PFA
MT3270 Problem Set 2 – G2 Clearly write your name on top of each page. Use Adobe Scan to scan your pages as a single-page PDF and upload it to Moodle. If you use other scanning software ensure that resolution is set to 72 dpi (or something of that order) to ensure the images are not too big, while still being readable. Instructions can be found in the Resources block of the MT3270 Moodle page. G2:1 Consider a thin disc of radius R and consisting of a material with constant mass density (per unit of area) %. Use cylindrical coordinates, with the z-axis perpendicular to the plane of the disc, and the origin at the disc’s centre. We are going to calculate the gravitational potential, and the gravitational field, in points on the z-axis only. 1. Show that the gravitational potential ϕ(z) set up by that disc is given by ϕ(z) = 2πG% ∫ R 0 r′√ (r′)2 + z2 dr′; make sure to explain where the factor 2π comes from, and where the factor r′ in the integrand comes from. 2. Evaluate this integral. 3. Approximate ϕ(z), both for 0 < z="" �="" r="" (i.e.,="" for="" points="" very="" close="" to="" the="" disc)="" and="" for="" z="" �="" r="" (i.e.,="" for="" points="" very="" far="" away).="" you="" will="" need="" the="" following="" taylor="" approximation:="" √="" 1="" +="" x="" ≈="" 1="" +="" x="" 2="" +o(x2),="" applied="" in="" different="" ways.="" 4.="" using="" a="" symmetry="" argument,="" explain="" why,="" for="" points="" on="" the="" positive="" z-axis,="" the="" gravitational="" field="" points="" towards="" the="" centre="" of="" the="" disk,="" i.e.="" parallel="" to="" the="" z-axis.="" 5.="" starting="" from="" the="" gravitational="" potential,="" and="" using="" the="" answer="" to="" 4.,="" derive="" an="" expression="" for="" the="" gravitational="" field="" a="" for="" points="" on="" the="" positive="" z-axis.="" 6.="" what="" does="" this="" become="" in="" the="" limit="" r→∞?="" 7.="" (brain="" teaser)="" show="" that="" the="" potential="" ϕ="" “becomes="" infinite”="" in="" that="" same="" limit.="" how="" can="" you="" get="" round="" this="" problem,="" to="" find="" the="" gravitational="" field="" from="" the="" potential="" even="" in="" the="" limit="" r→∞?="" 1="" g2:2="" (to="" the="" memory="" of="" i.m.="" banks)="" consider="" a="" “planet”="" consisting="" of="" two="" thin="" concentric="" shells,="" as="" in="" the="" pic-="" ture,="" both="" made="" from="" the="" same="" material="" (same="" mass="" density="" %).="" the="" radii="" r1="" and="" r2="" are="" given,="" and="" the="" thickness="" of="" the="" inner="" shell,="" d1="" is="" also="" given.="" using="" the="" shell="" theorem,="" determine="" the="" required="" thickness="" d2="" of="" the="" outer="" shell="" so="" that="" the="" magnitude="" of="" the="" gravitational="" field="" for="" people="" living="" on="" the="" outside="" of="" the="" outer="" shell="" is="" the="" same="" as="" that="" for="" people="" living="" on="" the="" outside="" of="" the="" inner="" shell.="" you="" can="" assume="" that="" the="" thicknesses="" d1,="" d2="" are="" negligible="" in="" comparison="" with="" the="" radii="" r1,="" r2="" (why="" would="" that="" be="" helpful?).="" g2:3="" consider="" an="" infinite="" mass="" distribution="" with="" spherically="" symmetric="" mass="" density="" %(r′)="%(r′)" given="" by="" %(r′)="α" r′3="" ,="" for="" r′="" ≥="" r0,="" and="" %(r′)="0," for="" r′="">< r0. thus, there is no mass inside a sphere of radius r0, and outside of this sphere, the mass density decreases with the third power of r′. using the shell theorem, calculate the gravitational field a(r) at every point in space. 2 r0.="" thus,="" there="" is="" no="" mass="" inside="" a="" sphere="" of="" radius="" r0,="" and="" outside="" of="" this="" sphere,="" the="" mass="" density="" decreases="" with="" the="" third="" power="" of="" r′.="" using="" the="" shell="" theorem,="" calculate="" the="" gravitational="" field="" a(r)="" at="" every="" point="" in="" space.="">