Hanging Mass on an Incline lab

Mr. Johnson shows off his hanging mass setup

In my general physics labs, we follow a familiar formula each week.:

(1) Students collect data, constructing a linear graph as they go.

(2) They take the slope of a best-fit line using far-separated points on the line that are not data points, including units on the slope.

(3) They use a relevant equation to relate the physical meaning of this slope to a measurable physical quantity.

That's it.  I keep things as simple as possible so as to work on just these three skills.  The trick is, of course, finding straightforward yet interesting experiments which lend themselves to this approach.  Oh, and these experiments must stick within the topic areas in general- (approximately Regents-) level physics.

The procedure is essentially the same each lab day.  I demonstrate a method of data collection.  I write on the board the graph that students are to make.  Students collect and graph their own data in groups of two.  Once I've approved a group's graph, then I hand them a two-page homework assignment for them to work on either the rest of the lab period, or that night if they need more time -- no one leaves early.

Ideally, I'm designing an experiment such that I know what the value of each group's slope should be, but the students do not.  This is a bastardized version of the "recurrent lab" as described by Mikhail Agrest.  Students earn credit either for predicting something using their slope, or for matching their slope to an independent measurement.

So, Greg, give us an example of such a laboratory activity.  Sure -- see the picture.  I have students set up an inclined track, on which they place a Pasco cart.  The cart is connected over a pulley to a hanging mass.    The hanging mass is adjusted until the system hangs in equilibrium.  The angle of the incline is what they are eventually going to predict; I surreptitiously come around during the period to measure each incline with my iPad clinometer app.

We graph the mass of the hanging stuff on the vertical, with the mass of the Pasco cart on the horizontal.  Some mass is added to the Pasco cart, the hanging mass is adjusted to equilibrium, and another data point goes on the graph.  Rinse and repeat for an easily obtained straight-line graph in about 30-60 minutes (including setup and cleanup).

The homework assignment based on this lab activity is available here.  In sum, students take the slope, and then are guided to identify the slope as the sine of the incline's angle.  Most groups easily match my measurement.  

This is one of my better lab exercises, because (a) it fits the formula we've been using all year with no deviation, (b) it allows for accurate prediction of a measurable but initially unknown quantity, and (c) it reinforces the content in the problem solving portion of the course.  Try it... post comments or questions.

GCJ

HOW MUCH? The heck you say.

We're studying circular motion in regular physics, and I'm preparing my laboratory activity for the week after Thanksgiving break.  (Not sooner -- we have our trimester exams next week.)  I want to do the "swing a stopper on a string in a horizontal circle above your head" experiment, a classic developed in the PSSC era.  I discovered my setups for this experiment to be a tangled mess, with numerous missing pieces, broken strings, and not enough hollow tubes for the string to go through in any case.  Now you can tell why I haven't done this experiment with my class in about five years.

I looked on the PASCO site, hoping to find a reasonably priced set of hollow tubes with stoppers and the light, low friction thread that leads to quality data.  And what to my eyes did appear:  $39 big ones for a set of five stoppers, two tubes, ten zip ties, and some regular old string. 

Okay, my department's budget is nearly unlimited.  I can -- and do -- order any equipment I want or need for my class.  Nevertheless, there's something to be said for intelligent use of resources.  $39 for items available in the storeroom?  Neither frugal nor intelligent.  This is why a former Chief Reader for the AP exam defined "Pasco" as a Latin verb meaning "to rob."*

* Before the Pasco Police come my way, please note that I am a HUGE customer, and a huge supporter of the company in general.  They sent me two loaner heat engines for my summer institutes -- no charge, no hassle, no problem.  (Of course, I probably garnered them 5-10 orders for said heat engines, so they got their money's worth.)  I tell anyone who will listen how reliable PASCO's products are, and how good their technical support is.  But the downside:  they're expensive.  And in this case, obnoxiously expensive.

I had no trouble finding stoppers, thread, and zip ties.  I'm going to use thread from Burrito Girl's* sewing kit rather than regular string; the chemistry department has stoppers of all sizes.  The trick was finding the hollow tubes without a trip to the hardware store.

* Burrito Girl is my wife and sidekick. 

My classroommate Alex Tisch looked at the picture, and offered up a suggestion that would make the editors of the Tightwad Gazette croon:  what about a BIC pen with the ink part removed?  Two decades ago I used to take apart these pens when I was bored in class... now I could use that experience to save my department some dough. 

In the event, I used a papermate brand pen.  The pen-tip is connected to a thin tube of ink, all of which can be removed from the pen casing easily; the cap on the other side took some wedging, but I got it out with a fingernail in less than one minute.  Voila, a "hollow tube," at a cost of about a quarter.

