Welcome Back!!!

Here's to a wonderful 2014-15 school year!!

For my mechanics classes, if you need some vector addition review, check this out.
If you want a review of vector multiplication, check this one out.

A motto for us:

Learn a ton while having some fun!

Lab Activity: Series RC Circuit Behavior

Purpose: Observe general behavior of RC circuits before getting into the details in class. We'll look at both charging and discharging cases.

General Setup: You’ll have a power supply, some resistors, a large capacitor, and some multimeters. Set up a circuit with the power supply, resistor and capacitor all in series with each other.

Case A – Small resistance

• Connect a 10-ohm resistor in the circuit with the power supply set to zero.  Make sure there is no initial potential difference across the capacitor. 
• With one person measuring the potential difference of the capacitor and another measuring the potential difference of the resistor, quickly turn up the voltage of the power supply (to ~10 V or so).  Write down observations below:

What happens to the voltage across the capacitor?

What happens to the voltage across the resistor?

Try to explain what you think is physically happening in the circuit based on your observations:

Sketch graphs of V_Cap vs. time, V_R vs. time, and I vs. time, as the capacitor charges. *Challenge: What function might produce each graph? Try to guess before we find the details!

• When the voltage across the capacitor is a maximum, how much charge is it storing?  Use your max. voltage measurement.

• When the voltage across the capacitor is a maximum, how much energy is it storing? U = ½ QV = ½ CV².

• Calculate the product RC.  This actually has units of time and is called the time constant of the circuit.  Show that (ohms)(farads) = seconds. 

Quickly turn off the power supply.  What happens to the voltage across the capacitor?

What happens to the voltage across the resistor?  Explain what you think is happening now, and sketch graphs of V_Cap vs. time, V_R vs. time, and I vs. time, as it discharges.

Case B – Large resistance

• Measure the resistance of the other resistor: __________ W. Connect it in the circuit with the power supply set to zero.  Make sure there is no initial potential difference across the capacitor. 
• With one person measuring the potential difference of the capacitor and another measuring the potential difference of the resistor, quickly turn up the voltage of the power supply (to ~10 V or so).  Write down observations below:

What happens to the voltage across the capacitor?

What happens to the voltage across the resistor?

Is there anything different compared to your first circuit?  If so, what? Think rates.

Try to explain what you think is physically happening in the circuit based on your observations:

• Calculate the time constant, RC.  Based on the two circuits you have observed, interpret what the value of the time constant of the circuit refers to.

Wait a couple minutes to let the voltage build up on the capacitor.  Quickly turn off the power supply.  What happens to the voltage across the capacitor?

What happens to the voltage across the resistor?  Explain what you think is happening now.

Theory into Reality: (wait until we do this type of calculation in class)

What is the equation we derived for charge as a function of time for a charging capacitor?

For the first circuit (10-ohm resistor), how much charge is on the capacitor after 0.05 seconds?  After 0.5 seconds?

What is the equation we derived for charge as a function of time for a discharging capacitor?

When the capacitor discharges, how long does it take to lose 25% of its original charge?

The Need for Cybersecurity

For those who have an interest in computer science, politics, economics, international relations, business, the military, espionage, and many other subjects and topics, cybersecurity is something you have hopefully heard of, but does not get an enormous amount of coverage in the popular press. In my opinion, this is perhaps the most important topic for personal and national security that exists in the modern era. Here is a discussion with Richard Clarke, who has served multiple presidents on national security issues over many years, giving examples of what will almost certainly happen at some point somewhere in the world, via attacks on computer networks.

Lab Activity: Determine Speed and Error Analysis of Data

Purpose: Given the essential equipment, you and your partners will have to determine methods of measuring speed as precisely and with as little uncertainty as possible.  The focus is on data analysis, particularly on methods to determine uncertainties and propagate them to results.

Materials:      Marble             Ramp               Meter Stick      Stopwatch        Graph paper

Procedure:

            You are given the task of trying to measure the speed (distance / time) of a marble that rolls off a ramp.  You want to try and maximize accuracy and precision while limiting uncertainties.  As far as procedure, write detailed steps you decide on for a method that involves extracting a speed from a graph.

            Do not expect much help from me as far as measurements.  I suggest that you take a few minutes before starting to write down possible uncertainties (experimental, which you may be able to minimize) and then give this a try.

Data Table: Develop a neat, organized, and appropriate data table for all measurements for your procedure.  Your group will need to take some number of trials for time measurements for each distance that you use.  In your data table(s), include the standard deviation for each set of time measurements.

                             s_time = æ [S(t_avg – t_i )²] ö ^½                                           

                                          è       N – 1         ø

Questions/Analysis:

1.      What potential experimental errors did you and your partner(s) decide were the most significant?  Explain why these caused the most worry.  Do not just say something like “human error” or “calculation error;” but be detailed and thorough.

