Equilibrium: quantitative demonstrations (or, how I teach vector math without teaching vector math)



That's a 2 N weight hanging by two strings.  The left-hand
rope passes over a pully to a 1 N weight; the diagonal rope
is attached to a digital scale at the top of the picture.


In Honors or AP Physics, I begin the year with equilibrium, not with motion.  In a few days, the class gets comfortable with free body diagrams, forces, and two-dimensional vector analysis. 

Next comes motion in one dimension (graphs first, then algebra), followed by projectile motion.  Finally, we cover Newton's second law.  Since we did equlilibrium already, students are comfortable with free body diagrams and writing a vector sum of forces; since we did motion already, they already have some idea of what acceleration is.  Rather than a tough *new* topic, the second law becomes a way to review and solidify the first two topics.

The picture shows my third or fourth demonstration of the school year.  We begin by equating horizontal tensions in ropes pulling on a stationary block.  Next, I hang a 200 g mass vertically to find the tension in the supporting rope.  Easy stuff, so far.

And then, I attach a horizontal rope over a pulley attached to a 1 N weight. (In the picture, the hanging weight on the left is below the table, and out of the frame.)  When we make the free body diagram, everyone's comfortable with the weight of the 2 N weight acting down, and the horizontal rope pulling leftwards.  We draw the arrow representing the diagornal rope's force at an angle, of course.  Then what?

The class has already been taught that in equilibrium, up forces = down forces and left forces = right forces.  I ask, which is this diagonal rope, an up force or a left force?  Someone always comes up with a reasonable answer:  "both."  I redraw the free body diagram, with the tension in the diagonal rope replaced by two arrows, one up, and one to the side.

The class is totally comfortable with the upward component being equal to the mass's weight of 2 N; and with the leftward component being equal to the weight hanging over the pully of 1 N.  (They're also totally comfortable with me using the term "component" without preamble.)  It only takes a suggestive diagram to get someone to suggest that the resultant tension in the rope itself will be not 2 N + 1 N, but the pythagorean sum of 2.2 N.

The clincher comes when  I call a student to the front to read the digital scale attached to the rope.  It reads... 2.2 N.  Physics works.  I can even predict (and then measure) the angle made by the diagonal rope. 

The key to this whole process is that I'm *not* just telling the class how to solve abstract vector addition problems.  I'm not telling them anything at all, really; I'm drawing diagrams and asking questions, getting someone in the class to suggest the next step wherever possible.  I model the correct problem solving method for equilibrium problems on the board, of course, but everything I do flows naturally from the fundamental principles of equilibrium: up = down, left = right.  I don't use any words like "vector" or "reference frame" or "coordinate system." The only technical term I'm introducing is "component," which was introduced organically.

The final demonstration with this setup involves adding a "mystery weight" to the stuff hanging over the pulley.  I measure the new angle that the diagonal rope makes, so I have a chance to suggest how to use sines and cosines to "break an angled force into components."  We predict the reading in the spring scale, and the amount of mystery weight that I added.  Once again, physics works. 

Buoyant Force problem and demonstration with a crazy answer

A fluid mechanics problem I often assign is based on a problem from, I think, one of the older Serway editions.  It shows a beaker full of oil sitting on a platform scale.  A bar of iron is suspended in the oil from a rope which is attached to a spring scale.  The problem asks:

A 1.0 kg beaker containing 2.0 kg of oil (density = 916 kg/m3) rests on a platform scale. A 2.0 kg block of iron is suspended from a spring scale and is completely submerged in the oil.

(a) Which scale reading should be larger (or should they be the same)? Explain conceptually.

(b) When the iron is in equilibrium, what is the reading in the spring scale?

(c) When the iron is in equilibrium, what is the reading on the platform scale?

Most everyone gets the idea that the platform scale reads a bigger force -- after all, even without considering anything tricky (like fluid mechanics), the spring scale seems to read just the 20 N weight of the iron, while the platform scale seems to read the 30 N weight of the beaker/oil.  A bit more logic with buoyant forces convinces the students that the spring scale must read LESS THAN 20 N, because of the upward buoyant force on the iron.  No problem.

 

Part (b) is similar to a demonstration from class, and numerous example and practice problems in texts.  They know to draw a free body, calculate the buoyant force using Archimides' principle, and use the free body to calculate the tension in the string connected to the scale.  The only halfway tricky part is finding the volume of the iron, which is easily done once the density of iron is looked up.  The buoyant force is about 2 N in this case.

 

Part (c) is the part that causes trouble.  Most of the class, at least initially, says that the reading on the platform scale is just 30 N -- the weight of the oil plus the weight of the beaker.  Others get the right answer of 32 N, but for crazy reasons.  Some come to the conclusion that since the oil "lost" the 2 N buoyant force, that we must return these 2 N to the oil through the reading on the platform scale by conservation of force.  Others simply draw the buoyant force acting down directly on the beaker.  Many make no argument whatsoever, but just add in 2 N, presumably because their friends told them to and they couldn't quite explain it. 

