Doubts about Gravitation, Force and Motion

[rktpro]

• My book says r^3/t^2 was found constant by Kepler. What this constant is called?
• How is acceleration equal to v^2/r in case of circular motion?
• Why is rate of change of momentum or say product of mass and acceleration equal to force? Why not was it ma^2? How was it proved that it is equal to ma?
• How newton showed that a spherical body of uniform density behaves as if whole of its mass is concentrated at its center? Also, why is it that a symmetrical body of perfect density balances about the point of center of gravity?
• What are the latest and advanced means to calculate the value of acceleration due to gravity?
• How do we estimate mass of a double star?
   
  These questions remain unanswered in my book. Maybe because they are little tough to understand.
  Please keep the replies as simple as you can so that I have no problem in understanding them.
   
  Thanks for reply!

[Cap'n Refsmmat]

My book says r^3/t^2 was found constant by Kepler. What this constant is called?

I'm not aware of a name, if there is one.

 

How is acceleration equal to v^2/r in case of circular motion?

http://farside.ph.utexas.edu/teaching/301/lectures/node87.html

 

Why is rate of change of momentum or say product of mass and acceleration equal to force? Why not was it ma^2? How was it proved that it is equal to ma?

Well, we know that [math]p=mv[/math]. So:

 

[math]\frac{dp}{dt} = \frac{d}{dt} mv = m \frac{d}{dt}v = ma[/math]

 

Saying that this is equal to force is the definition of force, rather than a derived result. It's how we define what force is: it's something which causes a change in momentum or an acceleration.

 

How newton showed that a spherical body of uniform density behaves as if whole of its mass is concentrated at its center? Also, why is it that a symmetrical body of perfect density balances about the point of center of gravity?

That's pretty much the definition of the center of gravity.

 

What are the latest and advanced means to calculate the value of acceleration due to gravity?

To measure local acceleration due to gravity, one can use a gravimeter:

 

https://en.wikipedia.org/wiki/Gravimeter

[DrRocket]

• How newton showed that a spherical body of uniform density behaves as if whole of its mass is concentrated at its center?!

 

calculus

[rktpro]

calculus

 

I wanted the derivation.

[swansont]

I wanted the derivation.

 

http://en.wikipedia.org/wiki/Gauss'_law_for_gravity

[D H]

My book says r^3/t^2 was found constant by Kepler. What this constant is called?

It's called Kepler's constant, but only after the fact. Kepler did not find that r^3/t^2 is constant. That's an algebraic expression. He, like Newton, preferred to use geometry, where proportions (but not constants of proportionality) are the rule. Kepler said "The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit." No mention of a constant.

 

There is an implied constant in there, but that's because nowadays we prefer to use algebraic reasoning over geometric reasoning. Algebra was fairly new and somewhat suspect in Kepler's and Newton's time. Newton neither trusted nor liked it. Newton's Principia is almost entirely done using geometric reasoning. Very little algebra, and hardly any calculus.

 

 

 

Why is rate of change of momentum or say product of mass and acceleration equal to force? Why not was it ma^2? How was it proved that it is equal to ma?

Saying that this is equal to force is the definition of force, rather than a derived result. It's how we define what force is: it's something which causes a change in momentum or an acceleration.

I disagree with Cap'n Refsmmat's answer to this question. Saying that force is defined as the product of mass and acceleration means that Newton's second law is not truly a law of physics. It's merely a definition. IMO, Newton's second law of motion is a law of physics, not just a definition. Newton vaguely defined forces up front in his Principia in Definition IV: A force is an action on some body that changes the state of motion of the body. Newton's second law relates this already-defined concept of force to how force does change a body's state of motion.

 

Note that F=ma^2 is in line with that vague definition of force. This equation certainly does mean that applying a force will change a body's state of motion. It is not in line with experimental evidence. Newton and his predecessors, particularly Galileo, did lots and lots of experiments that showed that for a given mass, acceleration was proportional to force and that for a given force, acceleration was inversely proportional to mass.

 

So how to "prove" it? You can't. Scientific laws cannot be proven to be true. They can be proven to be false. All it takes is one experiment to blow away the lifetime's work of some theoretician. In fact, Newton's laws of motion is not universally true. Newton and his predecessors didn't have access to the particle accelerators. They worked in the regime of speeds that are very, very low compared to the speed of light. Particle accelerators are but one demonstration that Newton's laws are not correct at relativistic speeds.

 

 

 

How newton showed that a spherical body of uniform density behaves as if whole of its mass is concentrated at its center?

calculus

Continuing my contrarian mode, I disagree with DrRocket's answer to this question. Newton proved his shell theorem in propositions 70 to 74 of book 1 of his Principia with not one smidgen of calculus. Instead he used geometrical reasoning, wall-of-text page upon wall-of-text page of it.

 

There's a dirty little secret of science education here. It took a long time to make that nice, all wrapped up in a bow presentation of science that students receives up until the senior year of college or so. Newtonian mechanics is a prime example. Newton, as I mentioned, largely relied upon geometric reasoning. It wasn't until 50 years after his death that Newtonian mechanics was reformulated in terms of algebra and calculus. It took another 150 years or so to boil Newtonian mechanics down to the simple forms we use now. Vectors, for example, are a late 19th century refinement to Newtonian mechanics.

 

Interestingly, just when physicists had finally come up with this all wrapped up in a bow presentation of Newtonian mechanics, other physicists were starting to find some rather big problems with Newtonian mechanics in the form of quantum mechanics and relativity.

Edited February 2, 2012 by D H