2.      Is it possible to “minimize” errors in this experiment?  Why or why not?  (Think about your equipment)  What did you try in order to minimize uncertainty on time measurements?

3.      Why is it important to be able to accurately determine speeds?  Think in terms of everyday events or situations; try to come up with two examples.

4.      Why is it important to be able to precisely determine speeds (i.e. measure down to more decimal places)?  Think in terms of everyday events or situations; try to come up with two examples.

5.      Make a graph of distance (y) versus time (x) using Excel.  Find the best-fit line and R² value for your data.  Then, put on error bars in the time dimension (this will likely have to be done by hand after printing out your graph.  Don’t forget to label axes, include units on the axes, and so on, when making a graph).  The error bars for time will make use of the standard deviation for that particular set of time trials.  It is OK for different points on a graph to have different sized error bars; in fact, you should expect to have different sized error bars for each data point.

6.      If you started your marble at the same height on the ramp each time you can assume it is going the same speed when it rolls off on a flat surface.  When you made your graph of distance versus time, this is why you would expect a straight line.  Are all your points on the line?  Don’t feel bad if they are not; rarely will data lie on the same line.  Determine and report the “best” speed of your marble as well as estimates of uncertainties on your speed result from your graph, using max and min slope lines as determined by your error bars.  In class, you will see an example of this.

7.   Calculate the uncertainty for your speeds by using the quadrature method for

      independent measurements (it is safe to assume that distance and time measurements

      are independent of each other and random since you’re using different measuring  

      devices for each).  Do this for each individual point of distance and time.  How do

      these uncertainties compare to those from your graph, using the max and min lines?

                       

                        dv = v [(dx/x)² + (dt/t)² ] ^½   , where dt is the standard deviation for that set

           of times, and dx = 1 cm = 0.01 m.

Note that dx/x and dt/t are called the fractional uncertainties of the distance and time measurements.  

We are adding the fractional uncertainties in quadrature (looks a lot like Pythagorean theorem!).

Grading:

Title (always have this)              1 point

Purpose (always have this)        1 point

Materials and Methods (Procedure)      3 points

- procedures should be written such that someone who has not done this before can recreate your lab

Data and Graph            5 points

Analysis Questions        10 points

Above and Beyond: If interested in learning the theory behind least-squares fits to linear data, look here.

Lab Activity: Air Friction

Purpose: You will investigate how air friction causes terminal velocity using coffee filters.  Part of this will include Interactive Physics computer simulations for multi-dimensional motion and air friction; this program is only on school computers.

Materials:      Meter stick                   Stop watch and/or video               Coffee Filters

For your report: You will need purpose; materials; data; and analysis sections for your write-up. It is always a good idea to organize data in tables so they are clear and neat, and include units on all measurements and results.

Keep in mind the BIG IDEA is that air friction (and friction in fluids in general) depends on how fast you move, f_air = -kv, where k is a positive constant.

Read through each analysis part below carefully, because it will guide you through what we are looking for.  Write things up using complete sentences.  I recommend Google Docs for your report (just need a single report for the group).

Procedures:

Make some predictions prior to actually measuring the terminal speeds of the falling coffee filters.

Question: Does mass affect the terminal speed?

            You can control the mass by using different numbers of filters.

Predict: What should happen to terminal speed as the mass increases?

Do it…make an appropriate data table with terminal velocity as a function of mass.  Do several time trials and include standard deviations.  Bonus: Determine the uncertainties on the terminal velocity results.  You will need to do this using propagation of uncertainties as we have done in the past; see Above and Beyond below. 

Do your best to estimate how long it takes for the filters to reach terminal velocity upon release.  You’ll probably want to drop the filters from 2-3 meters high, so you get terminal velocities. 

Questions/Analysis:

1.      Determine the terminal speeds for at least five different masses of coffee filters.  Estimate all measurement uncertainties and record those with your data. 

Above and Beyond: This includes using the quadrature method for determining dv values for each terminal speed.  Remember units are important on data and results. 

dv = v [(dt / t_avg)² + (dd / d)² ] ^½  

You and your partners need to come up with a reasonable estimate of uncertainty on the distance that the filters will fall; think of how well you can read the metersticks. 

2.      Use your measurements of terminal speed to determine values for k.  Include these in a data table.  What are the units of k?

3.      Write concise conclusions of what your data suggest about the effect of mass on terminal speed.  Make a graph in Excel of terminal speed as a function of mass (# filters) from your data.  Use as large a range of mass as possible, up to a point where it does not have a measureable terminal speed (where it continues to accelerate before hitting the floor). 