 

I'm glad that so many students have the physics instincts to recognize that 30 N can't be right.  A few will say they made a lucky guess, but I consider such a guess good physics intuition.  However, only a very few students get the justification for why the platform scale reads 32 N.  Do you know?

 

It's Newton's Third Law.

 

The buoyant force is the upward force of the oil on the iron.  Therefore, there must be a downward force of the iron on the oil.  When we consider the oil-beaker system, the downward forces sum to 32 N, including the weights of the oil and beaker, and the third law companion force to the buoyant force.

 

Do you believe me?

 

My students don't, at least not if they didn't get the answer right in the first place. So I set up a similar situation.  The picture at the top (credit to Frederic Lamontagne, WFS class of '11, for the photography) shows a beaker containing a submerged aluminum weight, just like in the problem.  When I remove the weight from the water, the reading in the spring scale increases, but the balance scale goes out of balance!  I have to rebalance the scale to make up for the removal of that downward force of the aluminum on the water.  Since the spring scale reading increased by 0.2 N, I had to add about 20 g to the balance scale reading.  Physics works.

 

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

An equilibrium quiz

Here's a classic question for the end of the equilibrium unit. I say "classic" because although I got the picture from Giancoli's text, I first encountered the question in Dave Ledden's physics class when I was a senior in high school.

A bear sling, as shown above, is used in some national parks for placing backpackers’ food out of reach of the federal bears. Is it possible to pull the rope hard enough so that it doesn’t sag at all? (Obviously, justify your answer in a couple of sentences… just “yes” or “no” doesn’t cut it :-) )


A good explanation points out that with no sag, no upward force would counteract the bag's weight; thus the bag would not be in equilibrium, and the bag must fall. An even better explanation would explain that the vertical components of the rope's tension provide the upward force to counteract the bag's weight; in order for vertical components of tension to exist, the rope must pull somewhat upward, and so must sag.

An outstanding explanation shows that the vertical components are each Tsinθ, where θ is the angle of the rope measured from the horizontal. If the rope approaches purely horizontal, the angle goes to zero. The vertical equilibrium statement is 2(Tsinθ) = mg. As θ goes to zero, sin θ also goes to zero... meaning that the equilibrium statement for a horizontal rope is mg=0. Impossible!

First day of school -- DO PHYSICS

Think about the experience of high school students on the first day of school. They will likely attend four to six academic classes, each for somewhere between 40 and 90 minutes. What will happen in those classes?

Most teachers will take care of administrative minutia. Pass out and read the syllabus, hand out and sign for textbooks, go over rules of the class, how grades are assigned, and blah b-b-blah blah blah. Perhaps a few perceptive teachers might undertake a short discussion about the class’s overall goals, like “What was the most important event in American History?” But I would hazard that in most classes, the actual content covered on the first day is minimal, passive, and non-essential.

Physics can be different.

I teach juniors and seniors only, in general and in AP physics. Presumably the 16-18 year olds in my classes can read; so I send them the syllabus ahead of time via email, and make them read it. Presumably my upperclassmen have learned how to behave in a high school class; so I consider it unnecessary and condescending to discuss a list of class rules such as “respect one another” or “no chewing gum.” (How would YOU feel if you attended a conference which started with a litany of restrictive, prescriptive rules behind which is the underlying assumption that you will do all of these naughty things but for the recitation of said rules?)

Within fifteen minutes of my students’ arrival on the first day of AP, I dive into physics. We define a force as a push or a pull, measured on a scale; I write the definition of an object in equilibrium, and show how to solve equilibrium problems. By the end of the first day, the class is ready for the following quiz (which leads off day 2):

The box pictured above moves at constant speed to the left. Which of the following is correct?

(A) The situation is impossible. Since more forces act right, the block must move to the right.
(B) T3 > T1 + T2
(C) T3 < T2 + T2
(D) T3 = T1 + T2
(E) A relationship between the forces cannot be determined.

And then on day 2, I show with a quantitative demonstration how to deal with a force that acts at an angle. We’re off and running, such that the problems on the SECOND NIGHT OF CLASS are at the AP-level.

The same principle applies to general physics – on the very first day we are making position-time graphs with the motion detector, such that the second night’s problems can involve serious graphical kinematics.

And since most other teachers are talking about the penalties for late work while I’m holding an active class complete with demonstrations, I instantly capture attention. I do think that, in general, physics is more entertaining than most other subjects. But if nothing else, on day one I’ve made students FEEL like my class is special.