(Oh, you want to know about the actual experiment?  Attach the stopper to the string with the zip-tie, thread the string through the tube, and hang a mass from the bottom end of the string.  Hold the tube and swing the stopper in a horizontal circle at constant speed such that the hanging mass hangs in equilibrium.  The radius of circular motion can be measured with a ruler.  The speed of the mass can be determined with a stopwatch, knowing that speed is circumference divided by the time for one revolution.  A graph of speed squared on the vertical axis and radius on the horizontal axis yields a line whose slope is the centripetal acceleration of the stopper.  This acceleration can be shown to be equal to g times the ratio of the hanging mass to the stopper mass.)

GCJ

General Physics Lab: Time of Flight for a Projectile

I've been working this year on developing laboratory exercises for general physics that involve simple data collection and straight lines.  I want to drill the idea of taking a slope of a best-fit line, and interpreting the physical meaning of that slope.

I've already discussed the classic ball-off-of-a-table projectile lab, for which David Moore recommended using a photogate to measure the ball's horizontal speed.  The traditional experiment controls for the horizontal speed, and asks students to predict the landing spot on the floor. 

I tried something different, because I wanted a graph.  I had the students measure the initial horizontal speed with the photogates... but then I had them measure the horizontal distance that the ball travels after flying off of the table.  They could make these measurements for a wide range of initial ball speeds.  I asked them to graph the landing distance on the veritical axis of a graph, and the velocity read from the photogates on the horizontal axis.

My class did a pretty good job of data collection. At this point, they're already quite good at drawing a best-fit line, at calculating the slope of the best-fit using two points on the line that are not data points.  Some even put proper units on the slope.  (The idea that a slope has units is, for some reason, a difficult concept to get across.)

What we're NOT yet good at is understanding the physical meaning of the slope of an experimental graph.

When I ask someone what the slope of this graph means, he invariably says "the slope is the change in the distance divided by the change in the speed."  Well, sure, but that's a mathematical answer.  I know that a slope is rise over run.  What's interesting and exciting is that the slope has a meaning beyond rise over run, which can only be determined with reference to the relevant equation.

The horizontal speed is constant; so speed = distance / time.  Some algebra rearranges this to time = distance / speed.  Well, distance / speed in this case is rise / run -- The slope is the time for the ball to fall to the ground.

One student's data is shown above.  The slope of his graph was 0.37 s.  The table from which he launched the ball was 72 cm high, predicting a time of fall of 0.38 s -- not bad, eh?

GCJ

Lab idea for general physics: introduction to force components

 I had just introduced the idea of a force acting at an angle in my general physics course.  I needed an experiment for lab day.

In general physics lab, I want as often as possible to be able to make a linear graph, and use the slope or intercept to calculate a verifiable physical quantity.  The data collection process should be as simple as I can make it -- very little messing with computers, with minimal calculation before the linear graph appears.

My first thought was to use the PASCO fan cart.  The fan can blow straight ahead, or can blow at any angle all the way to 90 degrees off of straight ahead.  If we place this cart on a PASCO track --  with grooves keeping the cart from sliding -- a spring scale attached to the front of the cart can measure the force the cart experiences in the direction of the track.  This force would be equal to the force of the fan times the cosine of the fan angle.  A graph of the scale reading vs. the cosine of the fan angle would make a line whose slope is the force of the fan.

Problem is, I have 22 students and 11 lab groups.  I don't even have two fan carts, let alone 11; that many fan carts would bust my budget.

After considerable brainstorming with some AP physics alumni, a new thought occurred to me... what if I could keep the angle of an applied force constant?  Look at the lower picture, the picture of the red PASCO cart.  This is one of the newer plastic "pascar" models.  It has several convenient raised thingamabobbers (that's a technical term) useful for attaching strings. 

One string is attached to the middle of the left-hand side of the cart.  This string is passed over a pulley, and a hanging weight is hanged from the string.  A second string is held across the diagonal of the rectangular pascar, as shown in the picture.  The geometry of the pascar itself determines this angle -- students are instructed to ensure that the rope stays along the diagonal to control the angle.  A spring scale is attached to the angled rope.  The top picture shows the Nachoboy holding the spring scale at the correct angle; the hanging weight is visible above the garbage can.  Component analysis shows that the hanging weight is equal to the tension in the angled rope times the cosine of the string's angle.

Students vary the hanging weight, and measure the tension in the angled rope with the scale.  They graph the hanging weight vs. the tension in the angled rope.  The slope of this graph should be the cosine of the angle of the string!

Each student used the slope of his graph to figure out the angle of the rope.  I measured the angle with a protractor... I got between 21 and 24 degrees, depending on where the rope was anchored.  Those who did the experiment carefully (most of the class) got a slope of 0.90-0.95... giving an angle between about 18 and 25 degrees.  Woo-hoo!

