Friday, August 23, 2024

Solutions to Rotational Dynamics Problems (4)


1) Consider the table shown below, applied to circular satellite orbits around the Earth:

Distance from Center of Earth

2 r

3 r

3r/ 2

4 r

5 r/2

5 r

Acceleration of gravity

g/4

 

 

 

 

 

 Complete the Table above given that r = 6.4 x 10 6 m. Hence or otherwise, deduce the acceleration of gravity g at a distance 10r.

Solution:

The solution here is based on noting that the gravitational field intensity g ~ 1/r2

That is, it is inversely proportional to the distance from the center squared. Hence, the remaining entries in the table as follows, in ascending order:

 
g/9, 4g/9, g/ /16, 4g/25   and g/25


Given this relationship (g ~ 1/r2) then at a distance 10r we have: g/100,
Thus, if r = 6.4 x 106 m, then at a distance:

 10 r = 10 (6.4 x 106 m) = 64 x 106 m = 6.4 x 107 m

And given, g = 9.8 ms-2, we would find a new gravitational intensity  (corresponding to the much greater distance from the center) of:

g/100  =   (9.8 ms-2)/ 100 = 0.098 ms-2

2) Newton’s version of Kepler’s 3rd law is usually written as:

 (m1 + m2)P2 = 4p2/G (r1 + r2)2 

where G is the Newtonian gravitational constant: 

(G = 6.7 x 10-11 N-m2/kg2) and (r1 + r2) is the distance between the centers of the two bodies.

 Use this to find the period P of the Moon if its mass m1 =  m2/80 where Earth’s mass m2 = 6.0 x 10 24 kg and (r1 + r2) = 384, 000  km.    Compare this to the value obtained by using Kepler’s simpler version of the 3rd law with (r1 + r2) =  a1= 0.0025 AU.


Soln.

m2/80 = (6.0 x 10 24 kg)/ 80 = 8.3 x 10 22 kg

(r1 + r2) = 3.84 x 10 5  m

P = Ö4p2/G (a1)3  (m1 + m2)

G =  6.7 x 10 -11   m 3  s-2 kg -1

(m1 + m2) = 6.0 x 10 24 kg  +  8.3 x 10 22 kg » 6.0 x 10 24 kg


P =

 Ö 4p2/ (6.7 x 10 -11   m 3  s-2 kg -1) (0.0025 AU ) 3  6.0 x 10 24   kg

P »  0.002 yr.


3) Verify the conservation of angular momentum applies for a spacecraft in orbit around the Earth if its velocity at perigee is  10.7 km/ sec, its distance from Earth at perigee is  6.6 x 10 3 km, its velocity at apogee is  0.75  km/ sec  and its distance at apogee is: 9.3 x 10 4 km.  Find the period of the spacecraft.

Soln.

For conservation of angular momentum L in an elliptical orbit (with ra  the radius vector at aphelion,  and  with rp the radius vector at perihelion)

 L = mva ra =  mvp rp

Or: va ra =  vp rp

va ra =    (0.75  km/ sec) (9.3 x 10 4 km) » 7.0 x 10 4 km2/sec

vp rp =    (10.7 km/ sec) (6.6 x 10 3 km) » 7.0 x 10 4 km2/sec

Given:  va ra »  vp rp

To within the limits of errors, the principle of angular momentum conservation  is validated.

The period is obtained from: T = 2π (a3/m)½

Where 
m  = G (m1 + m2) 


And:  a = [
ra +  rp ] /2 =  (9.3 x 10 4 km + 6.6 x 10 3 km)/2 »

= (9.3 x 10 7 m + 6.6 x 10 6 m)/2 » 4.9 x 10 7 m

T »

2π [(4.9 x 107 m)3/[6.7 x 10-11 Nm2/kg2) (6.0 x 1024 kg)]½

T 
» 4.3 x 10 4 s » 12.2 hrs.



4) A gyroscope is made from a flywheel made to spin about a rotation axis (bar) as shown in the sample diagrams in the text. If each of the two forces applied = 20 N and the distance d is 0.15m, then find the magnitude of the torque t.

