All Experts Redux: The Basics Of Escape Velocity, Gravity And Orbits (& Video of Bill Nye's Experiments on Orbits)

 Question -

Why do planets and other bodies (like artificial satellites) stay in orbit instead of crashing straight into the Sun or the Earth?

Answer:

Planets, other celestial objects, and satellites already in orbit are really in a state of free fall. Note also an object like an artificial satellite has to be launched into orbit, which means it must attain a certain escape velocity.(  v_esc  )

In the case of  v_esc  attained for an object with respect to Earth, we can work it out. The Earth's radius is 6396 km or call it 6.4 x 10⁶ m.

G, the Newtonian gravitational constant is G = 6.7 x 10^-11 Nm²/ kg²

The mass of the Earth is 6 x 10²⁴ kg

Then:

[2MG/ r]^1/2  =  [2 (6 x 10²⁴ kg) (6.7 x 10^-11 Nm²/ kg²)/ 6.4 x 10⁶ m ]^1/2

 This comes out to v_esc = 11,200 m/s or 11.2 km/s

Now consider a satellite already in orbit around Earth. Why doesn't it just fall? Well, firstly because it is moving too rapidly to do so. The speed of the satellite with respect to Earth ensures it will stay in orbit. What does this mean?

Although at each second the satellite has an acceleration toward the center of the Earth (called 'centripetal', denoted  F_c  )   - note from the diagram below :

it has NO vertical or downward velocity . The reason is that it falls
from each position at the same rate the Earth's surface falls away
underneath it.  Basically, the inward directed force  F_c    keeps the

satellite moving in its orbit and also falling from successive tangents denoted in the diagram by v bar.  These are given in the diagram as the product:

w r  =   (v/r) r

Thus, relative to Earth's surface, the velocity in a VERTICAL (downward) direction is ZERO, since the distance between the satellite and Earth's surface remains constant. Hence, we say the satellite - or any orbiting body - is in a constant state of "free fall'.

Perhaps a better diagram to convey the falling, from successive tangents, T, T' T"' etc is shown below:

Note that the successive vectors (from P, P' in upper diagram) denote successive tangents to the orbit identical to the T, T' etc. of Fig. 2. The 'force' denotes the centripetal (inward directed) force  F_c  . 

Note also that  if the motion is uniform then the only way that the centripetal acceleration (a_c = v²/ r) arises is via the change in direction of the velocity vector, v. Thus, the acceleration is: D v/r   or:  (v’ – v)/ r, but the magnitude of each vector is: |v| = rq/ t = r w. By similar triangles one would obtain:

D v/v  = s/ r  and   D v = v(s/r)  but s = (rq)/ t

So: D v = v(q/ t) = vw

And since: w = v/r then:

a _c = D v/ r = vw/ r = = v²/ r

And: GMm/ r ²   =    m v²/ r

Thus we see the gravitational force of attraction (based on Newton's law of gravitation, e.g. F =  GMm/ r ²)   is what supplies the centripetal force to keep the object in orbit. This condition of perpetual free fall is what we mean by "being in orbit".

The same applies to other bodies orbiting larger ones. Thus, a planet
orbiting the Sun is also in a similar state of free fall, with respect to
the surface of the Sun. By the same token its gravitational force of 

attraction (to the Sun) supplies its own centripetal force, in effect a condition of force balance. So long as this is sustained the object remains in its orbit.

While it is being pulled in toward the Sun (by the Sun's gravity) it has a speed in its orbit large enough so there's no vertical (e.g. downward) velocity component. 

Thus, planets orbit  in a circle - or ellipse - rather than falling into the Sun - which they would do if their orbital speeds suddenly halted or slowed drastically.  In the case of elliptical orbits we have:

Where  the points A and P denote the aphelion (farthest point) and perihelion (closest point) to the Sun (S), respectively. We let   V_A, V_P  be the respective velocities at those orbital extremes. As may be deduced here, points A and P are the only ones in the whole orbit for which the velocities are truly tangential or at right angles to the radius vectors for those positions. Consequently, we can write: 

V = (2π/T) r

where r is the radius vector at the point, and T is the period. 

