Feynman’s Lectures Exercises 2.16

I spent more time on this exercise than needed because I didn’t notice that the masses are equal. Otherwise, the application of the virtual work principle is straightforward.

The work is done by the gravitation force and the force accelerates the entire system, that is $M = m_1 + m_2 = 2m$.

Let’s use the positive sign the direction of gravity acting on $m_2$.

The work can be calculate as:

${\displaystyle \begin{array}{rl}{\displaystyle W}& {\displaystyle =F\mathrm{\Delta }s}\\ {\displaystyle }& {\displaystyle =Ma\cdot \mathrm{\Delta }s}\\ {\displaystyle }& {\displaystyle =Ma\cdot \mathrm{\Delta }{y}_2}\\ {\displaystyle }& {\displaystyle =2m\cdot \mathrm{\Delta }{y}_2}\end{array}}$

This work is equal to the change of the potential energy in the system as follows:

${\displaystyle \begin{array}{rl}{\displaystyle \mathrm{\Delta }{E}_k}& {\displaystyle =m_2\cdot g\cdot \mathrm{\Delta }{y}_2-m_1\cdot g\cdot \mathrm{\Delta }{y}_1}\\ {\displaystyle }& {\displaystyle =m\cdot g\cdot \mathrm{\Delta }{y}_2-m\cdot g\cdot \mathrm{\Delta }{y}_1}\\ {\displaystyle }& {\displaystyle =m\cdot g\cdot (\mathrm{\Delta }{y}_2-\mathrm{\Delta }{y}_1)}\\ {\displaystyle }& {\displaystyle =m\cdot g\cdot (\mathrm{\Delta }{y}_2-\mathrm{\Delta }{y}_2\cdot \sin \frac{\pi }{4})}\end{array}}$

Thus, we get:

${\displaystyle \begin{array}{rl}{\displaystyle a}& {\displaystyle =g\frac{m-m\cdot \sin \frac{\pi }{4}}{2m}}\\ {\displaystyle }& {\displaystyle =g\frac{(1-\sin \frac{\pi }{4})\cdot m}{2m}}\\ {\displaystyle }& {\displaystyle =g\frac{1-\frac{\sqrt{2}}{2}}{2}}\\ {\displaystyle }& {\displaystyle =\frac{1}{2}(1-\frac{1}{\sqrt{2}})g}\end{array}}$

Feynman’s Lectures Exercises 2.14 and 2.15

If the weights $W_1$ and $W_2$ move the distance $s$ along the plane to the left, their corresponding difference in height are:

${\displaystyle \mathrm{\Delta }{h}_1=s\cdot \sin \theta ,\mathrm{\Delta }{h}_2=-s\cdot \sin \theta }$

Then, the difference in potential energy of the system is:

${\displaystyle \mathrm{\Delta }{E}_p=\mathrm{\Delta }{h}_1{W}_1-\mathrm{\Delta }{h}_2{W}_2}$

By the definition of work:

${\displaystyle F\cdot s=\mathrm{\Delta }{E}_p}$

Then:

${\displaystyle \begin{array}{rl}{\displaystyle F}& {\displaystyle =\frac{\mathrm{\Delta }{E}_p}{s}}\\ {\displaystyle }& {\displaystyle =\sin \theta \cdot m_1\cdot g-\sin \theta \cdot m_2\cdot g}\\ {\displaystyle }& {\displaystyle =\sin \theta \cdot g\cdot (m_1-m_2)}\end{array}}$

For the acceleration along the plane, using $a =$ and the total mass $m = m_1 + m_2$, the acceleration is:

${\displaystyle a=\sin \theta \cdot g\cdot \frac{m_1-m_2}{m_1+m_2}}$

For the motion over distance $D$, starting from rest:

${\displaystyle D=\frac{a{t}^2}{2}}$

Thus:

${\displaystyle \begin{array}{rl}{\displaystyle t^2}& {\displaystyle =\frac{2D}{a}}\\ {\displaystyle t}& {\displaystyle =\sqrt{\frac{2D}{a}}}\end{array}}$

The speed at time $t$ is given by:

${\displaystyle \begin{array}{rl}{\displaystyle v}& {\displaystyle =at}\\ {\displaystyle }& {\displaystyle =a\cdot \sqrt{\frac{2D}{a}}}\\ {\displaystyle }& {\displaystyle =\sqrt{2Da}}\\ {\displaystyle }& {\displaystyle =\sqrt{2D\cdot \sin \theta \cdot g\cdot \frac{m_1-m_2}{m_1+m_2}}}\end{array}}$