4.      Sketch graphs (i.e. do not need numbers on the graph) of velocity as a function of time and acceleration as a function of time.  Put graphs for different masses on the same set of axes so you can show a comparison of the effect of mass on terminal velocity and acceleration. Use different colors, or solid-dashed-dotted lines, to distinguish the different graphs.

5.      Do a few trials for the other sized coffee filters, and draw any conclusions about how the size of coffee filters affect the terminal velocity.  Explain/support your conclusions in terms of data and observations.  Try to do this by holding mass constant between the filters as best you can. 

6.      For two of your small coffee filter examples, determine the percentage of kinetic energy that is lost due to air friction. Hint: think about how fast a filter should land if there is no air friction, and compare to your terminal speed at which it lands.

7.      Log into your school account. Unfortunately, Interactive Physics is not online, only on school computers. Go to Programs, and go into the Science group of programs. You should find Interactive Physics. Go into IPFiles, and then Physics Experiments.  In that folder find the Air Resistance folder.  There should be 4 computer simulations, and run all four. 

In each one, you can select different values of k. In some you can change mass, and in some you can change surface area.  Run a series of controlled computer experiments for each simulation, and write summaries of observations/measurements and your conclusions about the effect of the various parameters on the trajectories of projectiles when varying air friction, terminal velocity, and so on. 

Are these computer experiments consistent with what you see with the coffee filters?  Explain.

The point of all this is to gain a good conceptual understanding of what air friction is all about, and gain a better understanding of the complexity of reality, as opposed to the ‘physics land’ we tend to visit in most problems. Still, keep in mind that we are using a highly simplified model for air friction, and reality is still quite a bit more complex than we are treating air friction for things like cars, planes and rockets moving through the atmosphere (aerodynamics).  Aerospace engineers need to deal with the complexities in a major way. J

Lab Activity - Simulate Half-life with M&Ms

Radioactivity - Mmmm & Mmmm, Good

Purpose:

Radioactivity was a phenomena that helped change the nature of science as the twentieth century began (helped lead to fundamental changes in chemistry, nuclear physics, biology, archeology and geology).  And one of the most important ideas to come from the studies of radioactivity is the concept that it is a statistical and probabilistic process.  This means that one can apply statistical and probabilistic methods to the study of radioactivity, and one of those statistical measures is the half-life.  You will begin to gain understanding of half-lives by using some pretty sophisticated equipment.

Equipment:     M & M’s (pretty sophisticated, huh?!)

Procedure:

Work your way through the following questions with your M & M’s; choose one side of your M & M’s to represent a ‘living’ radioactive atom, and the other side to represent a decayed atom that will be removed from your sample.  Dump your ‘living’ pieces on your table and remove those that have just decayed. Continue this until all have decayed.

Questions and Analysis:

1.       Make a data table to keep track of how many M & M’s decay each time you dump them on the table.  Make sure to count the original number of pieces before you begin.  After all the pieces have decayed, make a graph of the number of pieces left after each ‘half-life’ versus the number of the turn you dumped them.  Is the shape of your graph linear?  If not, how would you describe it?

2.       The half-life for carbon-14 (this is an isotope of normal carbon-12; it just has two extra neutrons that make it radioactive) is about 5700 years.  What does this mean if you have a sample of 1000 carbon-14 atoms?  What does this mean if you have a single carbon-14 atom?

      3.   Make a graph of the number of carbon-14 atoms (suppose you have an initial

            sample of 1000 atoms) as a function of time, knowing the half-life is 5700 years. 

            For an initial sample of 1000 carbon-14 atoms, approximately how long would it

            take to have only 100 carbon-14 atoms remaining from the original sample? 

           Make your approximation from your graph.  Then, calculate it from the decay law:

          N  =  N_oe^{- t / t}

          where t = t _½ / ln2; this is sort of like an average time that is used for radioactivity

          and is different for each radioactive material.  Work together to figure this out, just

          like a biologist or archeologist would have to do.  Some call τ the decay constant or

          time constant for the material being considered.  We will see time constants again for

          certain types of circuits in E&M.

4.  Explain how scientists can use carbon-14 as a way of measuring the age of bones.  

     Would you trust carbon-14 dating for objects that may be millions of years old? 

     Why or why not?

5.       Name three other phenomena, events, activities, etc., that require probability to

     describe them.  Also, in your own words, define “probability.”

Above and Beyond: If curious, check out details about radiometric dating techniques and the math behind those techniques.

Lab - Video Game Physics Analysis

Lab: Video Game Physics Analysis

Purpose:

The research question is: does your video game follow physics phact or phiction? You and a partner will need to play about 10-20 seconds of an online video game that involves some type of collision, and then analyze the motion to determine speeds, accelerations, and whether energy and momentum are conserved.   