For the perihelion velocity we have:

V_P = h/ a(1 - e)

where a is the semi-major axis, and e is the eccentricity.

For the velocity at aphelion: 

V_A = h/ a (1 + e)

We see from this that:   V_P   >   V_A

So the planet is always moving faster at perihelion because it is closer to the Sun.

Then the ratio of velocities is:

(V_P/V_A) = (1 + e)/ (1 - e)

In the same exact fashion, the Moon (in its own elliptical orbit)  orbits the Earth (free fall) rather than 'crashing down' into it, because the speed of the Moon in its orbit is large enough to overcome the inward pull of Earth gravity. The Moon keeps continually falling in its orbit relative to the Earth - marking out an elliptical orbit- rather than falling ONTO the Earth.

Note in the case of the Moon's elliptical orbit,

We refer to the points P and A as the perigee and apogee, with its velocity greater at perigee than at apogee.

See Also:

Bill Nye demonstrates experiments that break down Artemis II mission

WSJ Regular Pages Expose WSJ Editors' Claim of "Democratic Assault on Medicare Advantage" As Codswallop

 

An Oct. 9, 2022, NY Times piece by Reed Abelson and Margot Sanger-Katz for the first time described in glaring detail how Medicare Advantage plans are bleeding Medicare dry.  They are basically "insatiable cash monsters," quoting the words of one physician whistle blower from Kaiser Permanente.

But The Wall Street Journal's hyper-capitalist editorial nabobs have been raging against this portrayal from the beginning. Case in point the April 9^th Journal Editorial, The Truth About Medicare Advantage - featuring the (subheader) claim “About half of  seniors use a program that is under Democratic assault.”

 And then leading into the lead paragraphs:

"The Trump Administration is considering automatically enrolling Medicare beneficiaries in private insurance plans—aka Medicare Advantage—as a default. This may be the single most effective way to reduce Medicare waste, fraud and abuse. So why do Democrats oppose it?

Democrats are trying to flip the script by claiming that Medicare Advantage is wasting money. Insurers, Democratic Senators last week wrote to the Centers for Medicare and Medicaid Services (CMS), “are endangering the solvency of the Medicare Trust Funds.

 This misinformation deserves a rebuttal."

No, YOUR misinformation deserves a rebuttal, and from another section of your own paper! The real mischief is the unsupported claim that Medicare Advantage is "the single most effective way to reduce Medicare waste, fraud and abuse."   

Contradicting that bunkum is the fact -  in a later WSJ regular article (Exchange, (April 18-19, p.B12) - that:

"For two decades, making money in Medicare Advantage followed a simple formula: Sign up seniors and document as many diagnoses as possible. The more codes for patient sickness on paper, the more the government paid."

There in blunt, no-nonsense words was (and still is) the template for Medicare Advantage wasting money in colossal fashion:  by deliberately running up the diagnoses for seniors which the government must then cover to fill up the insurers' coffers. So, of course, contrary to the WSJ editors' spin, it is wasteful - MEGA wasteful. And these editors have been screaming about it ever since the Biden administration finally decided to derail the Medicare Advantage gravy train and demand more accountability from insurers. 

 As the piece describes the Biden move:

"The government has been trying to rein in aggressive coding for years. Starting in 2024 the Biden administration phased in a new payment model, V28, which reduced or eliminated the  impact of certain diagnostic codes."

Surely, this is something the Journal's editorial Pooh Bahs ought to be cheering, but no - instead they heap scorn on the Dems by claiming that "Medicare Advantage is wasting money." One wonders if these illustrious twerps even read the content of the more recent regular page WSJ article. 

Well, what the hell do you call upcoding and piling on diagnoses? 

