The Ladder Problem

I like to solve physics problems.  Here is one for you (I just made it up).

A 4 meter ladder leans against a frictionless wall at a 30 degree angle.  The mass of the ladder is 10 kg.  A human stands 1 meter up the ladder and has a mass of 70 kg.  What is the minimum coefficient of static friction between the floor and the ladder so that the ladder doesn’t slip?

Here is the solution—in video form.

But wait! There’s more.  Let’s do another problem.  

Suppose you have the same ladder and the same human.  However, in this case the coefficient of static friction between the ladder and the ground is 0.55.  How far up can the human move before the ladder slips?

I like this question because I don’t know how the answer will turn out.  That makes it fun.  So, let’s do it.

But what kind of problem is this?  I’ll make this a multiple choice question.  Here are your options.

  1. A friction problem.
  2. An equilibrium problem.
  3. A ladder problem.
  4. A work-energy problem.

Your answer?  Go ahead and answer.  It’s important to think about things like this if you want to become an expert problem solver.

Did you pick?  OK, I’ll tell you that there will be quite a few students that will say this is a friction problem because it has a coefficient of friction.  That’s not untrue—but it’s not a good way to classify the problem.  You could also say this is a “ladder problem”—again, not untrue but not helpful.

This is an equilibrium problem.  We are trying to find the point at which the ladder slips—the point it leaves equilibrium, but it’s still sort of in equilibrium.

For an object in equilibrium, there are two main ideas—especially for a rigid object.  First, it must have zero net force.  Second, it must have zero net torque about any point.  In two dimensions, I can write these two conditions as the following three equations.

F_\text{net-x}=0

F_\text{net-y} = 0

\tau_\text{net-o} = 0

Let’s talk about the torque stuff.  I don’t want to get into a whole thing about torque, so let me just say that torque is like a “rotational force” and it can be calculated as:

\tau = Fr\sin\theta

In this expression, F is the applied force, r is the distance from the point at which you are calculating torque, and \theta is the angle between r and F.  This equation is equivalent to F-perpendicular times r or F times r-perpendicular.

Oh, torques that would make an object rotate clockwise are negative.

One last thing about torques—especially the sum of the torques.  If an object is in rotational equilibrium about some point (point “o”) then it is also in rotational equilibrium about any other point.  When you set up the torque equation, you can pick whatever point you like to sum the torques—it’s your choice (but choose wisely).

Now we can start setting up some equations for equilibrium.  I’ll start with a force diagram for the ladder.

I know that’s a little busy—but it will have to do.  Here are some comments.

  • There are two normal forces.  One from the wall (labeled 1) and one from the floor.
  • There are two gravitational forces—and this is a cheat.  There is a gravitational force on the ladder and it is as though it is one force acting at the center of mass for the ladder. If the ladder has a uniform density, the center of mass is the center of the ladder.
  • The other gravitational force is fake.  This force m_2\vec{g} is there to represent the weight of the human.  But that gravitational force acts on the human, not the ladder.  The ladder pushes UP on the human the same as the weight.  Since forces come in pairs, an upward pushing force from the ladder means there is a downward pushing force from the human on the ladder.  It’s equal to mg, but not mg.
  • The friction force is parallel to the floor.  The maximum magnitude of this friction force would be: F_f = \mu_s N_2 where \mu_s is the coefficient of static friction.

Now for some equations.  First, this is the sum of the forces in the y-direction.

F_\text{net-y} = 0 = N_2 - m_1 g-m_2g

Just a quick reminder.  These are not vectors.  These are components of vectors in the y-direction.  That’s why the two gravitational forces have a negative sign.  I guess I can simplify this a little bit.

N_2 = (m_1 + m_2)g

Next is the sum of forces in the x-direction.

F_\text{net-x} =0=N_1 - F_f

Since we are at the point of maximum friction, I can include the expression for the frictional force in terms of the coefficient.

N_1 = \mu_s N_2

Notice that I know the value of N_2 since it only depends on the mass of stuff—but I don’t know N_1.  I’m going to need another equation. That’s where the sum of the torques comes in.  

In order to write down the sum of the torques, I need to pick a point about which I calculate the torque.  I’m going to go with the bottom of the ladder.  At this point, there are two forces applied.  Since they are applied at the point about which the torque is calculated, they contribute zero torque and they won’t appear in the equation.  Winning.

Here is the sum of the torques about point O (at the bottom of the ladder).

\tau_\text{net-o} = 0 = m_2gs\cos\theta +m_1g\left(\frac{L}{2}\right)\cos\theta - N_1 L\sin\theta

What do I want to solve for here?  I want the distance the human goes up the ladder.  This is the value “s” in the expression above.  Really, I have all the values I need in that expression to solve for s.  I just need to set N_1 to the maximum frictional force.  But that would be boring.