Materials:

Students will need a computer with Internet access.  Students also need an analysis program such as Logger Pro or Tracker, which will be used to better analyze features of the game being played.  A screencast video of the student playing the game will be made using the online program called Screencast-o-matic.

Background:

Using video technologies of all kinds have opened up new ways of making connections between the physics we study and any phenomenon that can be captured on video, including video games.  Any game that has moving objects, interactions between objects, projectiles, orbits, collisions, and any other physical event, can be analyzed to see if our real laws of physics are being used to create the video game environment, or if the game is simply, for lack of a better word, being ruled by nonsense as far as the physics is concerned. 

By measuring distances, times and paths, and estimating masses, and scaling them by some appropriate scaling factors to better match them to real object and events, you will be able to see if those measurements and results are realistic, or if your game is being played in some other world or parallel universe, with different laws of physics.  You will be able to calculate speeds, accelerations, forces, momentum, and so on.

Pre-Lab: What do you need?

You will need to make sure you have Internet access on your computer.  This will allow you to use Screencast-o-matic (found at http://screencast-o-matic.com/), as well as access an online video game. 

It is possible to analyze a video game without additional software, just by using the screencast video directly and playing it frame-by-frame.  You would need a small ruler and possibly a protractor for 2-D interactions to make measurements directly on the computer monitor.  

However, a more accurate analysis can be done with other software, such as Vernier’s Logger Pro or Tracker, which is a free, open source piece of software off the Internet.  You can download Tracker at http://www.cabrillo.edu/~dbrown/tracker/, if you choose to use it for your lab.

Procedures:

I.        Set up Screencast-o-matic:

Go to the Screencast-o-matic site, http://screencast-o-matic.com/, and simply click on Start Recording.  There is nothing to download.  When you get a small setup window, the default is to record the whole screen, which will be fine. 

To start recording, all you will need to do is type ALT-P (press these two buttons at the same time), and you will get a countdown to recording.  That’s it!

II.      Before typing ALT-P, use a different tab/window in your Internet browser (this should work well in either Chrome or Firefox) to go to a site of a favorite video game.  Be sure the video game has motion and interactions between multiple objects, because that makes for a more interesting and useful analysis and discussion of how realistic the game is.  For example, one of my favorites is a game I grew up with, Asteroids.  There is a free online version at http://www.play.vg/games/4-Asteroids.html, if you are interested.

III.    Start Recording!  All you need to do at this point is type ALT-P, wait for the countdown, and then when Screencastomatic tells you it is recording start playing the game.  All you want to record is about 30 seconds – that will be more than enough for an analysis sample.  When you have what you want, type ALT-P again to pause Screencastomatic.

IV.    To complete the screencast video, after pausing Screencast-o-matic, you should get a window with an option to be Done with the screencast.  Click on Done.

This will take you to a screen to publish the video.  It is best to choose Publish to Video File.  This will then give you some options.  First is the type of file you want.  It is recommended to use the default option, a QuickTime MP4 file.  This will work with other analysis software if you have it.  Keep the size Full Size (the default setting).  Then click on Save Video. 

You will be asked where to save the file and to name the file.  It might be easiest to save on your Desktop, or start a folder for video files – that is all your choice.  Choose an appropriate name for the file, and then save it!  Congratulations, you just made a screencast video, and hopefully this entire process took no more than about 5 minutes!  

I.        Play and Analyze the video.  Double-click the video file to make sure it works and that you captured the action you wanted.  You can always go back through this procedure as many times as you want, with any and all video games you enjoy playing. 

KEY GOAL: Determine with your partner if energy and momentum are conserved?  Is your game following physics phact or phiction??

If you want to do a ‘quick and dirty’ analysis and have no access to other analysis software, then by pausing the video you can scroll it frame-by-frame.  Each frame will be one-tenth of a second of time, so this is the time step we will use for your analysis.  Use a small ruler and set a reasonable distance scale for your video game. 

For example, on my Asteroids video, I assumed the spaceship was a one-person ship that was 10 meters in length (about 30 feet seemed a reasonable, realistic size – this is 1/5^th the length of a full space shuttle, and I assumed a single, small fighter spacecraft for the game.  See Wikipedia.).  Below is how one would analyze the Asteroids video:

Measure the length of the ship with your ruler.  Let’s say the ship is 1 cm on your monitor.  So the scale is 1 cm of the video = 10 meters in reality; this is no different than a scale on a map, where 1 inch = 10 miles, for example.  The large asteroids flying around might be 2.5 cm on the monitor, so this would make them large rocks of about 25 meters in diameter.  You’ll need to set a scale for any video game, to find lengths, distances, masses, speeds, etc.

