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Using the technique shown in Satellite Orbits and Energy , show that two masses m 1 and m 2 in circular orbits about their common center of mass, will have total energy E = K + E = K 1 + K 2 G m 1 m 2 r = G m 1 m 2 2 r . We have shown the kinetic energy of both masses explicitly. ( Hint: The masses orbit at radii r 1 and r 2 , respectively, where r = r 1 + r 2 . Be sure not to confuse the radius needed for centripetal acceleration with that for the gravitational force.)

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Given the perihelion distance, p , and aphelion distance, q , for an elliptical orbit, show that the velocity at perihelion, v p , is given by v p = 2 G M Sun ( q + p ) q p . ( Hint: Use conservation of angular momentum to relate v p and v q , and then substitute into the conservation of energy equation.)

Substitute directly into the energy equation using p v p = q v q from conservation of angular momentum, and solve for v p .

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Comet P/1999 R1 has a perihelion of 0.0570 AU and aphelion of 4.99 AU. Using the results of the previous problem, find its speed at aphelion. ( Hint: The expression is for the perihelion. Use symmetry to rewrite the expression for aphelion.)

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Challenge problems

A tunnel is dug through the center of a perfectly spherical and airless planet of radius R . Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit at the surface.

g = 4 3 G ρ π r F = m g = [ 4 3 G m ρ π ] r , and from F = m d 2 r d t 2 , we get d 2 r d t 2 = [ 4 3 G ρ π ] r where the first term is ω 2 . Then T = 2 π ω = 2 π 3 4 G ρ π and if we substitute ρ = M 4 / 3 π R 3 , we get the same expression as for the period of orbit R .

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Following the technique used in Gravitation Near Earth’s Surface , find the value of g as a function of the radius r from the center of a spherical shell planet of constant density ρ with inner and outer radii R in and R out . Find g for both R in < r < R out and for r < R in . Assuming the inside of the shell is kept airless, describe travel inside the spherical shell planet.

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Show that the areal velocity for a circular orbit of radius r about a mass M is Δ A Δ t = 1 2 G M r . Does your expression give the correct value for Earth’s areal velocity about the Sun?

Using the mass of the Sun and Earth’s orbital radius, the equation gives 2.24 × 10 15 m 2 /s . The value of π R ES 2 / ( 1 year ) gives the same value.

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Show that the period of orbit for two masses, m 1 and m 2 , in circular orbits of radii r 1 and r 2 , respectively, about their common center-of-mass, is given by T = 2 π r 3 G ( m 1 + m 2 ) where r = r 1 + r 2 . ( Hint: The masses orbit at radii r 1 and r 2 , respectively where r = r 1 + r 2 . Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expressions for the kinetic energy.)

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Show that for small changes in height h , such that h < < R E , [link] reduces to the expression Δ U = m g h .

Δ U = U f U i = G M E m r f + G M E m r i = G M E m ( r f r i r f r i ) where h = r f r i . If h < < R E , then r f r i R E 2 , and upon substitution, we have

Δ U = G M E m ( h R E 2 ) = m ( G M E R E 2 ) h where we recognize the expression with the parenthesis as the definition of g .

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Using [link] , carefully sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass, m . Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal acceleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle ε , defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. tan ( λ + ε ) = g ( g ω 2 R E ) tan λ , where ω is the angular velocity of Earth.

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(a) Show that tidal force on a small object of mass m , defined as the difference in the gravitational force that would be exerted on m at a distance at the near and the far side of the object, due to the gravitation at a distance R from M , is given by F tidal = 2 G M m R 3 Δ r where Δ r is the distance between the near and far side and Δ r < < R . (b) Assume you are falling feet first into the black hole at the center of our galaxy. It has mass of 4 million solar masses. What would be the difference between the force at your head and your feet at the Schwarzschild radius (event horizon)? Assume your feet and head each have mass 5.0 kg and are 2.0 m apart. Would you survive passing through the event horizon?

a. Find the difference in force,
F tidal = = 2 G M m R 3 Δ r ;
b. For the case given, using the Schwarzschild radius from a previous problem, we have a tidal force of 9.5 × 10 −3 N . This won’t even be noticed!

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Find the Hohmann transfer velocities, Δ v EllipseEarth and Δ v EllipseMars , needed for a trip to Mars. Use [link] to find the circular orbital velocities for Earth and Mars. Using [link] and the total energy of the ellipse (with semi-major axis a ), given by E = G m M s 2 a , find the velocities at Earth (perihelion) and at Mars (aphelion) required to be on the transfer ellipse. The difference, Δ v , at each point is the velocity boost or transfer velocity needed.

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Questions & Answers

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
Atega Reply
introduction to econometrics
Husseini Reply
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Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
explain inflationcause and itis degre
Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
Mercy Reply
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money as unit of account means what?
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A unit of account is something that can be used to value goods and services and make calculations
Jim
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Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
Kalombe
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Kalombe
hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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