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23 Nov 2019

Why do books say that the farthest galaxies we can see in opposite directions are not within each other's visible universe, when Einstein's addition of velocities law shows that any two velocities add up to a velocity less than that of light?

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Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

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Consider a satellite of mass ms in a circular earth orbit. Anysatellite orbiting in the upper reaches of our atmosphere willexperience a small drag force, Fd, opposite to itsdirection of motion. This drag force will cause the satellite tospiral inward toward smaller orbital radii. In the process, theorbital velocity of the satellite actually increases! This problemwill investigate how a drag force can cause a body in orbit tospeed up.

The situation is sketched in Fig. 1. The diagram shows separateforce and velocity diagrams for the satellite. The mass of theearth is denoted me, while the mass of the satelliteis ms. The tangential velocity of the satellite isvt and the small radial velocity is acquires because itis spiraling inwards is vr. The size of vr isgreatly exaggerated in the diagram, and we can take vr<< vt and use vt as an adequateapproximation to the magnitude of the total velocity, v. We willcall θ the ‘angle of de- scent’.

Turning to the force diagram, we see that the total force actingon the satellite is the vector sum of F⃗d andF⃗g. Again, the drag force is much weaker than thegravitational force, so Fd << Fg . Wecan use this to our advantage by just considering the effect ofFg along the line of descent. We can see from thediagram that Fg has a component along the line ofdescent given by Fg sin(θ).

(a) First show how we arrive at the following two basicequations governing this system, and carefully describe thephysical meaning of the quantity msad inwords:

Fgsinθ−Fd = msad (1)

Fg = (ms*vt2)/r= G(msme)/r2 (2)

(b) We can find out more about ad and its relation tothe various forces and velocities by powerful

use of the chain rule for differentiation:

Explain Eq. 3 in your own words. Where did vr comefrom and why is the ≃ sign there?

ad = dv/dt = (dv/dr)*(dr/dt) =vr(dv/dr) ≃ vr(dvt/dr) (3)

(c) We can now make use of Eqs. 2 and 3 to show that:

msad = (1/2)Fg sinθ

A) A siring is tied to a book and pulled lightly as shown at right. The book remains in contact with the table and does not move. Draw a free-body diagram for the book. Label each of the forces exerted on the book as you were instructed in the tutorial Forces.

B) How do the forces exerted on the book, in this case, compare to the forces exerted on the book when the string is not present? List any forces that are the same (i.e., same type of force, same direction, and same magnitude) in both cases. Make a separate list of forces that change (or are not present) when the string is pulled.

C) Consider the following statement made by a student about a book at rest on a level table: "The two forces exerted on the book are the normal force directed up and the weight of the book directed down. These are equal and opposite to one another. By Newtons third law they are (action-reaction) force pair, so the normal force is always equal to the weight of the book." Do you agree with the student? Explain why you agree or disagree.

D) Consider a book on top of a level table while the book is being pressed straight down by a hand. In the space below, draw a free-body diagram for the book. Label the forces as you did in the tutorial Forces. How do the forces exerted on the book in this case compare to the forces exerted on the book when the hand is not pushing? List any forces that are the same (i.e., same type of force, same direction, and same magnitude) in both cases. Make a separate list of forces that change (or are not present) when the hand is pressing down on the book.

E) When the hand is pushing down on the book, is the magnitude of the weight equal to the magnitude of the normal force exerted by the table on the book? How can you tell?

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