A quantitative demonstration with springs... one that DOESN'T work!

All year long, the majority of my in-class time is spent doing quantitative demonstrations, in which the solution to an example problem is verified live in class via measurement.  Each demonstration seems to end the same way: I perform the measurement, we all see that the measurement matches the prediction within 10% or so... everyone exhales, and I say "Physics Works."

It's not a problem, per se, but by November or so the novelty of my approach has worn off.  Instead of bright shiny faces anticipating whether or not the experiment will match the prediction, I start to see resignation:  "Yeah, physics works, I know, we've done this a million times."  I've got to throw a changeup.

Once we've introduced the force and potential energy of a spring, I set up what looks like a routine quantitative demonstration.  I hang a mass from a spring, as shown in the diagram.  I determine the spring constant via the 5-second method that I discuss in this post.  This particular spring (the one on the left in the picture) has a spring constant of about 7 N/m.

Next, I hang 100 g of mass from the spring, and allow the spring to come to rest.  Using F=kx, I derive that the mass should stretch the spring by 14 cm; sure enough, a ruler placed alongside the spring shows 14 cm of stretch.  Physics works.

Now, I use the ruler to measure an additional 5 cm of stretch beyond the equilibrium position.  The question: After I release the mass from rest, what will its speed be when it passes through the equilibrium position again?

[Note to readers before you flame me about these next couple of paragraphs:  Keep reading.  Trust me.]

I use the force on a spring equation, F=kx, to calculate the force on the mass to be (7 N/m)(0.05 m) = 0.35 N.  Then, Fnet = ma, so the acceleration of the mass is (0.35 N)/(0.10 kg) = 3.5 m/s².

I take a brief interlude, if someone asks, to (correctly) explain that with a vertical spring, it's completely correct and most simple to treat it as if it were a horizontal spring, but with the x = 0 position at the place where the mass would hang at rest.  In other words, it is acceptable to consider that the spring force kx is the net force, as long as we define x = 0 at the resting position.


Now we have enough information to make a kinematics chart.  The mass starts from rest, its acceleration is 3.5 m/s² and it will move a distance of 5 cm.  Using the kinematics equation v² = v_o² + 2ax, the speed at the equilibrium position is 0.59 m/s, or 59 cm/s.

"How do we verify this prediction?" I ask the class.  They quickly and accurately tell me to place a motion detector below the mass, and to look at the velocity-time graph.  I do so, I read the v-t graph, and I see that it says about 40 cm/s.  Physics wor------  oops.  Physics DOESN'T work.


All year, I've conditioned the class to expect measurements accurate to 10 or 15%.  I'm not even getting within 30% this time... that's not a matter of "experimental error," especially when I try again and get the same result; or even more especially when I try it with the other spring, and I'm still off by more than 30%.  So, why didn't physics work?

Despite the desperate grasping at straws from some students, someone usually comes up with the right answer:  I used kinematics when acceleration was not constant.  The force of the spring was only 0.35 N immediately after release.  As the spring compressed, the spring force got smaller by F = kx.  So the acceleration got smaller.  So the mass didn't speed up as rapidly, and ended up going less than 59 cm/s.

How do we solve this problem correctly?  Using conservation of energy, of course.  Spring potential energy is converted to kinetic energy, 1/2 kx² = 1/2 mv².  Solving, I get v = 41 cm/s.  Now, physics works.

Side note:  Sure, occasionally a student will try to stop me from using kinematics with changing acceleration.  My response is a quick "shhh!!!," Dr. Evil Style, but with a smile.

GCJ

Mail Time: Approaching energy concepts in junior-level general physics

Erik Born, a colleague here at Woodberry, teaches our junior-level general physics course.  We aim this course essentially at the New York Regents exam, with a few tweaks of topics (e.g. we teach an astronomy unit, and we don't teach electric or magnetic fields).  Erik wrote to me the other night:

I was wondering if you can give me a short overview of how you approach energy concepts. I am used to starting with work, and moving from that into energy, but I'm following the order of your problem sets, which has started out with introducing KE and PE first. I was wondering if you could give me some advice about conversions between KE and PE without talking about work or total mechanical energy. Maybe you have a specific blog post on it?

Well, I do now, I suppose :-)

At the AP / Honors level, Erik knows that I introduce the definition of work, and then my own version of the work-energy theorem:  W_NC = (KE_B – KE_A) + (PE_B – PE_A). You can see my reasoning for why and how I use this formulation at this post.  

But at the general level, I approach energy concepts very differently.  I ask the class to state and write out all energy conversions IN WORDS.  