Again, let's review the basics: The government pays Medicare Advantage insurers a set amount for each person who enrolls, with higher rates for sicker patients. And the insurers, among the largest and most prosperous American companies, from the get-go developed elaborate systems to make their patients appear as sick as possible, often without providing additional treatment.  See graphic (from NY Times, Oct. 9, 2022):

As a result, a program originally proposed to help lower health care spending has instead become a voracious financial gobbler. Now even more costly than the traditional government program it was meant to improve.   So the Biden administration, reclaiming fiscal sanity, proposed adjustments to halt the bleeding of standard Medicare.

And now the insurers (and WSJ editors) are squealing like stuck pigs.

But the monstrous money grabbing doesn't end with hyper diagnosing to bill the government, it includes massive denials of service as well. 

A fall, 2022 Kiplinger Retirement Newsletter was clear on what is now happening to new Medicare Advantage beneficiaries: being denied access in record numbers.  Indeed, according to Kiplinger, at a rate nearly twice that before the pandemic struck in 2020.  Why?  Because the number of new Advantage beneficiaries has itself nearly doubled as we're now approaching the stage of nearly half of Medicare being privatized. (E.g. NY Times, 10/9/23): 

The advance of this profit first malignancy and private perversion of traditional Medicare is clear from the graphic above.  Why profit first? Because Kaiser Family Foundation, a research group unaffiliated with the insurer Kaiser, found the insurance companies typically earn twice as much gross profit from their Medicare Advantage plans as from other types of insurance.

Again, it's important to process what the WSJ editors are bitching about. Prior to the changes adopted by the Biden administration doctors in the  MA system diagnosed illnesses which either:  (a) never existed in the first place, or (b) diagnosed conditions that were much worse than they actually were.  The effects of either tactic were to amp up the money collected by the MA plans. But also to siphon money from traditional Medicare to bolster the parasitic Advantage plans.

   This despite critics complaining for years that insurers' tactics (a, b  above) led to overpayments and more rapid approach of traditional Medicare to insolvency.  As the NY Times reported (10/9/22),

 

"Eight of the 10 biggest Medicare Advantage insurers — representing more than two-thirds of the market — have submitted inflated bills, according to the federal audits. And four of the five largest players — UnitedHealth, Humana, Elevance and Kaiser — have faced federal lawsuits alleging that efforts to overdiagnose their customers crossed the line into fraud."

This is exactly why Medicare itself is heading for insolvency, likely within the next 3 years. The government now spends nearly as much on Medicare Advantage’s 29 million beneficiaries as on the Army and Navy combined. If the nation operated under an economically sound system the Bush -incepted private plan MA system would be abandoned in a heartbeat. But, of course, the predatory capitalist system under Trump won't hear of it.

See Also:

Corporate America | Healthcare Policy

Biden Just Took the First Step to Overhaul Medicare Advantage. Many More Must Be Taken.

by Diane Archer | April 8, 2023 - 6:05am | permalink

Excerpt:

The corporations that run Medicare Advantage plans are engaged in widespread waste, fraud and abuse, resulting in tens of billions of dollars of overpayments to them every year. The advocates and government agencies overseeing Medicare Advantage have spent nearly two decades reporting on this fraud and waste and urging Congress to overhaul the program. Few in Congress or the administration were listening. Now, the Biden administration is finally taking action, but it’s only a first step.

Healthcare Policy | Insurance

Here is the Truth: Medicare Advantage Is Neither Medicare Nor an Advantage

by Wendell Potter | January 15, 2023 - 7:23am | permalink

Excerpt:

Right now, well-funded lobbyists from big health insurance companies are leading a campaign on Capitol Hill to get Members of Congress and Senators of both parties to sign on to a letter designed to put them on the record “expressing strong support” for the scam that is Medicare Advantage.

But here is the truth: Medicare Advantage is neither Medicare nor an advantage.

And I should know. I am a former health-care executive who helped develop PR and marketing schemes to sell these private insurance plans.

And:

Stop the Medicare “Advantage” Scam Before Medicare is Dead

Thom Hartmann

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Corporate America | Healthcare Policy

by Thom Hartmann | August 6, 2022 - 6:08am

---

— from The Hartmann Report

Excerpt:

Congress must pass a law to stop the deceptive advertising of Medicare Advantage plans. Only Medicare should be able to call itself Medicare.

