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! 

Lab Activity: Hydraulic Jump => finding parameters

Lab Activity: ‘Deep Thinking’ and Parameterizing a Phenomenon: What Does it Depend on?

Goal: We want to gain practice and experience of how to think about and break down a phenomenon or physical situation/event as a scientist would. We will do this using the hydraulic jump.

Why do this?

Most people are not ever formally trained to think about things or observe things as a scientist. Part of the reason is there is no one correct way of doing this. However, one thing scientists tend to do naturally is, when seeing something that is interesting and perks the curiosity, not just to be satisfied with saying it is cool, but rather begin thinking about the parameters the phenomenon depends on.

To really know and understand how something works or behaves the way it does, a scientific study will require observations and measurements of controllable variables, i.e. experiments, which are determined by the physical parameters of the problem. If we can get good at determining the parameters, then this opens the door to a variety of possible studies and experiments. In physics in particular, doing this also opens the door to the development of mathematical models of the system.

Here I am using the definition of parameter = measurable factor on which the phenomenon depends. If it helps, think of it as an independent variable.

 How will we do this?

We will spend a few minutes looking at and thinking about a simple, everyday phenomenon, which is also one that we do not fully understand to this day: hydraulic jump.

What to do:

Not much in the way of procedures on this one. To form a hydraulic jump, pour water on a hard, flat surface. That’s it! Then, observe, ‘play’ with it, and think about it.

What does your team need to do?

Observe the jump and ‘play’ with the experiment in any ways you can think of, and develop a list of as many possible parameters as you can think of. That is, what physical quantities might affect the properties and behaviors of the hydraulic jump?

 One example that will probably stand out immediately is the flow rate of the stream of water. But think deeper than the obvious – what else might affect the jump? Who cares if it ultimately turns out some more obscure parameter does or does not affect the jump, write down anything and everything that comes to mind, even if you are completely unsure of its effect! There are no wrong answers to this part of the activity!  J

Team Brainstorm: Possible Parameters (just list them! You have 10 minutes)

Class Brainstorm: Possible parameters (anything other teams came up with, and you did not)

 This is a process many scientists use to develop their research programs. By carefully dissecting a single phenomenon or event or situation, into an ideally exhaustive list of possible parameters, they can then begin to figure out appropriate experimental designs to determine the effect of one or more of the parameters on that phenomenon.  From the data of those experiments, mathematical fits can be done and empirical mathematical models developed. And so on. We will gain experience throughout the program at doing all of this!

Now it is your turn to try this – you knew this was coming!

In the next three days and with a partner, find a phenomenon, event or physical situation that you are curious about. It needs to be something that is ‘everydayish,’ like a hydraulic jump. It needs to be something that we could do experiments on with accessible equipment, without the aid of something like the Hubble space telescope, a cancer research facility, or a particle detector at Fermilab or CERN. 

Spend a small amount of time and play with that phenomenon. Make your list of physical parameters that may have an effect on the phenomenon. Be as thorough as you can, using what we just did as a class as a model.

In addition, based on your initial ‘play’ with the system, which parameter has the most significant effect? This may or may not be correct, but which parameter stands out as the most important?

Have fun, and try to find something interesting, perhaps unusual!

To turn in: A description of your phenomenon/situation/event, your brainstorm list of parameters, and a preliminary guess of the most prominent parameter (think about what would you want to test first). Feel free to add photos, diagrams, etc., to show the rest of us what you are talking about.

Job Option for Physicists and Other Scientists: Hollywood Movie Making!

Think of the skills scientists, engineers and mathematicians are trained to develop: high-level problem solving, mathematical modeling, careful observation and experimental designs, use of trial and error to troubleshoot experiments and theoretical structures, high-end technology use, innovation and creativity in both hardware and software design, computer programming and computational thinking skills, and so on. These obviously are what is necessary to becoming a productive and contributing STEM professional. But these skills have, over the past two decades, become transferable to many other areas of study and work. Back when I was in graduate school (early to mid-1990s), several graduate students who I knew and received their PhD's in high energy physics, actually left the field and went to Wall Street, where they use Monte Carlo modeling techniques to try and predict the market, both in the near and long-term timeframes. Another student at the time, who stayed in particle physics and is now on faculty at Berkeley, is a consultant for the hit comedy series, "Big Bang Theory." And now, here is an article outlining physics professors who are now working for some of the major Hollywood studios to help improve special effects and the artistic side of movies, using advanced physics theories and simulations, which are making the most realistic computer graphics we have ever seen! So if you have a fancy for hard core science and math and computer applications, along with a love of art and/or movies, there is yet another different type of career that is possible, along with many others in this new technology and information driven age.

A Clever Way of Thinking About Gravity, Gen. Relativity Style

Thanks to Jack C. for sending me the link to this video - I had never seen it, but it is a clever way of thinking about and picturing how gravity works according to Einstein's masterpiece, general relativity. The essence of this theory is gravity is not a true force at all, but rather the consequence of warped space-time, in the presence of mass and energy (technically, mass-energy density). Check it out!

Magnetic Field of Earth Weakening

Here is a topic we will study during the E&M portion of our AP Physics C course - the magnetic field of the Earth. Over the past few centuries, scientists know the magnetic field, created from the motion of the molten iron core of the planet, has been weakening. New data from the Swarm satellite show the field is declining about 10 times more rapidly than previously thought, or perhaps it has been happening at an accelerating pace over the past century of measurements. Either way, this declination of strength suggests the possibility of the Earth moving into another episode of magnetic pole reversal.  Computer simulations show that a decline begins prior to when a reversal takes place, and the Earth has not seen a reversal for some 800,000 years - this is quite a bit longer than the 200,000 year average over geologic time, so some would say we are 'overdue' for a reversal.  So, perhaps within the next 500-1000 years, a reversal will occur.

We do not know the full consequences for civilization, since the last time it happened we have no witness accounts or accurate data. We would expect higher doses of cosmic radiation, but it is not clear (as far as I can tell) what the health and biological consequences of this would be. Other concerns would be the effect of higher radiation rates on the electronics on the ground, such as telecommunications networks, wireless technologies, and the power grid.

Check out the Scientific American article.

Evidence for Inflationary Models of Big Bang

Check out this MIT video of Max Tegmark, professor of physics at MIT, who shows experimental evidence for some of the predictions of inflation models of the Big Bang.  Inflation is a theory that says, when the Big Bang happened, very shortly after there was an exponential expansion of the volume of space.  This was created by 'dark energy,' and led to today's observable universe.  Well, the theory predicts a number of things, and we are beginning to see many of these predictions verified through direct measurements, to the point where Prof. Tegmark solidly states those in the field believe inflation happened.

How the ATLAS Detector works at CERN

Students often ask me how it is possible to detect tiny subatomic particles, most of which live only a tiny fraction of a second before they decay into other particles.  I worked on the CDF experiment at Fermilab back in graduate school, and most of these collider detectors work the same way.  This is a video that has a focus on the ATLAS experiment at CERN, over in Switzerland. ATLAS is similar to CDF, only a whole lot bigger!  Check it out!