If the mass of the flywheel is 1.0 kg and its radius r = 0.1 m find the magnitude of the angular momentum if  w =  p rad/sec. Thence or otherwise find the rate of precession of the angular velocity W.

Soln.

t    =   2F d = 2 (20 N) 0.15m =  6 Nm

The angular momentum:

L = Iw =  (½ m r 2 ) ( p rad/sec) =

½ (1.0 kg) (0.1m) 2  ( p rad/sec)  =    0.016 kg m2 /sec

The rate of precession:

 W   = 2F d/ Iw  =   6 Nm/ 0.016 kg m2 /sec =  375 kg m2 /sec


5) (Inquiry problem)

A proponent of Immanuel Velikovsky's "colliding worlds" nonsense claims that all that was needed for Earth's rotation to stop (when Venus came so close to Earth in passing from Jupiter to its present orbit) was a torque defined:  N = Fr X Rc.  as used in his diagram below:




The problem is that an OFF-axis torque  (e.g. perpendicular force ) is needed to effect this, and the only "torque" he has is mythical, based on: N = Fr X Rc. But Fr is a fantasy force that doesn't exist! The only actual force acting is an attractive (gravitational) one through the centers! Hence no external agent is available to act perpendicular to the line of centers to halt the rotation! 


6) Inquiry Problem

In the early morning hours of June 14, 2002, the Earth had a  remarkably close encounter with an asteroid (2002 MN) the size of a small city. It remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was 73,600 miles from the center of the Earth—about a third of the distance to the Moon. Observations indicate the asteroid to have a diameter of about 2.0 km. Estimate the kinetic energy of the asteroid at closest approach, assuming it had an average density of 3.3 g/ cm 3
------------------
Hint:  Use  E tot = K + V  = mv2 / 2 -   GMm/r =  -   GMm/2r

And take  the asteroid as coming from 'infinity' with r the distance to Earth at closest approach.

Soln.

E tot = K + V  = mv2 / 2 -   GMm/r =  -   GMm/2r

To obtain velocity:

E tot  = mv2 / 2 -   GMm/r =  -   GMm/2r

And: v2 / 2 -   GM/r =  -   GM/2r

v2 / 2  =  -   GM/2r + GM/r =   GM/2r

Then velocity: v = Ö GM/2r

Where r =   73,600 miles =  117, 600 km = 1.17  x 10 6 m

We know M (Earth's mass) = 6.0 x 10 24   kg

G =  6.7 x 10 -11   m 3  s-2 kg -1

Average density of asteroid given as  3.3 g/ cm 3  = 3300 kg/ m 3 

Mass of asteroid = density x volume

Assume spherical volume so V = 4/3  (p  r 3 )

Where r = 2.0 km =  2.0  x 10 3 m

The Volume V = 4/3  (p (2.0  x 10 3 m)3 ) = 3.3  x 10 10 m 3 

Mass of asteroid = (3300 kg/ m 3   ) 3.3  x 10 10 m 3 

= 1.1 x 10 14  kg

v = Ö GM/2r =  where M = 6.0 x 10 24   kg

G =  6.7 x 10 -11   m 3  s-2 kg -1

v = Ö (6.7 x 10 -11   m 3  s-2 kg -1)  6.0 x 10 24   kg/ 2(1.17  x 10 6 m)

= 1.3 x 10 4 m/ s

The kinetic energy of the asteroid is:

K = 9.6  x 10 21 J

 For reference, this magnitude approaches the energy of the 

Chicxulub asteroid at  300 zetta joules.

Wednesday, August 21, 2024

Newsflash To WSJ Editors: "Price Fixing" Is Not The Same As Protecting Consumers From Price Gouging

 


Frustrated that Kamala Harris hasn't yet submitted to media interrogations, the Reich media (like the WSJ,  Newsmax, FOX) has been barking nonstop about its own takes on her positions.  All wrong. For example, the irrepressible WSJ stooge Holman Jenkins Jr.  in a weekend piece (Aug. 10-11), yapping about Harris "adopting Trump's plan to stop taxing tips."  As I corrected the clown in my Aug. 16 post she didn't copy it she created her own REAL version to benefit working people, e.g. waitresses, busboys, casino assistants etc. 

