How to truly understand the math in physics

[aiop]

I want to know if there's a book or a direction of research that explains math in a way that will give me the tools to be able to look at math and truly understand why each piece is used in each spot like the physics who invented it must of understood.

For example: Just for something as simple as the universal gravitational equation Fg= Gm1m2/r^2 how can I understand that, and more so how can I understand why a basic square root sign changes that to mean the required orbiting velocity of a satellite. Orbiting v= square root Gm/r

I am an undergraduate student studying theoretical physics in my first year.

Thanks in advance!
Aiop

 

[Drakkith]

I want to know if there's a book or a direction of research that explains math in a way that will give me the tools to be able to look at math and truly understand why each piece is used in each spot like the physics who invented it must of understood.

No, there is no single source that can teach you this. You'll learn the rules of math as you progress in your studies and your physics classes should teach you how to apply the math to the physics to solve problems and derive different rules and laws Both processes happen simultaneously and are ongoing.

 

[aiop]

Do you have any recommendations in material in the mean time? I am dying to understand it.

 

[Drakkith]

Other than a standard college-level textbook, I can't say that I have any. The physics textbook I used was an older version of this one.
In case the link breaks in the future, the book is: University Physics with Modern Physics, by Hugh D. Young (Author), Roger A. Freedman (Author)
You should be able to buy a copy that doesn't have the "modern physics" part, since you shouldn't need anything from that section for several years.

 

[DS2C]

Other than a standard college-level textbook, I can't say that I have any. The physics textbook I used was an older version of this one.
In case the link breaks in the future, the book is: University Physics with Modern Physics, by Hugh D. Young (Author), Roger A. Freedman (Author)
You should be able to buy a copy that doesn't have the "modern physics" part, since you shouldn't need anything from that section for several years.

Had a good laugh looking at one of the reviews. Damn loose leaf textbooks are HORRIBLE. Always rip, crinkle, fold no matter how careful you are.

 

[SchroedingersLion]

For example: Just for something as simple as the universal gravitational equation Fg= Gm1m2/r^2 how can I understand that, and more so how can I understand why a basic square root sign changes that to mean the required orbiting velocity of a satellite. Orbiting v= square root Gm/r

It doesn't work like this. A priori, you could not know or see why using the square root will give you the desired quantity.
But you can mathematically derive the desired expression by using physical relations.
You start with the physical relation Fg = Fz, the zentripetal force Fz=mv^2 /r (m is the mass of the satellite, either m1 or m2).
And then you solve the equation for v using mathematical rules.

In the end you see: 'ah, that looks almost like the original term, just with the square root'.
Sometimes, in a script or in a book, the calculation is not carried out, and there is just an arrow or a phrase like 'it follows', so you have to check the derivation yourself if you don't understand it immediately.

 

[Mark44]

I want to know if there's a book or a direction of research that explains math in a way that will give me the tools to be able to look at math and truly understand why each piece is used in each spot like the physics who invented it must of understood.

For example: Just for something as simple as the universal gravitational equation Fg= Gm1m2/r^2 how can I understand that, and more so how can I understand why a basic square root sign changes that to mean the required orbiting velocity of a satellite. Orbiting v= square root Gm/r

I don't believe there's a direct connection between these two formulas. The first formula gives the gravitational force between two objects, and the second gives the velocity required for a satellite to remain in orbit.

The thinking that went into the first formula is that the attractive force F_g is proportional to each of the masses involved, but is inversely proportional to the square of the distance. In symbols, this is F_g ∝ $\frac{m_1 m_2}{r^2}$. (The symbol ∝ means "is proportional to.") So given two masses m₁ and m₂ whose centers are r units apart, if you double m₁, the gravitational force will double. If you double both masses, the gravitational force will quadruple (be four times as large).

If the two masses (m₁ and m₂) are moved so that they are at a distance of 2r, the gravitational force will be one fourth of what it was when they are r units apart, since we're now dividing by (2r)² = 4r².

The constant G is what allows us rewrite the proportionality above as an equation; i.e., $F_g = G\frac{m_1m_2}{r^2}$.

For your equation on the orbital velocity, see https://en.wikipedia.org/wiki/Orbital_speed#Mean_orbital_speed. In the derivation, they make a simplifying assumption, that the mass of the orbiting satellite is negligible in comparison to the mass of the object the satellite is orbiting.

 

[Dr. Courtney]

Personally, I prefer an experimental/experiential approach to understanding the math rather than a proof-based approach.

But Gauss's law (electricity and magnetism), quantum mechanics, and my study of the hydrogen atom's symmetries revealed more about the mathematical beauty (symmetries) of Newton's Universal Law of gravitation than studying it in isolation likely would have.

Patience grasshopper. Keep working hard. "True understanding" comes in layers like peeling an onion. Great beauty awaits.