I need the masses for each object.  Here we may need to do some research.  I looked up the mass of the space shuttle on Wikipedia, and it is listed as about 2000 tonnes = 2 x 10⁶ kg.  One-fifth of this would be 4 x 10⁵ kg, which is the mass of the spaceship I would use. 

For a large asteroid in the game, I wanted a good, realistic estimate for the mass of a 25 meter asteroid. I looked up 'Asteroid' on Wikipedia and found that this is a reasonable size, as the vast majority of asteroids are under a 100 meter diameter.  The data for density of some asteroids averaged around 3 g/cm³ = 3000 kg/m³, so assuming a 25 meter diameter (volume of a sphere is (4/3)πR³) spherical asteroid, this gives a mass of around 2.5 x 10⁷ kg. 

We would need to do a similar estimate of mass for smaller asteroids, when they are blasted apart in the game.  The diameters would need to be measured, and then scaled in a similar way to get the masses. 

Now, think about the physical quantities that can be measured/calculated.  Speed = d/t, so suppose you run move the paused screencast video 10 time steps.  This would be one second of real time.  You measure one of the asteroids moving 1 cm on the monitor in that time.  This means the asteroid moved 10 meters in one second, and its speed is 10 m/s.  You can measure the speed of the bullets the spaceship shoots in a similar manner, or the speed of the ship if you have it drifting around.  Or suppose the ship is at rest on the screen, and you accelerate in a straight line.  You can measure how many cm it moves in a certain number of time steps, and then calculate the acceleration (assume a constant acceleration) with d = ½ at².  You could also calculate its final speed once the acceleration ends. 

When you blast an asteroid, if you know its initial speed and direction (you can measure angles with a protractor on the monitor or with the analysis software), you can measure the final speeds and directions of the smaller chunks of rock – determine if momentum was indeed conserved. 

With the masses and speeds, calculate the momenta of each object, and the kinetic energy of each object.  Is momentum conserved when you shoot an asteroid?  Is energy conserved?  If not, is the change in kinetic energy negative or positive? 

Calculate the gravitational forces between different objects. Are those forces relevant?  That is, are they strong enough to affect each other’s motions? 

How much would you weigh if you stood on one of the asteroids? 

Would you be able to jump off the asteroid by exceeding the escape velocity with your jump? 

What would the acceleration of gravity be on one of the asteroids? 

How does the mass and speed of one of the alien spaceships compare to your spaceship?

Do all of these calculations and see what the results are! 

Summarize your results: how realistic are they?  

Does the video game seem to follow more closely to phact or phiction? 

For teachers, you may want to specify a game, or at least a type of game, and have students focus on figuring out if one or two physical quantities are realistic or not, such as is energy conserved, or are speeds and accelerations reasonable, rather than having students look at numerous quantities.

Have phun!!   J

Lab Activity: Distance vs. Displacement Vector

Goal: To find the total distance and displacement vector to your house from ETHS!

Problem:

 Here is the question for you – how can you determine the total distance you walk from ETHS to your house, as well as the displacement vector from ETHS to your house?  Think about what information you need to find these, and how can you use Google Maps to help. ETHS is 1600 Dodge Avenue. 

How:

On graph paper, make a map (to scale) from ETHS to your house and use data from Google Maps to get the total distance and displacement vector.  These will all be different for each person in class.  Be sure to define your scale, such as 1 cm on the map is equal to 100 feet, or whatever you want it to be for your map.  Also orient your map with N, S, E, W, so you can calculate the angle of the displacement vector from ETHS to your home!  Give your angle in terms of how many degrees from 0-degrees, which is the positive x-axis.  Let ETHS be the origin of your graph. 

Analysis/Questions:

1.       In your own words, what is the distinction between ‘distance’ and ‘displacement?’

2.       Write down your results for distance and displacement vector measurements from your map.  You’ll need the total x- component (along East-West axis) total y-component (along the North-South axis) of your displacement vector, as well as the direction (i.e. angle) of the displacement vector from ETHS to your house.  Express the results in feet, meters and miles.    1 mile = 5280 feet = 1609.3 meters

3.       Can the magnitude of displacement ever have a larger magnitude than distance?  Explain.

4.       Explain how displacement vectors would be used in the navigation of ships or planes.

5.       What about global travel – motion over a 3-D surface – does anything change? If so, what new variables do you need to consider? Explain any thoughts on this (talk it through with others if you wish). Think of the Sphereland movie.

Bonus: How do space scientists and engineers navigate space probes that are sent to other objects in the solar system? For instance, what navigation issues and considerations  come up with sending people to Mars?

Lab Activity: Use of Pendulum to Find g and Mass of Earth

            Purpose:  In this activity, you will determine the acceleration of gravity and the mass of the Earth using a simple pendulum. 

