classical mechanics

Cross and Dot product of vectors.

Someone asked a very interesting question on the role of vectors in Physics. He was curious to know if dot product of vectors is natural but vector product is just syncretism, — that is make shift or unnatural manipulation.

Every vector can be resolved into two components. The cosine and sine components (any two vectors would constitute a plane) while cos part can represent the projection defined through dot PDT we can’t leave out the sine part. It plays its role through the vector or cross PDT.

The vector direction is no more along same direction as original vectors because of orthogonality. To preserve symmetry of both orthogonal components (or equal footing of both vectors, vector a and b eg) we need the 3rd dimension. Hence such a definition of cross PDT.

Eg the emf generated in a changing “mag field” (Faraday’s law) depends on change in mag field if area is held const. It also depends on change in “area vector” if mag field is held const. So there are two vectors involved and their transverse values matter (and not their longitudinal values). To preserve equivalent role of both area and mag field vectors the resultant vector must be in a 3rd orthogonal direction …

Also think of this; a scalar is not necessarily directionless. (think electric current or even temperature or heat gradient etc) They just do not have the full fledged capacity of vectors. Its like flower bud vs fully blossomed flower.

So scalars can’t be added like vectors. We tend to make a mistake here. We say scalars don’t have a direction. That’s totally erroneous. They do have direction and it matters. Which direction you want to stick to if the current flows along certain direction only?

Lets make it still more clear.

If there are two directions in which there are electric currents, we say they are both equal, the direction won’t matter. That’s where we make the mistake. We should say they are equivalent and not equal. Equality is ideal, its mathematical. But equivalence is physical. Its the effects of both currents in a certain sense that make them equivalent, but their strict equality does not follow.

Coriolis Force; an interesting idea

Coriolis Force
The article is 5.34 K worded. 

I have been thinking of writing a basic introduction, on this important physical concept — of, what’s known as a Coriolis force.

It is one of the interesting ideas of physics, which does not get as much of a mention perhaps, unless you just happen to know this, because of your advance footing, in the discourses of physics.

This is perhaps so because its two orders of magnitude smaller, in its strength, compared to relevant forces, in a situation, that involves this small — but its a significantly measurable force.

So, what’s a Coriolis Force?

I presume that most of us would be aware of, what’s a centrifugal force. Its in this context, that, a Coriolis force is most understandable.

So, what’s a centrifugal force?

A Centrifugal force, comes into picture, every time a centripetal force comes into consideration. A centrifugal force is the corresponding pseudo-force, of a centripetal force. So, for every centripetal force there would be a corresponding centrifugal force.

So, we need to shed light on a few things, before we understand, what is a Coriolis force. We need to, therefore, discuss; what are pseudo forces and what’s a centripetal or a centrifugal force, then a Coriolis force would be clearer, to the root.

So lets begin with force. 

Force has a basic standing in physics, in connection to, in what frames of reference we need to measure such a physical quantity, as are all other physical quantities to be measured, have a significance, as to in what frames of reference we are measuring those quantities in.

Solutions to Irodov problems

Problems In General Physics, 
I.E. Irodov
Part-I Physical Fundamentals in Mechanics.

Chapter 1.1 Kinematics.  

10 interesting problems in elementary mechanics

This post intends to provide 40 interesting problems in elementary mechanics from IE Irodov, Problems in general physics that I solved in last couple weeks. Note that another 30 problems (additional 12 at hand) which has been solved will be uploaded after they are scanned, shortly.

Basic Concepts.
In this class today we will discuss the subject of “kinematics” briefly and solve some problems, based on the same. 

The motion of objects are studied under the heading “mechanics”.

♣ Mechanics is called “kinematics” if we study about the “nature of motion” without regard to what factors are causing such motion. 

♣ In addition to kinematics, when we focus our attention to study the factors that cause motion, such is named as dynamics. 

In kinematics today we will discuss a few problems that will cover the topics of …

How to add speeds; Galileo and Einstein won’t agree.

How to calculate the speed of anything, when their speed becomes closer to the speed-of-light. 
This article was originally a comment in the linked article;  Why Nothing Moves Faster Than Light.

— In order to correct the comment I have made earlier  ” unless something is completely mass-less in its rest-frame ” I also add the following. This is a fact which I have realized lately — or rather trapped myself to commit an inconsistent remark, by following the same comment in making other remarks elsewhere. 

But it’s better late than never to realize; when something is mass-less, it will never have a rest-frame, because by Einstein’s transformation rules, known as Theory of Relativity, to be consistent, a mass-less particle will always move at the speed of light c, no matter which frame we are looking at it from. This then leads to the velocity addition formula of Einstein.

Now we will discuss in a slightly more detail the two kind of velocity addition formula, one prior to Einstein and one that came from Einstein’s work. 

Prior to Einstein. 
According to Newton and Galileo ( Galileo Project ), known by a name Galilean Relativity, the following follows; if C moves at speed

Why is energy conserved?

Here is the way; that hasn’t been changed in a long time. We, start with a simple object, and we note that, such an object is defined for its motion by whats called, its location in space = x, or, the increments in its location, which is called, an infinitesimal distance = dx, the instantaneous time at which its motion is referred, t, or, the increments in its time, called dt.

Thats it. And, we would like to know; all that the object does in terms of x, t.

I would not like it, if my dear people sit in a car, and it vanished into thin air and never came back. I would like to keep track of it, the car, because I know my people would still be in it. I would like to, keep track of satellites, and, missiles and airplanes, I would like to know, whats happening around me, and why its happening.

It all started, with the quest, to solve for the trajectory, and then, became more complicated, as the complexity of these objects or systems grew. Collectively, they satisfy greatly, the quest we had set on, since the millennium and more, to understand, whats all, that goes on, in our universe, in our close vicinity, and in situations far off from us, as far as the extraneous bounds of the galaxy, in which we live, and more and more and more and deeper and deeper.

Then, x, t are not sufficient, to describe such situations. But, since its all systematic, we know all that has been defined, its not a party or Ramstein Music Band, where you forgot what happened yesterday. Its Hello Physics Inc. Pay Attention.

Now, as we defined dx and dt, we also note that, their ratio, or as-is-called, rate of x wrt t, called speed = v, is often formally written, as, x with a dot on it, … Its the first order time derivative of x = …

We also form, two quantities;

1. from v we form m.v = p = momenta, by multiplying the mass m into the velocity or speed v.

2. from a, we form in the same way, F = m.a = force.

But, force is also defined to be the time-rate of change of momentum p, or in other words, the ratio of the increments dp and dt, …

This latter is called Newton’s 2nd law; …

Its called a law, but strictly speaking, its a mathematical law so far, and not one which describes, universe’s phenomena so it cannot be called, a physical law or principle as of yet.

This point was originally raised by Feynman; as far as I know.