Using Complex numbers in Classical Mechanics

When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions ($v_r , v_{\theta}, a_r, a_{\theta}$) are. Here’s one failsafe way using complex numbers that made things really easy :

$z = re^{i \theta}$

$\dot{z} = \dot{r}e^{i \theta} + ir\dot{\theta}e^{i \theta} = (\dot{r} + ir\dot{\theta} ) e^{i \theta}$

From the above expression, we can obtain $v_r = \dot{r}$ and $v_{\theta} = r\dot{\theta}$

$\ddot{z} = (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} ) e^{i \theta} + (\dot{r} + ir\dot{\theta} )i \dot{\theta} e^{i \theta}$

$\ddot{z} = (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} + i \dot{r} \dot{\theta} - r\dot{\theta}\dot{\theta} )e^{i \theta}$

$\ddot{z} = (\ddot{r} - r(\dot{\theta})^2+ i(r\ddot{\theta} + 2\dot{r}\dot{\theta} ) )e^{i \theta}$

From this we can obtain $a_r = \ddot{r} - r(\dot{\theta})^2$ and $a_{\theta} = (r\ddot{\theta} + 2\dot{r}\dot{\theta})$ with absolute ease.

Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!

How does sand from Sahara end up in your windshield ?

TBH cleaning your car is a rather mundane task. But when you fill your head with some interesting physics the task actually gets rather pretty interesting. Here’s some good for thought on such an occasion :

The dust on your windshield might actually be from the Sahara desert

The stacked ball drop

You basically take couple of balls, align them up and drop them to the ground. The ball at the top reaches the most highest due to the subsequent transfer of energy from the other balls.

Source Video : Physics Girl

Here is an exaggerated but amazing slow motion of the same energy transfer with a water balloon. Notice how the transfer of energy takes place between the water balloon and the tennis ball.

Source Video : Slow Mo Lab

Sandstorms in the desert

Sandstorms/ Dust storms as you might be aware, are pretty common in the desert. . Dust storms arise when a gust front or other strong wind blows loose sand and dirt from a dry surface.

And this can cause something phenomenal to happen:

If the wind speed is sufficient then larger sand particles can propel finer ones high into the atmosphere. ( just like the stacked ball )

Then these fine particles are caught in the global wind pattern and are transported across the globe until they fall down to the earth as rain.

How cool is that ! Have a great day!

** Wiki on Saltation

Chain Reaction.

Everyone knows that a line of standing dominos creates a fun chain reaction when you knock the first one over; but did you know you can use increasingly larger dominos and get the same result?

The setup.

Professor Stephen Morris knocks over a 1-meter tall domino that weighs over 100 pounds by starting with a 5mm high by 1mm thick domino.He uses a size ratio of 1.5, meaning each domino is one and a half times larger than the last one. This is the generally accepted maximum ratio that dominos can have to successfully knock each other over.

Hans Van Leeuwen of Leiden University in the Netherlands, published a paper online showing that, theoretically, you could have a size ratio of up to two. But that’s in an ideal (and probably unrealistic) situation.

Fun fact.

There are 13 dominoes in this sequence. If Professor Morris used 29 dominoes in total, with the next one always being 1.5x larger, the last domino would be the height of the Empire State Building.

Source: Physics Buzz.

Behold, Physics!

In this stunning demonstration, the Myth-busters fired a soccer ball at 50 mph out of a cannon on a truck riding at 50 mph in the opposite direction.

The ball just falls down as if it is free falling!! This is a consequence of the fact that in Newtonian mechanics, opposite vector quantities cancel out each other. You probably have heard that a legion times, but here’s the visual proof.

Sometimes I feel that textbooks butcher elegant concepts by excluding visuals during the explanation. But again, if one could incorporate gifs onto books the world would definitely be a better place. 🙂