For example, a ball drops to the floor -- how fast is it going?  They must start by writing, in longhand, (gravitational potential energy) converts to (kinetic energy).

Or, a waterskier takes off from water level at 14 m/s, and ends up going 13 m/s at his max height.  They must write: (kinetic energy) converts to (gravitational potential energy) and (kinetic energy).

If a block is moving fast and stops on a rough surface, they must write (kinetic energy) converts to (thermal energy).  If you want them to call this (kinetic energy) converts to (work done by friction) that's fine, too.

We haul up a block by a rope:  (Work done by the rope) is converted to (potential energy).

In each case, once they write the words, then they translate the words into an equation:  KE = 1/2 mv² , PE = mgh, spring PE = 1/2 kx², and work = Fd.  Then they solve for whatever needs solving.

Erik continues:  How do you deal with positive and negative work?

In regular physics, I *don't*.  If we're writing the energy conversions in words, like I described, then we don't have to worry about positive and negative work.  The sign of the work comes out in the wash of the equation written from the description.  Examples:

Block is moving, comes to rest on a rough surface.  (kinetic energy) converts to (work done by friction). *   1/2 mv²  = (Ff)d.  Yes, I know that kinetic friction does negative work, but that's not necessary in this formulation.  The equation and statement in words provide both conceptual understanding AND a path to solution for any relevant value, with no negative signs involved.

* I let them know that a source of "thermal energy" is "work done by friction."  The distinction isn't important at this level; however, recognizing that the relevant equation for thermal energy here is (force of friction) x distance *is* important.

Block is pulled up by a rope.  (Work done by the rope) converts to (gravitational potential energy).  (tension)d = mgh.  Once again, it's not important that work done by the rope in this case is positive:  The sign comes out in the wash.

As a comparison for a teacher (not for the students), many textbooks teach a similar approach to calorimetry:  don't use positive and negative signs, write out the heat gained or lost in words.  For example, write (heat lost by hot iron) = (heat to raise liquid water to 100 degrees) plus (heat necessary to convert water to steam).   Then mcDT_iron = mcDT_water + ml_steam.

An unusual formulation for the work-energy theorem

Most textbooks define work and kinetic energy, then show the connection: the net work done on an object is equal to the change in that object’s kinetic energy.

My students have always had difficulty with this principle. I spend a LOT of time explaining how to calculate the net work on an object, either by taking the scalar sum of the work done by each of several forces, or by calculating the work done by the net force. It doesn’t matter – students will continue to set the work done by any old force equal to a change in kinetic energy (or sometimes to just kinetic energy, without the “change” part.)

The most confusing type of problem asks something like, “How much work must you do to raise a 1 kg block 1 m high?” Students don’t know whether the answer should be +10 J, -10 J, or zero. And don’t get me started on “How much work must be done by an external force to move a charge from point P to infinity.”

These problems are tricky because of the implicit assumption of zero kinetic energy at each position. Using the standard textbook definitions, I can get students to understand that since there’s no change in kinetic energy in these cases zero net work was done. But that doesn’t answer the question! This idea of work done by a non-conservative force is difficult in the context of the work-energy theorem.

My answer this year: I didn’t technically teach the work-energy theorem, as described in textbooks, at all.

I defined work properly, including how to find the sign of work done by a force. (If the force has a component parallel to displacement, work done by the force is positive; if the force component is antiparallel to the displacement, work done by the force is negative.) I defined kinetic energy, and gravitational potential energy.

Next I jumped straight into the conservation of energy, but I wrote it in a weird way. Consider an object moving between two positions A and B. We know the total mechanical energy at position A must equal the total mechanical energy at position B, less the work done by any non-conservative force that dissipates mechanical energy to thermal or other energy. Instead of using one side of an equation for position A and one side for position B, I wrote the following:

W_NC = (KE_B – KE_A) + (PE_B – PE_A)

How did I define this W_NC term? I called that the work done by a non-conservative force. (How did I define a “non-conservative force?” At this stage, I said a non-conservative force was anything aside from the force of gravity. Bear with me.)

In the standard conservation of energy without friction problem, W_NC goes to zero. If a block slides with some friction or air resistance, then W_NCbecomes a negative value; if we’re talking, say, an airplane with a propeller, then W_NC is the (positive) work done by the propeller. The class had little trouble with this formulation, especially as they quickly recognized that most of these terms will usually go to zero.