Instead, let me make a plot of the frictional force as a function of human distance up the ladder.  That will be more fun, right?

Here’s what I get.

From this plot, the human can go up 1.16 meters before the required frictional force exceeds the maximum.  Oh, here is the code for that plot.

Now that you have the code, you can change stuff—like the angle of the ladder or the mass of the human or whatever.  

The end.

Intro to Chaos in Mechanics

This is really just for me so that I won’t forget.  I mean, I will forget—but then I can look back at this post and remember stuff.  Here’s to you Future Rhett.

What is a chaotic system?  Really, that’s the question—isn’t it?  There is the classic example of the double pendulumHere is some code for a double pendulum. And this is what it looks like.

Nov-11-2018 15-57-59.gif

But this isn’t the best system.  The problem is that there are two coordinates—the angle for the top bar and the angle for the bottom bar.  Sure, it’s cool—but what if you want to plot angle vs. time or something.  You have to plot both angles vs. time and that’s a bummer.

OK, how about a model of bounded population growth?  That’s just one dimensional, right?  Actually, it doesn’t even have to be population, it’s just an equation—something like this.

x_{n+1} = 4rx_n(1-x_n)

In this expression, r is some parameter—it really doesn’t matter what.  Let’s just model this expression for different values of r.  I’ll use a starting x value of 0.1 and r values of 0.7 and 0.9.   Here is the code.

GlowScript IDE 2018-11-11 17-55-15.png

Notice that when r = 0.7, the population reaches some stable value—but this is not true for r = 0.9.

Bifurcation Diagram

Now for another way to look at a chaotic systems—the bifurcation diagram.  Honestly, I didn’t really understand these things until I made one.  Here’s what we are going to do.

  • Start with some initial value of x (just pick something—I’m going to use 0.5).  Pick a value for r also.  Let’s just start at 0.1.
  • Run the model for 200 iterations and throw out that data.  This should allow us to look at the long term behavior for that particular value of r (throws out the transient behavior).
  • Now run the model for 100 additional iterations and save these.
  • Create a plot of these final x values vs. r.
  • Next increase the r value a little bit (I will increase it by 0.001)
  • Repeat until you get bored.

So if the model is stable after the initial stuff, then it will just keep plotting the same value of x after the first 200 iterations and you will just get a dot.  If it’s not stable after the first stuff, then you will get a bunch of dots with different x values.

OK, let’s do it.  Here is the code.  Oh, I made a function to iterate the model.  I probably should put more comments in there.

This is what it looks like.

GlowScript IDE 2018-11-12 09-08-29.png

Up to an r value of about 0.75, you only get one final x value.  After that, you get two different values . With r over 0.9, it gets crazy.

OK, that’s enough for now.  I just want to make sure future Rhett knows how to make a bifurcation diagram.

Angular Momentum and the Moment of Inertia

Let me be clear—something isn’t working with my program.  However, I think the idea is solid. Also, by writing this I might be able to figure out my problem.

Two Forms of Angular Momentum

I’ll just just to the main point. I’m trying to make a connection between the two forms of angular momentum.  The first is the point-particle definition.  This says that the angular momentum of a point is defined as:

\vec{L} = \vec{r} \times \vec{p}

In this expression, L is the angular momentum, r is the position vector, and p is the linear momentum.

The other form of angular momentum is defined as:

\vec{L} = I\vec{\omega}

Here ω is the angular velocity vector for a rigid object and I is the moment of inertia tensor.

Ok, let’s get started.  I’ll begin with a simple case and move to more complicated stuff.

A free particle.

A particle has a mass m and moves with a constant velocity (no external forces).  What happens to the angular momentum?  Here is a python program for a free particle in which I also calculate the angular momentum.  This is just a picture – you need to go here to run it.

Trinket 2018-11-05 10-46-12.png

Here is a plot of the z-component of the angular momentum in this case.

Trinket 2018-11-05 10-48-10.png

Notice that it’s constant.  Oh, I assume you know about cross products—they are in vpython, so you don’t have to do it manually.

So, yes—angular momentum is conserved.  No big surprise there.  But what if you change the origin?  Is angular momentum still conserved?  Try it.

There is very little point looking at the moment of inertia.  I’ll do that next.

Single object constrained to circular motion.

I want to get an object moving in a circle—but I don’t want to make some crazy constraint.  This sounds like a job for THE SPRING.  Yes, I will model an object moving in circular motion by using a spring attached to the object and some stationary thingy.

Here is the code.  Here is what it looks like.

Nov-05-2018 11-19-46.gif

If you calculate the angular momentum (from the linear momentum), here is what you get.

Trinket 2018-11-05 11-21-20.png

It doesn’t look constant—but I think that’s just a rounding error.  Reminder: this is the angular momentum from \vec{L} = \vec{r} \times \vec{p}.  What if I use the moment of inertia and the angular velocity?  Do I get the same thing?

I can calculate the angular velocity as:

\vec{omega} = \frac{\vec{r} \times \vec{v}}{r^2}

For objects moving about a fixed axis, the moment of inertia is a scalar value that is calculated as:

I = \sum_i m_i r_i^2

Notice that if you put these two definitions together, you get (essentially):

I\vec{\omega} = \vec{r}\times \vec{p}

So it should work.

Two masses.

Using a stationary pivot point can cause some problems.  Since the pivot point is stationary, there must be some external forces on the system.  This means that calculating momentum and angular momentum can be difficult.

Here is the program.  Here is what it looks like.

Nov-05-2018 12-54-39.gif

Yes, these are two unequal masses but the center of mass is stationary.  Also, I will skip the graph, but angular momentum (the z-component) is constant.

Here is a different (but similar) version of the program in which I also calculate the moment of inertia version of the angular momentum.  Everything seems to work—until it doesn’t.

What happens if the center of mass of the system is not zero?  In this case, I need to redo the angular momentum calculation.  First, for the point model, it would be still be r cross p, but I can write it two ways:

\vec{L} = \vec{r}_1 \times \vec{p}_1 = \vec{r}_{com} \times \vec{p}_{com} + \vec{r}_{1r} \times \vec{p}_{1r} +\vec{r}_{2r} \times \vec{p}_{2r}

In this case, the 1r subscript means the position relative to the center of mass and the momentum relative to the center of mass.  The com subscript means center of mass.

For the moment of inertia method, I have:

\vec{L} = \vec{r}_{com} \times \vec{p}_{com} + I\vec{\omega}

But this is where I will stop. For some reason, I can’t get a constant angular momentum using the moment of inertia.  Here is the plot of the component of momentum for the case when the center of mass is moving.

GlowScript IDE 2018-11-05 15-08-36.png

I feel like I am making some silly mistake.  So, here are some notes and comments.

  • Maybe I am calculating the relative velocity incorrectly.
  • Maybe it has something to do with my definition of the angular velocity.
  • Note that the two masses can have slightly different angular velocities since this isn’t actually a rigid object—it’s just mostly rigid (stiff spring).
  • I feel like I have so many different programs, that I’m losing track of what works (that’s why I wrote this blog post).
  • What’s the next step?  Well, after getting this calculation to work—I have big plans.  The ultimate goal is to have a 4 mass rotator (4 masses connected by springs) and calculate the moment of inertia and the angular momentum.  I would be very happy if I could show that the angular velocity vector doesn’t have to be in the same direction as the angular momentum vector. That would be cool.

 

Trip Report: Texas AAPT/APS Section Meeting

Since this is just a normal plain blog, I can do silly things like this report on my recent trip.  Why not?

Where and Why?

I was invited to give the keynote address as well as a workshop on python at the AAPT/APS section meeting at the University of Houston.  Since this isn’t too far away, I decided to just drive there – it’s about a 5 hour trip.  Not bad, plus I can bring as many pairs of shoes that can fit in my car.  I brought one pair of shoes.

I drove in on Friday and arrived Friday evening – I stayed at hotel on the outskirts of Houston.

A note regarding section meetings.

I really like section meetings.  They are smaller, cheaper, and it’s easier to get around and see everyone.  Oh, national meetings are cool too – but sometimes they are just too big.  Also, who likes paying 500 dollars just for registration?  Not me.

Python Workshop

For the workshop, I used my python material.  This is essentially the same stuff I used at the Chicago Section of AAPT.  Here are some notes.

  • The material basically this stuff on trinket.io.
  • I also have instructor materials and other files posted on the PICUP site.
  • It seems there were about 15 participants. The room had computers for people to use – that helps out a bunch.
  • There was an issue with the projector – it wasn’t quite working.  Someone brought in a backup, but it wasn’t bright enough.  It’s funny how small problems like this can make a big difference when people are learning.
  • Another issue for python workshops – variety of people.  Some people have never used python and some have experience. This makes it slightly difficult.
  • Other than that, I think the workshop went well.  I had one person ask me afterwards how to become an expert with python.  My response was to just keep practicing.  The best way to learn is to learn python to solve particular problems.  It’s pretty tough if you try to learn stuff without a purpose.  Oh, also – sloppy code is fine.

Keynote: Science Communication with MacGyver and MythBusters

Normally, I give a talk that focuses on physics of science fiction or video analysis or something like that.  I’ve talked about science communication before – but in this case I wanted to include a bunch of examples from MacGyver and MythBusters – so I had to make a new talk.

Check out the venue (maybe it’s difficult to see from this pic though):

This is the “club level” of the University of Houston football stadium.  No, there wasn’t a game going on at the time (but that would have been funny).  It was a nice place – the screens were in a weird position, but still it was nice.  Oh, I did make one fairly big mistake.  I was having trouble with the projectors and I ended up with “mirroring” on my computer.  This means that I didn’t see the next slide and and I didn’t have a clock. I really like seeing a clock.

For the talk, I focused on 4 “rules” of science communication:

  • You can’t be 100% correct, but you can be 100% wrong
  • Build a bridge from the science to the audience (complicated, conceptual, or shiny physics).
  • Science fiction is still fiction.
  • Use mistakes as a foot in the door to talk about what you want.

Overall, I think it went well.  Oh, there was one great question at the end.  “How do we use science communication to help people understand climate change?”  My response: we need to focus on the nature of science and understanding of what exactly science is all about.

Finally, here is another picture. This is me on the football field (which was kind of cool).

Update on Python Physics Curriculum

So here is the deal.  I had this idea.  The plan was to include numerical calculations into the intro physics curriculum by writing a sort of online textbook.  Or maybe just redo my Just Enough Physics ebook to include more numerical calculations.  Anyway, this is what I came up with. It’s written with trinket.io – an online implementation of python that pretty much rocks.

Here is my curriculum (it’s incomplete – but totally free).

Introductory Physics with Python

Here are some of my own thoughts on this curriculum (including using trinket.io):

  • It’s free and online.  That’s mostly good – but I don’t know if online is the best format for physics.
  • There is one thing about trinket.io that makes this rock.  There is python RIGHT IN THE PAGE.  Readers can view and run code – no logging in, no saving, nothing.  Just edit and run.  No barriers.
  • It has the same idea as Just Enough Physics in that it goes over the basic stuff – but doesn’t overload the student with tons of different ideas (no fluid dynamics, waves, buoyancy, sound…).  It’s not that those are bad topics, it’s just too much.  Too much.
  • Homework.  Students want homework questions.  I sort of added those in – but students seem to want traditional homework questions.

Now for the part that needs work.  Well, all of it needs work – it’s not complete.  But I made an error – I figured I would finish this curriculum as I was using it to teach the summer session of physics, but the pressure was too much.  In the end, I think I made it too much like the traditional format of a textbook (with the traditional order of topics).  Really, I started along the best path – but went off the rails when I wanted to do a problem that involved new physics.  So, I just added that new stuff in there.

I need to rethink just what I want to cover – and here is my new plan.

  • Kinematics in 1-D and 2-D. I like starting with kinematics because students can model motion and this works great with numerical calculations.  The one problem is that you have to use acceleration instead of change in momentum – and this messes up with my momentum principle.  Actually, maybe I will just do 1-D motion so that I don’t need vectors.
  • Forces. I don’t really want to focus on forces and equilibrium, but the students need this to do more stuff.  In this, I need to do the following.
    • Vectors.  Boom – need vectors.
    • Special forces: gravity, real gravity, maybe Coulomb force.
    • What about friction, and forces of constraint (like the normal force)?  Here you can see how it gets out of hand.  Friction is super crazy if you think about it – so are normal forces.
    • What if I just did simple forces – like pushing with your hand or rockets?
  • Momentum Principle.  Here I need to make a connection between forces and motion.  Since I used acceleration before, I need to make a connection between the momentum principle and \vec{F}_\text{net} = m\vec{a}.  Honestly, I hate calling this Newton’s Second Law – it seems wrong.
    • But what about circular acceleration?  How do you deal with that?  I don’t know.  Maybe just avoid it for now.
  • Work Energy Principle. I think this is mostly ok – except I need to introduce the spring force and spring potential energy.
  • Angular Momentum Principle.  My initial idea was to cover “Three Big Ideas” – momentum principle, work-energy, angular momentum principle.  However, there is SO MUCH baggage associated with angular momentum principle.  Much of this stuff is just beyond intro-level students.

I think I have a new plan.

  • Start with kinematics in 1-D.
  • Forces – but simple stuff.  No friction.  No normal forces.  All the examples will be in space or something.
  • Momentum Principle and acceleration. Again, normal stuff.  No forces of constraint.  Mostly space stuff because that will be fun.  Projectile motion stuff too.
  • Work-Energy Principle.  Springs, gravity, dropping objects.  Orbits.
  • Special cases.  Instead of Angular Momentum, I’m going to go over forces of constraint, friction, normal forces, circular acceleration.

The end.  Oh, I need to make sure there are plenty of exercises for students.  Rewrites coming.

Numerical Calculation Collection

The following are some of my best posts about numerical calculations.