Trump's version, as Robert Reich pointed out last Monday ('Last Word') would in reality benefit Hedge Fund operators, investment bankers and private equity honchos who'd use his mutation to justify "tips" for their assorted machinations, selloffs, buyouts, what not.  

Then there was the WSJ editorial appearing Aug. 17-18 (p. A12);

Kamala Harris Endorses Nixonomics - WSJ

 'She Wants Price Controls on Groceries, Venezuelan-Style'

Trying to portray Harris's vow to combat price gouging (made in her NC appearance last Friday) as being "price fixing". No, it wasn't!. She is committed to stop all the price gouging that one beholds in the super market aisles, not "fixing" prices by mandate.  

Price controls, on the other hand, is something I saw in Barbados in the early 1970s, when the Errol Barrow administration imposed a list of set prices on grocers, supermarkets.  The array of targeted goods included: eggs, chicken, plantains, cucumbers, bananas, bread and cereals.  For example, eggs had to be reduced from any arbitrary grocer pricing (usually above $5 Bds - or $2.50 U.S.) to a flat $2 a dozen.  Chicken had to be reduced from prices usually in the realm of $4.50 /lb. to $1.50/lb. Bread had to come down from $4 a loaf on average to $2.  Needless to say the grocers and supermarkets were not happy and many workers were laid off.

But this is not what Kamala was proposing. But maybe the anti-Kamala WSJ spin gurus forgot that in 2023 PepsiCo had become a prime example of how large corporations cynically used the pandemic-supply template to increase costs, and then some. Hugh Johnston, the company’s chief financial officer, actually went on record in February, '23 saying that PepsiCo had raised its prices by enough to buffer further cost pressures in 2023. At the end of April, the company reported that it had raised the average price across its snacks (e.g. Doritos) and beverages (Gatorade) by 16 percent in the first three months of the year.

According to Albert Edwards, a global strategist at Société Générale, referring to profit margins, quoted in a WSJ piece in April, 2023 (check it out, WSJ editors):

“Companies are not just maintaining margins, not just passing on cost increases, they have used it as a cover to expand margins."

So Kamala definitely has an unimpeachable policy position to run on, as well as others, including expanding the child tax credit - which of course the reactionary Right will spin as "socialism". But let's bear in mind these yahoos back in the 1930s were yelping that Social Security was socialism and in the 1960s that Medicare was one jump from communism. The point? The Rightists will always bleat and whine about any policy position Harris gives that benefits working or middle-class families, like increasing the child tax credit to $6,000. (It will revert to $1,000/yr. by 2026 - likely under a Trumper administration.)

 All of which means she needs to take care and only parcel them out a bit at a time. As Michelle Obama noted in her terrific speech at the DNC last night, "they (right media) will find a way to lie about Kamala".  And indeed they have! In the same editorial spieling nonsense that "the Biden-Harris inflation has made homes unaffordable for most young families".  Failing to note it was supply side issues in labor and construction materials - during the pandemic which began under Trump - that ushered in the inflation later.  

The WSJ editors then have the gall to ding Harris' proposal to provide $25,000 for first time home-buyers, calling it "a brainstorm of more subsidies." Of course, because these nattering nabobs would prefer the money go to another round of trillion dollar tax cuts for billionaires and corporations - in case enough dummies put the orange fungus back in office.

All of which suggests to me Kamala needs to take care and only parcel her plans and proposals out a bit at a time. Because whatever she offers, the reactionary yappers will either misrepresent them, or put them under the "socialist" rubric.

Still, I am confident, that as the next few weeks pass, more voters will become aware of the rank malarkey the Wall Street Journal and its TV cousins at FOX News are trying to peddle, i.e. that Kamala Harris plans to go the "Venezuela" route  in advancing socialism merely because she wants to help more working class Americans survive in the supermarket aisles, as well as in the restaurants, as service staff.