            Background:  A pendulum is a simple device that exhibits periodic motion when the bob is lifted and given potential energy.  Gravity does work on the bob and transforms the potential energy into kinetic energy.  As potential energy and kinetic energy take turns transforming into each other, the resulting motion is of a periodic nature.  The period of a pendulum, which refers to the time it takes to swing back and forth once (a full swing), is related to the length of the pendulum and the acceleration of gravity (this assumes a ‘small angle’):

                                          Period = T = 2p(L / g)^{1/ 2}

(Why do you suppose the factor of 2p comes up? We will eventually derive this.)

            Materials:       

Stop watch or video (phone, camera)

endulum

Meter stick                  Your brain

            Procedure:

            You and your lab partners need to determine an accurate way of determining the acceleration of gravity, from which you will also calculate a value of the mass of the Earth!  You will need to do this using a pendulum, which you know something about because of your last experiment in determining a pendulum’s properties.  Decide what the best measurements and methods are, as well as how to minimize any potential errors.  Then, write down your step-by-step procedure, and clearly and neatly present your data.

            Analysis (answer in complete sentences):

1.      Based on your measurements, determine the acceleration of gravity, g.  Show your calculation.

2.      Using your result in (1), calculate the percent error relative to the accepted g = 9.81 m/s².  Recall that %-error = {[your value – accepted value] / [accepted value]} x 100.

3.      Explain, in your own words, why the period of a pendulum is independent of the mass of the bob.

4.      What should the period of your pendulum be if you performed this experiment on the moon (gravity is about 1/6 that of the Earth)?  What about on Jupiter (gravity is about 20 times that on Earth)?  Is this a direct or inverse relationship between T and g?

5.      What are the main sources of error in this particular activity?  How did your group try to minimize those errors? 

6.   Explain what a Foucault pendulum is used for.  You may need to look this one up.

7.   How long would you need to make a pendulum in order for it to have a period of 1.0  

     second?  Show your calculation.

8.  From your value of g, determine the mass of the earth!  Show your calculation.  Other

     useful information may include the radius of the earth and G. 

G = 6.67 x 10^-11 Nm²/kg² and R_E = 6.4 x 10⁶ m.

*Note that by knowing the size (radius) of other moons and planets, you could take a pendulum, measure the period, and determine their masses, too.  Sensitive measurements of a pendulum’s period anywhere on earth also allows us to determine small difference in g, which tells us something about how the radius changes as well as densities of the earth at those locations.

Lab Activity: Develop Empirical Formula for Ohm's Law

 

Purpose:  Using measurements taken in class, you will develop an empirical formula for  Ohm’s law.

Background:  You don’t get any this time since it might “spoil the surprise.”

Materials:       

Power supply              Resistors          

Multimeter/Ammeter    Wires               

Breadboard

Procedures:

(A)   The data you want consist of measurements of resistance, voltage and current, the “Big 3” of electricity.  Set up a data table where you will have a single and constant resistance value.  Select a voltage value and then measure the current.  Do this for 5 different voltage values, and don’t go over 4 volts and do not max out the ammeter. You will be making a graph of voltage vs current. Remember that resistors get hot after a while!

(B) Change the resistance value.  Select a voltage and record the current.

After turning down the voltage, select another resistance value.  Set the power supply back to the same voltage (bus-to-bus) you just had, and record the new current.  Do this for at least 4 more resistance values, and re-set to the same bus-to-bus voltage value each time.  Try predicting what current you will measure as you change resistance values.

            Measure the diameter and length of the resistor material (the cylindrical portion of the resistor, and not the metal legs); use the digital micrometer for this.  You will need these measurements for the last analysis question.

Analysis:

1.      Make a graph of your data in part (A) above.  The graph should be one of Voltage (y) versus Current (x).  Also make a graph of your data in part (B) above.  This graph should be one of Current (y) versus Resistance (x).  Make your graphs using Excel, and find the best-fit functions for each graph.

2.      What shape are your graphs?  What type of relationship exists between voltage and current for a constant resistance?  Between current and resistance for a constant voltage?  Find the best-fit functions to your graphs.

3.      Based on your data and your graphs, combine the two best-fit functions from your graphs and write down an empirical equation that shows how current depends on resistance and voltage.

4.      For the following arrangements of resistors below, calculate the total resistance. (you’ll get these in class)

a.                                                                           b.

5.      Why do electronic devices get hot?  Where does the energy come from that you eventually feel as heat?

6.      Based on your observations, what role might resistors have in electronic circuits?  After all, resistors waste energy and increase your power bill…why use them?