For Exercise 2.15, the acceleration is given by:

${\displaystyle a=\frac{1}{2}\cdot g\cdot (\sin \theta -\sin \varphi )}$

Substituting this into the expression for speed, we get:

${\displaystyle v=\sqrt{g\cdot D\cdot (\sin \theta -\sin \varphi )}}$

I finally bought the Feynman Lectures on Physics, something I wanted to do for a very long time. Ever since reading You’re Surely Joking Mr. Feynman. After watching some of his interviews on YouTube and going through a few chapters online, I realized there’s something special about how Feynman explains things. He doesn’t just teach formulas — he teaches understanding. Once you grasp that, you realize what true understanding is. Here are a few quotes from the introduction to the New Millennium edition:

It was like going to church. The lectures were a transformational experience, the experience of a lifetime, probably the most important thing I got from Caltech. I was a biology major but Feynman’s lectures stand out as a high point in my undergraduate experience … though I must admit I couldn’t do the homework at the time and I hardly turned any of it in. I was among the least promising of students in this course, and I never missed a lecture. … I remember and can still feel Feynman’s joy of discovery. … His lectures had an … emotional impact that was probably lost in the printed Lectures.

The book is beautiful. I don’t choose books solely on looks, but when there are different options, it’s better to go for the one you like. I’ve noticed that this works for anything — if I like something that’s part of an activity, I’m more likely to engage with it. So, in that sense, it is rational to choose a book by its cover. One sad thing is the lettering from the original edition has been lost in the New Millennium version:

Almost all other popular physics books look very average. Compare another popular textbook with the austere design of Feynman’s lectures:

My plan is to work through all three volumes over the winter and solve all the exercises from the companion workbook. I will buy a refresher for math once I hit a wall. There are a few books I have in mind: Spivak’s Calculus and The Princeton Companion to Applied Mathematics. Both are as beautifully designed as Feynman’s lectures.

Made a few changes to the site, and now I can have posts without titles. It gives a sense of freedom, similar to what you might have unconsciously experienced on Twitter. Having a title adds a layer of seriousness to a post.

The Archive layout has changed to accommodate missing titles — now the dates link to posts, and the dates are in ISO format set in tabular figures for proper alignment.

With each iteration, it’s becoming more brutalist, and I like it.

Finally watched Carl Sagan’s Cosmos, a TV series produced in the 1980s. I don’t think there are many people like Carl Sagan. The show he created is so delicate, with many complex emotions. It’s captivating, fun, sad, and melancholic. The original music was composed by Vangelis, and it’s hard to imagine how it could be better.

We’re now rewatching the sequel by one of his students — Neil deGrasse Tyson. While it covers similar topics and has better graphics, it feels a little shallower.

THE GAP by Ira Glass

Feynman on Playing

A few hours after I wrote the last post, I remembered one chapter from Surely you’re joking, Mr. Feynman:

Then I had another thought: Physics disgusts me a little bit now, but I used to enjoy doing physics. Why did I enjoy it? I used to play with it. I used to do whatever I felt like doing - it didn’t have to do with whether it was important for the development of nuclear physics, but whether it was interesting and amusing for me to play with. When I was in high school, I’d see water running out of a faucet growing narrower, and wonder if I could figure out what determines that curve. I found it was rather easy to do. I didn’t have to do it; it wasn’t important for the future of science; somebody else had already done it. That didn’t make any difference. I’d invent things and play with things for my own entertainment.

So I got this new attitude. Now that I am burned out and I’ll never accomplish anything, I’ve got this nice position at the university teaching classes which I rather enjoy, and just like I read the Arabian Nights for pleasure, I’m going to play with physics, whenever I want to, without worrying about any importance whatsoever.

Within a week I was in the cafeteria and some guy, fooling around, throws a plate in the air. As the plate went up in the air I saw it wobble, and I noticed the red medallion of Cornell on the plate going around. It was pretty obvious to me that the medallion went around faster than the wobbling. I had nothing to do, so I start to figure out the motion of the rotating plate. I discover that when the angle is very slight, the medallion rotates twice as fast as the wobble rate — two to one. It came out of a complicated equation! Then I thought, “Is there some way I can see in a more fundamental way, by looking at the forces or the dynamics, why it’s two to one?” I don’t remember how I did it, but I ultimately worked out what the motion of the mass particles is, and how all the accelerations balance to make it come out two to one.

I still remember going to Hans Bethe and saying, “Hey, Hans! I noticed something interesting. Here the plate goes around so, and the reason it’s two to one is …” and I showed him the accelerations.

He says, “Feynman, that’s pretty interesting, but what’s the importance of it? Why are you doing it?”

“Hah!” I say. “There’s no importance whatsoever. I’m just doing it for the fun of it.” His reaction didn’t discourage me; I had made up my mind I was going to enjoy physics and do whatever I liked.

I went on to work out equations of wobbles. Then I thought about how electron orbits start to move in relativity. Then there’s the Dirac Equation in electrodynamics. And then quantum electrodynamics. And before I knew it (it was a very short time) I was “playing” — working, really — with the same old problem that I loved so much, that I had stopped working on when I went to Los Alamos: my thesis-type problems; all those old-fashioned, wonderful things.

It was effortless. It was easy to play with these things. It was like uncorking a bottle: Everything flowed out effortlessly. I almost tried to resist it! There was no importance to what I was doing, but ultimately there was. The diagrams and the whole business that I got the Nobel Prize for came from that piddling around with the wobbling plate.

Typeface Updates 003

Typeface design has turned out to be an activity that’s easy to start but very hard to master. It’s relatively straightforward to get to know your tools and learn how to draw basic glyphs. The hardest part is making the entire text look good. You run into problems when you like a glyph on its own, but it catches your eye when it’s part of the text. As Matthew Carter said, “Type is a beautiful group of letters, not a group of beautiful letters.”

The biggest challenge with personal projects like this is the lack of restrictions. When you have a list of requirements, it provides the necessary constraints that shape your work. Without them, you tend to stagger from one idea to another, with no clear goal to measure your progress against. As a result, the design changes frequently.

But this lack of restrictions also has its advantages, as it creates a lot of room for experimentation. You get to understand why some typefaces have a curved leg on an ‘R’ and others are straight by just trying it yourself. You might copy an element from a typeface you like and discover why it works. You end up learning more in a given time by trying different things.

However, it’s hard not to hit a wall when you don’t have a specific goal. Knowing that making it good will take more time than just learning how to do it adds another layer of frustration. Even though there was never any particular goal to start with and it was always about learning, It feels like a failure not to see your projects “finished”.

Note to self: Never set goals like “designing a typeface for your site.” It’s too concrete, too result-oriented. It undermines the playful aspect of the process. Instead, focus on the journey.

I don’t want this to become a project I feel obligated to finish; I want it to remain a toy.

Optimistic Pessimism

Even though I often feel pessimistic about what happens around me, I’m in love with life.

I notice how beautiful the light looks as it filters through the summer leaves.

I listen to a piece of beautiful music that turns everything inside me upside down.

I watch a movie and think, “Yes, that’s it, that’s exactly what I wanted to say!”

I go to the gym and train so hard I can barely stand, wondering what more I am capable of.

I feel the sun warming my skin as I cycle through the morning mountains.

I see a beautiful scene of an old couple sitting together on the street. Though I’m too afraid to take a photo, I still enjoy the moment.

Despite everything that’s fucked up in today’s world, I still love being part of it.

Friends

Friendship is much simpler when you’re a kid. You can simply approach someone and ask them to be your friend. You can visit your grandmother for the summer, come back, and pick up conversations as if nothing happened. Children’s friendships are about living in the moment and enjoying each other’s company.

As we grow up, friendships become more complicated. If you haven’t been in touch with someone, you might feel guilty—even when there wasn’t anything specific to discuss.

I realized that’s okay. Friendship isn’t supposed to be complicated. You can enjoy time together whenever you want without needing to be in touch constantly. This time together doesn’t have to be planned in advance. Meaningful friendships arise naturally anyway.

So stop overanalyzing. Don’t worry if you haven’t been in touch for a while. It’s normal to drift apart, and it doesn’t make you a bad person. The next time you’re in a place where an old friend lives, send them a message and ask to meet up. Think about how you want to spend time together and do something you used to enjoy. If they were a good friend before, that’s reason enough to enjoy their company again.

Don’t overthink it and enjoy what you have.