Unless you’ve been out of the country for the past few years, you’ve seen the ads on TV featuring Joe Namath, Jimmy Walker, or William Shatner hawking so-called “Medicare Advantage” plans.

Medicare Advantage is not Medicare.

It’s private health insurance being offered to people over 65, with the bill paid for by Medicare. Once you get on an Advantage plan it’s very difficult to get off, and if you’ve been on for more than a year you may not be able to go back to regular Medicare with a Medigap plan at all.

And:

How Medicare Advantage Scammers Get Away With It

Thom Hartmann

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Healthcare Policy | Insurance

by Thom Hartmann | June 15, 2022 - 6:13am

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Excerpt:

In my multiple writings on the Medicare Advantage scam, the most common two responses I get (besides, “Thanks, you may have saved my life!”) are, “I’ve never had a problem with my Advantage plan,” and “If it’s so bad, how come so few people are saying so?”

Both are honest, good-faith questions and highlight how easy it is for insurance companies to get away with their Medicare Advantage scams. The answer to both boils down to the unique nature of insurance being the only “product” we buy where we have no idea if it’s any good until something bad happens — which can take years.

Every state in the union has an insurance commissioner. But why?

Why would any state go to the trouble and expense of creating a new layer of bureaucracy?

We don’t have “auto dealership commissioners” or “big-box retailer commissioners”: only insurance has an elected or appointed overseer.

Why would a state want to elect or appoint a very well-paid person to a new position in state government? Why would they appropriate money for a staff, for offices, in some cases even for buildings for a state insurance commissioner?

It turns out the answer is quite simple. One of the easiest scams in the history of scams, going all the way back centuries before Alfred Ponzi set up shop in Pie Alley, is done with insurance.

Here’s how it works.

If you have insurance, you send them a check every month. You think you’re covered and they’ll be there for you when you need them.

But you have no way of knowing if they’ll really be there for you when you need them because you’ve never used the service in a real health crisis.

Mensa Inequality Intermediate Algebra Problem

 

Revisiting Operations-Computations In Basic Tensor Algebra

We already know tensor algebra is a crucial first step in getting acquainted with tensor calculus and also getting a better grasp of differential geometry. Let's also begin by noting that just as in linear algebra, e.g.

• Math Revisited: Looking At Linear Algebra

 matrices factor into this.  

I.e. every symmetric tensor will have the form (using dummy indices i, j):   

a _{i j} =

(a₁₁ ……a₁₂………a ₁₃)

(a₁₂….. a₂₂……….a₂₃)

( a₁₃……a₂₃………a₃₃)

The anti-symmetric or skew symmetric tensor will have the form (note how the order of the dummy indices changes):     a _ji  =

(0 ……a₁₂………-a ₃₁)

(-a₁₂…..0……….a₂₃)

( a₃₁……- a₂₃……0)

Or effectively only three distinct components.

The most basic tensor of all is the unit tensor, idemfactor or identical dyadic.  defined:

I  =  i^ i^ + j^ j^ + k^ k^  =

When I is multiplied by any other vector as a factor the vector is unchanged, i.e.

    I × C  =  i^ C_x + j^ C_y + k^ C_z  

The Kronecker delta, which we already saw in Introduction to Differential Geometry, Part 1, is:

d _ik = 

(1….0…..0)

(0….1…..0)

(0….0… ..1)

Example 1:  Compute:  a_{i j}  d i _j   

If a_{i j}  is a second order tensor with matrix:

Then we have:

And we let   x  _i  and   y _j   be first order (rank 1) tensors (i.e. vectors) respectively given by:

x _i    =  (2, 1, 4)

y _j    =  (3, 7, -1)

a)Find:   a _ij  x _i     +  a _ji 

b)Find:  ( a_{i j}    -  2/3  d  _ij   )

Solutions:

a) From Mathcad matrix computations:

b)

Diagonalizing tensors is analogous to obtaining the eigenvalues for a matrix in linear algebra.  Hence, we need to extract the eigenvalue equations via diagonalization and obtain the distinct eigenvalues.   Consider the object (solid tetrad) shown below for which we want to obtain  the principal axis.