7.      From a chemistry perspective, why are conductors and insulators so vastly different?  What is it about these various materials that make their electrical properties different?  Also, from chemistry, what are semiconductors, and why are they important to electronics?  Discuss in terms of band theory.

8.      In your own words, why is the relationship E = -dV/dr the key to understanding how an electric circuit works?  Be thorough, and write in terms of potential and electric fields, and what they do to delocalized electrons inside the wires/conductors of the circuit. Start with the battery/power supply of the circuit.

9.      Using the diameter and length measurements of the resistor, along with the actual resistance value, calculate the resistivity of the material used to make the resistor.  Include appropriate units.

On your own:

Post-Lab

Run the following PhET computer simulations, in order to get some good visuals related to what you directly observed and measured in this experiment.  The simulations are:

Ohm’s law - http://phet.colorado.edu/en/simulation/ohms-law

            Resistance - http://phet.colorado.edu/en/simulation/resistance-in-a-wire

            Basic Circuit - http://phet.colorado.edu/en/simulation/battery-resistor-circuit

Look at the Netlogo simulation for an electron moving through material. Follow a single electron, and note the path. What defines ‘more resistance’ versus ‘less resistance’ for an electron?

Lab Activity: Equipotential Lines and the Electric Potential Gradient

Mini-lab: Equipotential Lines and Gradients

Purpose:  We will measure equipotential lines on a sheet and determine the electric field patterns through the use of the gradient concept.

Procedure:  Hook up two cables from a power supply to one of the conductive sheets.  Turn up the voltage in order to set up a potential difference between the two ends of the sheet. 

Using a multimeter, measure the voltage going down the center of the sheet from one end to the other at 2 or 3 cm intervals. 

Now try to measure equipotential lines going side to side.  You may want to find the lines every 0.25 volts apart, for instance.  Draw these lines on a clean sheet of white recording paper in pencil, and once you have a pattern draw in the electric field lines in ink, indicating the direction of the E-field.

Observations and Questions:

1. In words, what does E = -dV/dr mean physically.  Some find it easier to think of this more generically as E = -DV / Dr, instead. Indicate the significance of the minus sign.

2. On your sheet, include arrows showing which way the electric field vectors are directed.  To do this you will have to determine which end of your sheet is at a higher potential, and which is at a lower potential.  Use measurements from the multimeter to determine this.     

3.  Why can an electric field vector not run parallel to an equipotential line?

4.  How can you tell on your ‘map’ where the electric field is strongest?  Weakest?  Explain your reasoning.

5.  Pick two equipotential lines on your map.  Using the voltage values you measured and distance measurements, determine the approximate strength of the electric field between those two lines. 

6.  How much work would be done if you moved an electron from one of the equipotential lines used in #5 to the second equipotential line?  The charge of an electron is 1.6 x 10^-19 C.  Also, what is the maximum speed an electron and proton could attain if energized by that voltage?  The mass of an electron is 

9.1 x 10^-31 kg, and that of a proton is 1.6 x 10^-27 kg. 

7. In problems like #6, what determines if positive work is done by the electric field? Give your answer both in words and with a mathematical argument.

8.  I’d like you to work your way through the ActivPhysics simulation 11.11, Electric Potential: Qualitative Introduction lesson (11.11).  Try the simulations with the questions being asked; I’ll be looking for feedback from you about whether you think this is helpful or not. 

When done with your lab, do the 1986 AP problem with equipotential lines.  

Lab Activity: Develop a Math Model for the Period of a Pendulum

Purpose:  In this activity you will investigate how hanging mass, length, the strength of gravity and starting angle affect the period of a pendulum. Then, form a math model for the period of a pendulum.

Materials:         
Pendulum apparatus              Stop watch or other timing device/technique

Meter stick                           Various masses for pendulum bob

Balance                                 Protractor or other means of measuring angle
Logger Pro or Tracker          Electronic force sensor         Video camera or phone

Background:

                A pendulum is a classic example of periodic motion, which is a motion that is repetitive, redundant, keeps repeating itself, keeps going back and forth…oh, I’ll stop now.  A pendulum only works when there is gravity, and of course can be used as a clock.  It is also an example of circular motion, which means there is a net centripetal force, mv²/R.  Because we began studying circular motion and will be getting deeper into gravity, we will try to figure out the basic properties of a pendulum and see how we can combine gravity and circular motion together in a few different ways, whether it is a pendulum, an amusement park ride, or satellite motion.  Keep the force diagram for a pendulum in mind (what does it look like?).

Research Questions:

What effect(s), if any, do mass of the bob, length of the pendulum, and angular amplitude of a pendulum, have on the period of the pendulum?  Our goal is to use mathematical fits to data to determine an empirical formula for the period of a pendulum.  Your group is looking to combine your graphical fits into the form T = f(L)g(angle)h(m)s(g).

