A computational paradigm

I was posed a question by an anonymous follower whether the following animation could be easily simulated on a computer. 

In today’s world, lots of research are being aided by using numerical methods. But it is quintessential to note that computational methods alone are not enough to dictate behavior of the natural world. It is with the amalgamation of experiments that it’s beauty exemplifies.

The butterfly effect

One of the many reasons its hard to predict behavior ( say the weather for instance ) is primarily because of the errors that are induced whilst recording it. 

And these errors evolve with time. Let’s take the trivial example of a double pendulum.

Notice how a slight variation in initial angle with horizontal axis of the blue pendulum causes a huge aberration in the result.

Flow past a cylinder

When experiments are carried out under controlled conditions, it is possible to observe and simulate phenomenon.

But like it was pointed out before,the simulation per se is proportional to the accuracy of the instruments used to make the measurements themselves.

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So, yeah it is possible to simulate a system such as the one asked, considering crucial boundary conditions are known to us with considerable precision.

And as the complexity of the problem evolves, the computation time and power required also increases exponentially.

Thanks for asking, Have a great day!

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I’ve always wondered how it is that the formula for a Pendulums periodicity was derived. Would it be possible for you to explain how the formula operates?

Ah,The Graceful Pendulum. 

Before I explain the pendulum’s periodicity, take a look at this demonstration. 

It is not of a pendulum, but a spring-mass system. But it is no different from that of a pendulum, in that the paper roll in placed on the bottom in the case of the pendulum.

We can see that the pendulum’s motion, when plotted with time resembles a sinusoidal curve. 

The sine wave is a periodic function, i.e it repeats itself after some time. 

From Classical mechanics, we know that:

Distance = Velocity x Time. 

For one oscillation: 

2π ( distance travelled in one revolution ) = ω ( Angular Velocity ) x T.

Therefore, we get the formula for the time period of a pendulum as:

I apologize for taking so much time to answer this question. But. Thank you so much for asking this question. Hope this helps you understand it better.

Cheers!

PC: xkcdgraphs, nuffiendfoundation, mit

Types of Damping

Damping is an influence within or upon an oscillatory system that has the effect of reducing, restricting or preventing its oscillations.

There are 4 types of damping:(in the order of the animations shown)

1. Under Damped System.

The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.

2. Critically Damped System.

The system returns to equilibrium as quickly as possible without oscillating.

3. Over Damped System.

The system returns to equilibrium without oscillating.

4. Un-Damped System.

The system oscillates at its natural resonant frequency

( Sources: xmdemo, timewarp,wikipedia)