For which we define a _{i j} = 

With T’ =  A × I × A^t

Where t denotes the transpose.  Then we obtain, T’ =

Which is to be diagonalized.  Writing this out:

(15 - l…..0……..0)

(0…….11- l ....-3Ö2)

(0……..-3Ö2…8- l )

This leads to a cubic equation with triple roots which are:

l1 = 15,  l2 =  5, and l3 =  14

Substituting l1 in the matrix we get:

 Now find the remaining two eigenvalues, l2 and l3, to diagonaliize the matrix and obtain solutions in c _x, c _y and c _z.

Thereby obtain the principal axes in terms of: e^’_x,        e ^’_y and e^’ _z .

Show in particular that:

  e^’ _z    = - 1/ Ö3   e1^ -  1/ Ö3   e2^ + 1/ Ö3   e3^

Solution:

Using  the eigenvalue l2  = 5 we arrive at:

For which:

c _y = c _z / Ö 2    and  

c = 0 e₁^  -  c _z / Ö 2    e₂^ + c _z  e₃^

Whence:

(c _z  ² / 2  +  c _z  ²)  = 1    Þ   c _z   =  Ö (2/ 3)

Further: 

e₂^’  =

 1/Ö 2   (Ö (2/ 3)) e₂^”  +  Ö (2/ 3) e₃^”  = 1/ /Ö 3

e₂^”  + Ö (2/ 3) e₃^” 

For  the eigenvalue  l3  =  14:

We arrive at:

Yielding:   c _y = -  Ö 2 c _z   and   c _z    =  1/ Ö 3

Þ   c   = 

Ö 2  c _z   e₂’’’   +  c _z   e₃^”’ 

=  -Ö (2/ 3) e₃^”  + Ö (1/ 3) e₃^”  

Then  e _j =  å³ _{i= 1}   a _{i j}   e _j ’ and:  e _i’ =  å³ _{j= 1}   a _{i j}   e _j

And:

e _x^”    =  - 1/ Ö 2 e ₁   - 1/ Ö 2 e ₂   

So: 

e _x^’    =  e _x^”    =  1/ Ö 2 e ₁   - 1/ Ö 2 e ₂    + 0

Similarly:  

e _y^”    =   1/ Ö 2 e^ ₁   +   1/ Ö 2 e^ ₂    + 0

And:  e _z^”    =   e^ ₃     Þ   e _y^’ = 

1/ Ö 3 (1/ Ö 2) e^ ₁   +  1/ Ö 3 (1/ Ö 2) e^ ₂  +   Ö (2/ 3) e₃^ 

=   e^ ₁   / Ö 6   +   e^ ₂   / Ö 6   +   2e^ ₃  / Ö 6  

And:  e _z^’    =  

 - 1/ Ö 3  (e^ ₁ ) - 1/ Ö 3  (e^ ₂)   + 1/ Ö 3  (e^₃)  

(The Principal axes are: e _x^’ ,    e _y^’   and  e _z^’ )

Suggested Problem:

In a certain rectangular coordinate system, the directions of whose axes are given by the unit vectors i, j and k, the inertia tensor of an object is given by:

I = K x= 

(1….0…..0)

(0….1…..1)

(0….1… ..1)

a) What are the principal moments of inertia of the object (the moments of inertia along the principal axis) relative to the origin of the above coordinate system?

b) What is the direction of the principal axis corresponding to the principal moment of inertia and equal to K?

c)If the origin of the above rectangular coordinate system is at the center of mass of the object and the total mass of the object is M, what is the change in the inertial tensor of the object if the rectangular coordinate system is displaced parallel to itself a distance r_o in the direction

 

 (1/ Ö2)j +  (1/Ö2)k?