Develop Hypothesis:

Before doing any measurements, state what you think the effect of mass, angular amplitude, length, and the acceleration of gravity have on the period of the pendulum.  Remember, reality always tends to be a bit complex!

Procedure:

                You will be trying to determine the best fits to data using Excel.  Your measurements will use four different techniques for collection. 

i.                     Use a stop watch to measure the period as a function of length of the pendulum;

ii.                   Use video to determine the period as a function of mass hanging from the pendulum. NOTE: when changing masses, you will need to adjust the length of the string accordingly to maintain the same total length to the center of mass;

iii.                  Use an electronic force sensor to measure the period as a function of angular amplitude;

iv.                 Use the PhET simulation for pendulums to determine the effect of gravity on the period – check out the blog post which provides the links and information for this portion.

 You will need to get data to make four graphs:

i.                     Period as function of length. Note the constant mass and angle;

ii.                   Period as function of mass. Note the constant length and angle;

iii.                  Period as function of starting angle. Note the constant mass and length;

iv.                 Period as function of g (using PhET simulation, keep constant mass and length and angle).

Keep in mind, you should always be thinking of estimating errors on all measurements (in any experiment you ever do!) as well as how to minimize those errors.  Think about what might be the best way of measuring a given quantity.  Also, do not forget to include units on all measurements, along with a reasonable estimate of the uncertainty of all measurements!  For anything measured in trials, think standard deviation.

                For your write-up: Purpose, Materials, Procedure, Data Tables, Questions and Analysis (with appropriate graphs; these should be titled and labeled with quantities being graphed and units!  Graphs need to be done on the computer.).  Do a group report, and consider Google Docs if that works best for the group.

Definitions:

                Period = time it takes the pendulum, when released from rest, to swing over and back to where it started; or the time for one round-trip.

                Length = length from the point where the string is free to swing to the center of mass of the bob at the end of the string.

                Mass: we assume massless string, so just the mass of the hanging bob.

                Angular amplitude: the starting, maximum angle for the period, as measured from the vertical.

Questions/Analysis:

1.       For the graph of period as a function of length, figure out errors on measurements and include them as error bars on your graph.  The error bar for period measurements (some number of trials with the stop watch) should be the standard deviation, as always.  Find the best-fit function for the graph in Excel. Include the equation and R² value.  Include the point (0.01,0.01) on your graph and in your fit, since (0,0) won’t allow certain fit options.

2.       For the graph of period vs. mass, use five different masses and use a video to get period measurements in Logger Pro or Tracker.  Make sure to adjust the length of string so the overall lengths of the pendulum are always the same.  For instance, when you hang a larger mass on the string, you will need to shorten the string a bit since the object will be longer (think about where the center of mass is).  Try to estimate errors in length and in period from your video measurements in LoggerPro or Tracker. Find a best-fit function for the data.  Include the equation and R² value.

3.       For the graph of period vs. angle, use the electronic force sensor and measure the period in the Logger Pro display. Plot your data using Excel and obtain a best-fit function to the data to get an idea of the relationship. Go from small angles around 10^o, and go up to around 90^o.

4.       Last but not least, graph period vs. g from the PhET simulation.  Find a best-fit function for the data.  Include the equation and R² value.  Keep in mind you will only have 3 points, so do what you can with this.

5.       There are only three significant forces acting on the mass at any time, those being tension, gravity and air friction.  Which of these are constant, which are non-constant?  If any are non-constant, when are they the strongest, and when are they the weakest?  Explain, and include a force diagram/free-body diagram for a pendulum.

6.       Using your graph in #1, how long should a pendulum be in order to have a period of 1.0 sec (on earth)?  How long should a pendulum be to have a period of 2 seconds?  Use your fit to do so.  Keep in mind the first mechanical clocks were made with the pendulum! (way to go Galileo)

7.       What exactly does F_centripetal = mv²/R mean?  What does the value of mv²/R tell us, and what does

             it depend on specifically for a pendulum?  Think about what force or forces keep the pendulum

             moving in that portion of a circle.  Consult our notes and the book for assistance.

Write a brief, to-the-point (i.e. one paragraph) summary of your findings/conclusions about what the period of the pendulum depends on and how it depends on specific quantities. 

*Use and combine the results of your individual best-fit functions for your four graphs to obtain an empirical mathematical model for the period of your pendulum as functions of length, mass, g, and angular amplitude.*

              Put it in the form: T = f(L)g(angle)h(m)s(g)

Assess the techniques.  Based on your experience:

-          Rank the ease of the four measuring techniques.

-          Rank the accuracy of the four measuring techniques.

Provide any other feedback you think is relevant to this set of experiments!