What we need is for voters this year to be more alert as well as more intelligent and critically thinking. So they know immediately when they see images or videos of women with 'Swifties for Trump' T-shirts, it is all AI -generated horse shit, e.g.


So why aren't the WSJ editorial blowhards bringing up any of this, especially as barely a week ago the orange Turd claimed Kamala was using AI to generate extraordinary crowds, in a Michigan appearance? Well, of course, because they are on Trump's bandwagon!


  Because no real American woman of any integrity or sense - especially a Taylor Swift fan - would so debase herself as to be willingly associated with the serial rapist, and female genital grabber Trump. 

Well, with one sad exception.  A kook named Jenna Piwowarczyk caught wearing a "Swifties for Trump" t-shirt at a Trump rally in Racine, Wisconsin, on June 18, and posted to X by Wisconsin Right Now.

Could someone get this forlorn fool an ECT jolt? 

But fortunately, such females with brains in their behinds remain in the minority. Meantime, we sorely need Taylor Swift to come out by the last night of the DNC and endorse Kamala Harris, to show she's on the side of the angels. As well as dispelling the AI-fabulism that any such entity as Swifties for Trump' exists in any serious capacity.

See Also:

by Maya Boddie | August 21, 2024 - 6:19am | permalink

— from Alternet

Since President Joe Biden dropped his reelection bid last month, Vice President Kamala Harris' campaign has raised $500 million, according to CNBC News.

CNBC reports, Harris' "political operation raised $310 million in July alone, with more than $200 million of that sum coming in after Biden’s withdrawal."

Furthermore, the campaign expects to see the cash haul reach $600 million by the end of this month.

Additionally, Punchbowl News' Jake Sherman reported via X that of the $105 million the House Democrats super PAC "House Majority PAC" raised in July, billionaire Mike Bloomberg donated $10 million, and Fox News' Rupert Murdoch's daughter-in-law Kathryn Murdoch contributed $750,000.

And:

by Amanda Marcotte | August 21, 2024 - 6:04am | permalink

— from Salon

I'm old enough to remember when the sexist stereotype of feminists was that they're "humorless" and "shrill." So it never fails to make me smile to hear Donald Trump, in ever-shriller tones, rant and rave about how much he hates Vice President Kamala Harris' laugh. Her guffaws haunt him. We cannot doubt he's kept up late at night, dwelling on his fear that a woman might be enjoying herself. At a rally in Asheville, North Carolina last week, Trump could not stop babbling furiously about Harris's laugh.

"That’s a laugh of a person with some big problems," Trump said, projecting his deep psychological issues onto his opponents, as usual. He insisted it's "the laugh of a crazy person."

The Harris campaign was unfazed, following the speech with an email that read, "Donald Trump delivered what was supposed to be a speech focused on his economic plan but ended up resembling more of one unhinged man’s public airing of grievances."

And:

by Heather Digby Parton | August 20, 2024 - 6:15am | permalink

— from Salon

All the attention is going to be on the Democratic Convention in Chicago this week but it's pretty clear that for the next few weeks, the center of gravity in American politics is going to be in Pennsylvania. All the strategists and political pundits tell us that both the Harris and the Trump campaigns see it as a must-win and according to the latest polls, it's a tight race there. Already both candidates and their running mates have held events in the Keystone State, with no doubt many more to come.

I recently spent some time in southern Pennsylvania, as I often do, visiting friends in the late summer and was struck by the fact that unlike 2016 and 2020 there were far fewer Trump signs out in the rural areas. For years they had kept their tattered big blue Trump flags flying even long after the elections but this year they seem to be few and far between. Sure there were some yards covered with signs and trucks festooned with bumper stickers but it's much rarer than it's been in the past. What that means, I can't say definitively but something has changed and it's fair to guess it has something to do with enthusiasm. It feels as though some of the air has gone out of the Trump balloon. ( I'm not the only one who sees that.)

