Writing Assignment 2: Can Newton stop worrying? Please read this document carefully and solve the six problems isolated in boxes. Then, create a write up (essentially an article] which introduces the reader to the material and resolves the main problem at hand: Can Newton Stop worrying? Your write up should contain all of the background information presented here (but in your own words} as well as presenting the various solutions to the six items. However, your write up should not enumerate the six items, but rather address {and solve them) as part of an organic presentation of the material. In particular, do not assume that the reader has access to the present document. More instructions and guidelines can be found on the course web page. Before publishing his groundbreaking theory of gravitation, Isaac Newton spent a great deal of time worrying]: Newton's theory of gravity says that the gravitational force on an object of mass m due to an object of mass M is inversely proportional to the square of the distance between those objects. In mathematical form, this is uid FzG r2 e, where e, is a unit vector in the direction from M to m, r is the distance between the objects and G is a constant of proportionality, called the gravitational constant. As it turns out, this force {which is a vector) is gotten from the gradient of a function gb : R3 > R dened by M M 7 = G (1) "'22 +3? +z2 where we shall assume that the object of mass M sits at the origin [0,0,0] and the object of mass m sits at the point [m,y,z), a distance r = 2:2 + 5:2 + z2 from (0,0,0). did?) 2 (m? 3'? Z) = _G 1. Show that F = mV. Hint: You should note that e _ (mvyazl _ 2: y Z .- Iiyvzll y/1'2+y2+z25\/2:2+y2+z2'Jm2+y2+z2 I This function at is called the gravitational potential for the mass M. In some sense, it determines the motion of all massive objects in space2. And for this reason, we focus our study on q'). We can now discuss the source of Newton's great worry. Though Newton's theory worked spectacularlya, it treats massive objects (e.g., our Sun) as points in space without volume. This is, of course, nonsense. Tb account for realistic massive objects, the gravitational potential must be dened using a triple integral where mass is treated as being spread continuously over the space it occupies. For example, if the object is a sphere or radius 1 centered at the origin (0,0, 0), the gravitational potential is 2w 7r 1 2 app0% L A %sin(

1, i.e., you're viewing the gravitational potential from outside the sphere, the limits of integration R4] and RI should simplify considerablyr upon noting that 0 g ,0 g 1 <: r. under this simplication compute the double integral and simplify. your resulting expression stir should now only be a function of r g p. finally using result from part put for in terms m. comparing what we obtained does gravitational potential look any different away point mass as it sphere comment on why newton can stop worrying. take granted that fubini theorem is valid case>