As we discussed other forms of potential energy – elastic potential energy and electrical potential energy – we merely changed our equation for PE, and we called W_NC the work done by any force that isn’t gravity, a spring, or electricity. This definition is entirely correct and consistent as long as we stick to problems involving a single form of potential energy. (At higher levels of physics we’d have to discuss the deeper meaning of a “conservative” force and how the potential energy is defined… but we’re not at a higher level of physics right now.)

The main advantages of this formulation of what I now call the “work energy theorem:”

• I spend essentially no time explaining how to find the NET work on an object. The concept of net work is not particularly important in this formulation. Good – that used to be confusing to my class.

• We only have to learn and use one overriding equation for all energy conservation-type problems. I used to teach Wnet = ΔKE, plus conservation of energy without friction, plus energy conservation with friction. That’s all accounted for in my newly formulated work-energy theorem.

• The problem in green in the third paragraph above no longer causes trouble! Because I’ve explicitly included how work done on an object can change both the object’s kinetic AND potential energy, there’s no confusion. Both KE terms go to zero, one of the PE terms goes to zero, and we end up with W_NC = PE_B. When we lift the object, we have to do work equal to mgh; when we move the charge from point P to infinity, since potential energy at infinity is zero, we do work equal to qV at point P.

My colleague teaching honors freshman physics tried this formulation, and he loved it. I’m going to do this again next year. Try it! Tell me what you think.

GCJ

More roller coasters -- do heavier riders go faster?

When I went to amusement parks as a middle- and high- schooler, my friends and I strategized carefully for the log flume.  We always put the heaviest people in the front.  Our reasoning:  we figured that way we'd go faster and get wetter.  Funnily enough, I don't remember ever performing a controlled experiment, or recording our results.  Perhaps I should return my Nerds of America membership card.

In retrospect, I think we had half of our conclusion right. 

I *think* that putting heavier people in the front does cause a bigger splash when the log flume hits the bottom, because the front end penetrates the water sufrace more easily -- fluid resistance generally depends on mass and surface area.  I have a wee bit of experimental evidence from last week's Disney trip. 

"Splash Mountain," pictured to the right, is essentially a sophisticated log flume.  As the temperature in Orlando had risen only to about 50 degrees farenheit, we wanted to minimize how wet we got.  And, as the temperature had risen only to about 50 degrees, we could ride pretty much as many times in a row as we pleased -- hardly anyone else was dumb enough to be in line.  We always sat in the last of the four rows.  It seemed as if there was a bigger splash when the front rows were occupied, though I cannot quantify the effect.  The only statement I can make for sure is that when people sat in the front row, they always got substantially wetter than we did.  So teenaged Greg was possibly correct about heavier people in front leading to wetter riders.

On the other hand, we were likely dead wrong on the issue of speed at the bottom of the ride.  The analysis will be different, I think, for a log flume and a roller coaster.  For a log flume, where the "log" is not particularly long compared to the length of the hill, one can probably consider the log as a point mass released from rest at the top.  The frictionless case is well studied in introductory physics -- the speed at the bottom will be root 2gh, where h is the vertical height of the hill, and so does NOT depend on the mass of the riders or the mass distribution.  (With air and water resistance the answer may change, but I'm not exactly sure how or how much.)

The roller coaster problem is more interesting.  Consider Big Thunder Mountain Railroad (covered in yesterday's post).  We do have to consider the mass distribution.  

On BTM, the train is hoisted up to the top of a hill, then released approximately from rest -- but not until most of the train has already gone well over the crest of the hill.  If the train is empty, or if the train has an approximately uniform distribution of riders from front to back, then its center of mass is somewhere near the center; the speed at the bottom will be root 2gh, with h measured from the center of mass's position at the top of the hill to the bottom of the hill.

Consider what happens, though, if we load up the heaviest riders in the front.  That moves the center of mass closer to the front.  When the train is released at the top, the center of mass will be lower than for a uniform rider distribution.  Thus, the h term is smaller, and the speed at the bottom is smaller.

If my analysis is correct, then the fastest speed should be obtained by putting heavy riders toward the back, so that they are right on top of the hill when the hoisting mechanism lets go of the train.  I don't have any experimental evidence, not even anecdotally.  (That's primarily because 37-year-old Greg can't ride even mild roller coasters more than once or twice without barfing.)

I know that amusement park physics is a popular endeavor.  A decade ago, I even took a few classes to the parks myself.  Now that we have wireless, portable, standalone probes like Labquest or the equpiment with Pasco's iphone app, it sure seems like hard data should be easy to acquire.  Has anyone done this?  I'd love to find out...

GCJ