Monday, August 19, 2024

The Best Explanation For Why The Stock Market Cratered Two Weeks Ago

 

                                          WSJ Finance columnist Jason Zweig


Anytime the stock market craters - like it did two weeks ago- you can be sure people (not just investors) want to have answers.  To refresh memories, on Monday, August 5th the Japanese stock market had its worst day   since 1987, dropping 12.4% and U.S. stocks cratered by over 1,000 points on the DOW.  At the same time, Wall Street's 'fear gauge' - the VIX index of volatility - shot up more than 50 percent to its highest level since the pandemic days of 2020.

The following day, August 6th, Japanese stocks bounced back by 10% while the S&P 500 gained 1 % and the VIX fell by 28%. By week's end stocks stood not far below where they were before the wild ride.  What happened? What spurred the hysteria?  Well, hysteria and lack of critical thinking as WSJ's Jason Zweig interpreted it.  From my own perspective, after decades of following the markets (ever since the crash of 1987),  it amounts to consistent over-reaction by investors and traders, to any and everything.

War brewing in the Middle East?  Watch out! The DOW is likely to dive. Too many tapping unemployment benefits, or too low a monthly jobs number - watch out!  Inflation too high or Feds mulling interest rate hikes, watch out! Any or all of these can bring a correction or a crash.  This is one reason many of us left the stock market ages ago. We opted instead to plow our savings into immediate fixed annuities to be able to have a dependable fixed lifetime income, no matter the world's events, weather disasters or any other outside factors.

As for the theories about the crash Aug. 5th, there were several trotted out. Among them,  big hedge funds had borrowed in cheap Japanese yen to buy U.S. stocks and other assets, then  panicked when the yen suddenly rose against other currencies. (Since this made borrowing more costly).

Another theory: Investors suddenly lost confidence - given the lower job number (114,000) that the Federal Reserve could prevent the economy from falling into recession.  As I told Janice, at least they got 'tuned up' for what might be the real thing in a year, if enough dummies vote Trump back into power.  Because the orange moron, vowing 10-20% tariffs on foreign goods across the board, will likely trigger massive layoffs, the highest inflation since the oil shock of the 70s and yeah, a deep recession. Or even Depression. And yet some 56% of American dupes (according to a poll this a.m.) still think him to be "better on the economy".

Anyway, I side more with Jason Zweig's conjecture about what happened to trigger the cratering of stocks two weeks ago. In his words:

"The extraordinary smoothness of markets over the past year and a half had goaded hedge funds and other big traders into taking ever escalating amounts of risk.  From Feb. 22, 2023 to this July 23, the S&P 500 never dropped more than 2% in a day, the longest such streak in more than 17 years, But you can only stretch a rubber band so far until it snaps, and when it snaps it stings.

The simplest explanation of all: Markets went haywire because markets consist of people and crazy behavior is contagious. To paraphrase Mark Twain: 'Truth is stranger than fiction because fiction has to make sense.' Markets don't."  

He then goes on to cite Nobel laureate in economics Paul Samuelson who argued that while markets are "micro-efficient" they are "macro-inefficient". Meaning that while investors are generally good at quickly integrating new information about individual securities, they are bad at sizing up macroeconomic developments that can affect assets like stocks, bonds or commodities.

Basically, as one expert pointed out to Zweig,  macro-inefficiency arises because while an individual security (e.g. bond) is fairly stable, broader bundles of assets are not. Thus, these "can be swayed by countless forces" making their value more subjective, as well as fragile.

This is basically why neither of us has any skin in the stock market, or the bond market.  We dislike the runaway emotions of traders, or flash trade investors dictating the value of our money - from day to day, or month to month. And since there is zero chance of the human sympathetic nervous system changing or becoming more resilient in the face of oncoming threats (e.g. climate change, Trump etc.) we are quite ok with immediate fixed annuities. At least we can sleep at night!

See Also:

And:

Immediate Annuities Remain The Best Bet to Secure A Decent Retirement 

And: