, as this is an eyeball measurement, whereas all other data was collected directly and live via LoggerPro software.

MAR-16-2015 Modeling Friction Forces

Modeling Friction Forces: Part 1

(Static Friction)

Purpose:  This lab is broken into five sections, each dealing with either static or kinetic friction between the felt on the bottom of a block and the surface it's traversing.  We will be gathering data using motion sensors and force sensors for specific sections of this lab.

Procedure:  We first determined the mass of a single block (the one with the felt bottom) which we measured to be 121 g.  We then tied a string to the block, with the opposite end running over a pulley at the end of a table attached to a bent paperclip, which is in turn attached to a Styrofoam cup.  This setup is depicted below.

(Part 1 initial setup)

We then began to add small amounts of water into the cup.  This was continued until there was JUST enough water in the cup to cause the system to begin to move.  Once this occurs, we measured the mass of the cup and water to be 34 g.  

(The cup, bent paper-clip, and water combined mass JUST starts to move the system)

Next, we measured the mass of a second block, and stacked it on top of the original.  We then added more water into the cup until this new system JUST began to move.  This process was repeated until a total of four blocks were stacked upon each other.  Depicted below is a graph which contains the Normal force as the x-axis, and the force of static friction as the y-axis.

(LoggerPro linear fit graph of data collected)

The above graph shows that the slope of this line is the coefficient of static friction, and that value is 0.3712.  Below is handwritten work regarding the same data.

(Work for Part 1 of lab)

Modeling Friction Forces: Part 2

(Kinetic Friction)

Procedure:  In this section of the lab, we used a force sensor in order to determine the force of a constant pull on a block of wood traversing a surface.  First, we calibrated the force sensor using a 500 g hanging mass.  We then determined the mass of the wooden block that has felt on it's lower surface.  We held the force sensor horizontally and zeroed it.  Afterwards, we tied the force sensor and wooden block together with string.  We then used LoggerPro software to record and analyze the average force exerted on the block during a pull at constant speed.  In order to get an average force value, we used the Analyze menu in LoggerPro, then selected Statistics, then recorded the mean value of the pulling force.  This process was repeated for a total up four block stacked upon each other.  The LoggerPro graph is depicted below.

(LoggerPro linear fit graph for part 2 of lab)

It appears the above graph is incorrect in nature, as I doubt the coefficient of kinetic friction is 0.020. The data I received from a lab partner was perhaps corrupted when transferred.  The above graphs were made at home without the aid of experimental equipment.  I'll attempt to rectify this portion of the lab post-haste.

Modeling Friction Forces: Part 3

(Static Friction From A Sloped Surface)

Procedure:  We placed a block on a horizontal surface, and then raised one end of the surface until the block JUST started to slip.  At this point of maximum static friction, we recorded the angle between the horizontal and tilted surfaces.  Below is the work done to calculate the coefficient of static friction between the block and the surface it was attempting to traverse.

(Determining the coefficient of static friction)

Modeling Friction Forces: Part 4

(Kinetic Friction From Sliding A Block Down An Incline)

Procedure:  We placed a motion sensor at the top of an incline which was steep enough to cause the weighed block to accelerate downward.  Recording the angle at 20.6 degrees above the horizontal.  The physical setup as well as the solution are pictured below.

(Inclined surface at 20.6 degrees above the horizontal)

(Solution for coefficient of kinetic surface)

11-MAR-2015: Modeling the fall of an object falling with air resistance

Part 1

(Determining the relationship between air resistance force and speed)

Purpose:  The purpose of this portion of the lab is to collect data regarding the velocities of falling coffee filters and relate it to the force of air resistance acting on said coffee filters.  We will use this data and plot it appropriately in order to determine the values of k  and n in the following power-law equation:

   Fresistance = k * v^n

Procedure:  We began by walking over to the design technology building.  Inside, there is a balcony on the second floor which can be easily viewed from the nearby staircase.  We then used video capture software on our MAC systems to record the falling coffee filters (first one, then 2 stacked, up to a total of 5 stacked at a time).  Utilizing the LoggerPro software back in class, we created a reference length distance within the recorded video.  We then plotted points on the video as the object fell (at 1/30th second intervals).  Gathering all this data together, we plotted position vs. time graphs for each individual run.  An example is seen below:

(Position vs. Time graph of first run; a single coffee filter)

Having measured the mass of a single coffee filter (m = 0.00092 kg) we were able to produce a Force vs. Velocity graph.  Such graph is seen below:

(Force vs. Velocity Graph)

It is from this graph, we can see the values of k and n for our power-law equation. 

Where k = 0.01193 and n = 1.951.

Part 2

(Modeling the fall of an object including air resistance)

We gathered and organized all of our data into an Excel spreadsheet in order to determine the terminal velocity of each run (that is, a single coffee filter, to five stacked together).  

(Excel spreadsheet containing all gathered data; this section represents a single filter)

If we scroll down the velocity (v) column, we'll see that it begins to become a constant value as time goes on.  This value, is the value of terminal velocity of the coffee filter.

(Excel sheet for run 1; terminal velocity = 0.866282 m/s)

Conclusion:  To be 100% honest, this particular lab write-up is (when compared to my others) not up to standards; I was sick, as I'm sure was noticed, most of the week this lab occurred.  Therefore, my understanding of exactly what went on during this lab is, somewhat fractured.  I do understand however, the concept of how terminal velocity is represented in the Excel datasheet.

06-MAR-2015: Propagated uncertainty in measurements

Lab 6: Part 1

 (Measuring Density of Metal Cylinders)

Purpose:  In this lab, the main purpose is to determine just how uncertainty in measurements can lead to uncertainty in final results of calculations which utilize said measurements.  This is done using methods of differentiation learned in calculus.  The first half of this lab focuses on determining density of three different cylindrical shaped objects (as well as determining our uncertainty in our density calculations).

Materials:  For the first portion of this lab, we were given three cylindrical pieces of metal.  We then used a Vernier Caliper to measure their height and diameter, and we used a scale which we recorded their masses from.  These items are depicted below.

(Vernier Caliper and the three unknown cylindrical shaped metals)

(Measuring the mass of cylinder #1)

Procedure:  Firstly, we measured and recorded each metal cylinders height (cm), diameter (cm), and mass (g), these results are shown below.

(Measurements of unknown cylindrical metals)

While we can calculate the density of each cylinder given these measurements, it is unclear just how certain our results are.  To get a more precise result, we need to take into account the uncertainty in each individual measurement and then determine our uncertainty in our density calculations.

We begin determining the uncertainty in our density calculations with the simple density definition.

     D = (m / V)

We know the volume of a cylinder is given by:

     V = pi* r^2 * h    Where r = (diameter / 2)

Combining these yields:

     D = [ m / (pi * (d/2)^2 * h) ]  --- > D = (4m / pi * d^2 * h)

We now have an equation representing density in terms of each metal's mass, diameter, and height.

Using methods of differentiation from calculus, we can determine the uncertainty in each measurement in order to determine the uncertainty in our density calculation result, one such instance is depicted below:

(Calculation of uncertainty in density measurement for cylinder #1)

So, using the method depicted above, we have:

  D1 = 8.80 g/cm^3 (+ -) 0.201 g/cm^3

  D2 = 2.79 g/cm^3 (+ -) 0.0561 g/cm^3

  D3 = 6.64 g/cm^3 (+ -) 0.145 g/cm^3

Comparing these outcomes with a table of accepted values of densities of various metals,

we see that metal 1 corresponds to Copper (Cu), metal 2 with Aluminum (Al), and metal 3 with Cerium (Ce).

Conclusion:  This lab is meant to develop the skills and procedures necessary to calculate just how uncertain experimental results can be.  We use methods of differentiation in order to find a range of values for our final result, which should (and did in this case) land within the scientifically accepted values for such calculations.

Lab 6: Part 2

(Determination of an unknown mass)

Purpose:  The purpose of this section of the lab is essentially identical to the part prior, only difference being tools of measurement, and instead of solving for density we will be solving for the mass of a hanging object placed in a system of static equilibrium.  Our final result will have the mass of the object, as well as a range of uncertainty associated with such result.

Materials:  The systems in which the hanging masses belonged were set up prior to us beginning this lab.  Depicted below are photos of 2 out of the 3 set ups.

(Unknown mass #7)

(Unknown mass #1)

Procedure:  The measurements required to determine the mass of the object are 1) The forces (N) and 2) The angles between a horizontal line and said forces (degrees).  It's important to remember, that when using these recorded angles in calculations, they must be converted into radians.

Depicted below is the work done in order to determine the unknown mass along with the uncertainty in it's determination:

(Solving for propagated uncertainty for unknown mass #7)

Using the same method above for unknown mass #1, we have:

   m #1 = 0.743 kg (+ -) 0.414 kg

Conclusion:  This second portion of the lab demonstrates how the propagated error becomes larger when the measurement tools become less and less precise, hence the larger range of uncertainty.

04-MAR-2015: Non-Constant Acceleration Problem

Purpose:  This lab is based on a non-constant acceleration problem detailed below:

     - "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 level ground.  At that point a rocket mounted 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."

The purposes of this lab include solving this problem analytically as well as numerically (using Excel).  Using the given data above we can obtain a function representing the elephant's acceleration at any given time.  

Materials:  The materials for this lab include: Microsoft Excel, integral calculus techniques.

Procedure:  Solving this problem analytically involves integrating the acceleration function with respect to time to obtain the elephant's velocity function, then integrating once again to obtain the elephant's position function.  Since the problem states to find the distance of the elephant once it is at rest, we set the derived velocity function equal to 0 and solve for t.  This gives us the time the elephant is at rest.  We then evaluate our derived position function at this time to give us the position of the elephant when it comes to rest (which is what the problem is asking for).  Relevant equations are listed below.

     - Mass of rocket over time function

       m(t) = 1500 kg - 20 kg/s*t

     - Deriving acceleration function

     a(t) = Fnet / m(t)

     a(t) = -8000 N / (6500 kg - 20 kg/s * t)

     a(t) = (-400 / (325 - t) ) m/s^2

     - Derived velocity function

     v(t) = [25 - 400ln(325)] + 400ln(325 - t)

     - Solving for time elephant comes to rest

     0 = [25 - 400ln(325)] + 400ln(325 - t)

     t = 19.70 s

     - Derived position function

     x(t) = [25 - 400ln(325)]t + 400[(t - 325)*ln(325 - t) - t + 325ln(325)]

     - Solving for position elephant comes to rest

     x(19.70) = 248.7 m

The equations above are simply a summary of the work involved in integrating this manually, the overall work would take up a significantly larger portion of this blog entry.  We were lucky the initial acceleration function was able to be integrated manually, as this is not always the case.  After solving this problem analytically, we then began to solve it numerically utilizing Excel to input and manipulate our data.  Depicted below is the beginning of said Excel spreadsheet.

(Top of Excel spreadsheet)

If we scroll down to where column A (time) is equal to our analytically calculated value of 19.70 s, we see that the value of column F (position) is in accordance with our calculated value of 248.7 m.

(Section of spreadsheet which reveals the position at 19.7 s to be 248.70 m)

Conclusions:  

1)  As it is displayed in the above screenshot, the value of the elephant's position we calculated analytically using integration (248.7 m) is equivalent to the position of the elephant in our spreadsheet's data table.

2)  You know the time interval is small enough when the velocity is extremely close to 0, (- 0.01 s in this case).  If there were no analytic result with which to compare our data table to, another way to determine the final position of the elephant is to examine the values of position in the table.  Taking a closer look reveals that there is a MAX number of 248.70 m, values for position before and after are all less than that at t = 19.70 s.  

25-FEB-2015: Free-fall Lab

Free Fall Lab

(determination of "g")

Purpose:  To examine and determine if the following statement is valid:  "In the absence of all other external forces except gravity (g), a falling body will accelerate at a constant 9.81 m/s^2.  Also, this lab is intended to introduce the usage and usefulness of numerical processing software (in this case, Microsoft Excel).

Experiment Equipment:  In order to perform this experiment, we will utilize a Spark Generator (which sparks every 1/60th of a second) held in place by an Electromagnet which is attached to the top of a Sturdy Column.  This set-up creates a 1.5 m falling distance (with the total height of the assembled apparatus being 1.86 m).  In place along the path of free-fall, is a strip of Spark-sensitive Tape.  The assembled apparatus includes a heavy tripod base with screws for leveling purposes, a free-fall body (object), a weighted clip to anchor the spark paper, as well as a power supply powering the spark generator.

Procedure:  Steps listed below

1.   Turn the dial hooked up to the electromagnet up slightly

2.   Hang the wooden cylinder with the metal ring around it on the electromagnet

3.   Turn on the power of the sparker thingamugug. 

4.   Hold down the spark button on the sparker box.  (This begins to spark at 60 Hz.)

5.   Turn the electromagnet off so that the object descends in free-fall.

6.   Turn off power to the sparker thingamugug.

7.   Tear off the spark-sensitive tape; replace with new strip for next student.

The spark burns on the tape indicate the position of the falling mass in intervals of 1/60th s.  This strip of tape essentially reveals the relationship between position vs. time of the free-fall object (with gravity being the ONLY force acting on it).

Data Collection:  In order to gather and interpret this data more clearly, we placed the spark-sensitive tape on the top of our desk and secured it in place (fully extended) with the aid of some adhesive tape.  We then acquired a 2 m stick in order to measure the displacement of the falling mass over continuous intervals of 1/60th s.  We recorded the 19 values depicted below.

 (recorded value of displacement)

Organizing Collected Data into Excel:  We began with a new, empty spreadsheet.  In the first column (A), we entered in the time interval of 1/60th s into 19 rows.  In the following column (B) we entered our recorded displacement values at each corresponding intervals of time.  In column (C) we utilized the formula capabilities of Excel to calculate change in position of the mass for each interval of time.  In column (D) we then calculated the mid-interval time (that is, the half-way point between each interval).  In turn, we used the same tool once again in column (E) to determine the speed of the free-falling mass at every mid-interval point in time.

(spreadsheet generated using recorded data from spark-sensitive tape)

Once all the data was organized neatly via Excel, we then selected the last two columns (D & E) and used Excel to visually graph the data in a Speed vs. Time Graph.  We also used the same method in order to graph a Position vs. Time Graph using the values of distance and time from the spreadsheet.  Both graphs have their corresponding equations displayed as well, these are useful in calculating our recorded value of "g".  The graphs are depicted below.

(Speed vs. Time Graph)

(Position vs. Time Graph)

Questions/Analysis:

1.  Show that, for constant acceleration, the velocity in the middle of a time interval is the same as the average velocity for that time interval

  -  To demonstrate this, we will use the time interval of: to = 0.1000 s and  tf = 0.1167 s. 

We know that the average change in velocity is equal to it's change in position divided by the change in time during it's position change.  So:

      Vavg = (xf - xo) / (tf - to)

               = (12.40 cm - 9.90 cm) / (0.1167 s - 0.1000 s)

               = 150 cm/s

If we look at the graph of speed vs. time, and examine the speed of the object at the mid-interval at time 0.1083 s, we see that the instantaneous velocity is 150 cm/s, which agrees with our average velocity calculation.

2.  Describe how you can get the acceleration due to gravity from your velocity/time graph.  Compare result with the accepted value of g.
  -  The graph displays the corresponding equation of the line.  This equation is velocity at any given time, or V(t).  We know that acceleration is the change in velocity over the change in time.  So,

    V(t) = 950.53t + 50.658
    V(0.3000) = 335.8 cm/s
    V(0.0167) = 66.53 cm/s

    a = (Vf - Vo) / (tf - to)
    a = 950 cm/s^2 = 9.50 m/s^2

  Our calculated value (utilizing the information provided from the graph above) of g seems to be off by 0.3 m/s^2.

3.  Describe how you can get the acceleration due to gravity from y our position/time graph.  Compare result with accepted value of g.
  -  We can derive an equation for the acceleration of the free-fall body at any given time by taking the second derivative of the equation displayed in the position vs. time graph.

   f(x) = 472.88x^2 + 50.937x + 0.0373
f ' (x) = 945.76x + 50.937
f ''(x) = 945.76

  The second derivative of the position function shows that no matter the time, the acceleration is constant at 945.76 cm/s^2, or 9.50 m/s^2.

Experimental Uncertainty:  In order to obtain an actual value for the uncertainty in this experiment(which tells the reader about the instruments used to makes measurements), we gathered the entire class' calculated values of g and inputted them into a new Excel spreadsheet.  What we did next, was use Excel to determine the Standard Deviation of the Mean of the class' values of g.  This spreadsheet is depicted below.

(Spreadsheet used to determine standard deviation from accepted value of g)