GCJ

Projectile lab with a marble: use a photogate!

Greetings from the AP reading in Fort Collins, Colorado.  I'd say 3/4 of my teaching ideas have germinated in  this enclave of friendly and professional physics teachers.  Today's thought comes courtesy of David Moore, who is part of the team grading this year's fluids experiment problem. 

A common  laboratory exercise asks students to predict the landing spot for a marble projected off of a table top.  Usually the marble is rolled down a ramp from the same height every time to ensure a consistent initial horizontal speed.

Measuring that horizontal speed is tricky.  Motion detectors don't pick up objects as small as marbles very well.  I suppose video analysis would work, but that's too intricate for a general physics class, I think.  In the past, I've had the class use stopwatches.  If the marble rolls across a flat tabletop, then the distance of the flat region divided by the time to travel that region gives the marble's horizontal speed.  However, my lab groups have made consistently incorrect predictions using this method.  Just a small reaction time issue can cause the marble to miss the target by 30% or worse. 

David says he uses a photogate placed near the end of the table!  Knowing the marble's diameter and the time during which the gate is interrupted, the marble's speed can be calculated.  Even better, use two photogates near each other:  the speed is the distance between beams divided by the time between the beams' activation times.  Reaction time or stopwatch clumsiness is not an issue when photogates are involved.

Revising my general physics laboratory program

Over the years, my AP physics labs have meshed into a true PROGRAM. That means not only are the individual experiments worthwhile, but that the year’s set of laboratory exercises serves to develop and reinforce a set of worthwhile skills. When the students start to roll their eyes at me as if to say, “yeah, yeah, we know what to do now, we’ve done it a dozen times,” I know that the lab program has been successful.

My AP lab program is designed to teach the lab skills necessary for the AP exam. Most prominent among these is the process of linearizing a graph, and using the best-fit line to determine an unknown quantity. That’s an important and useful skill…but one that is above the heads of my GENERAL physics students, at least at year’s beginning. For general physics, I have an entirely different set of experiments, each one solid, but without any guiding theme. In other words, my general physics labs in no way constitute a PROGRAM.

My summer mission is to begin to change this shortcoming: I want to revise my laboratory exercises so that they all mesh together. Specifically, I am going to try to use the “recurrent” lab model, as presented by Mikhail Agrest in The Physics Teacher, and as discussed earlier on this blog. My goal is for most of my experiments to ask students to use the result of their measurement to make a testable, high-stakes prediction.

[As a brief aside, note that no one is standing over me demanding that my lab exercises meet any particular “inquiry objectives” put together by “learning specialists.” No, I just have some time to invest for next year’s classes, so I’m working on general physics lab. I often see relatively new physics teachers become intimidated by the sheer volume of background work necessary to put together a strong physics course. The fact is, it takes many years before every aspect of your course will be successful. Partly this is because much trial and error is involved – I’ve wasted a couple of years here and there discovering why various physics teaching methods do NOT work for me. More to the point, no one has the time or experience to do everything perfectly right away. I’ve been teaching physics for fourteen years, and I’m only now coalescing my general physics lab program into something above the level of “adequate.” That’s fine – the labs have been adequate, and I’ve spent enormous amounts of time making other aspects of my course really good. Work on one thing at a time, and don't let anyone tell you you stink just because, say, your labs aren't perfect.]

The first topic of the year in general physics is position-time graphs. After nearly a week of lectures, demonstrations, and problems, the first lab exercise is to make a position-time graph for a “constant speed vehicle” using a stopwatch and metersticks. One goal of the experiment is to introduce my expectations for graphs in physics class, including proper labeling, as well as how to take a slope properly. Another goal is to give students kinesthetic experience with position-time graphs. They see that the slope of the position-time graph is the velocity of the vehicle, just as they’ve learned on homework.

What am I going to do differently this year? After each group has calculated the slope of their position-time graph, I will ask them to predict the position of the vehicle after it has been moving for, say, 10 s. Each group will give a location with uncertainty – i.e. 300 +/- 10 cm. Finally, *I* will measure the position of the cart after 10 s. Points will be earned for matching my measurement; more points, including some extra credit, can be earned for groups who match my measurement with the smallest uncertainty.

The second experiment uses “tape timers” to make a position-time graph for a cart on an incline. By taking the slope at several points, a velocity-time graph can be made, and used to find the acceleration of the cart. Except that this year, I will use a sonic motion detector to determine the acceleration of each group’s cart. Matching the acceleration that I measure will be a major part of the lab score.

You see? While every experiment won’t match this format, I’ll adjust as many as I can. You got good ideas? I’d love to hear them. Post a comment.

GCJ