# Finding Earth's Mass - Gauss Law

## Homework Statement

Problem 1. A block having mass m and charge +Q is connected to an
insulating spring having force constant k. The block lies on a frictionless, insulating horizontal track, and the system is immersed in a uniform electric field of magnitude E directed as shown in Figure
P25:7. The block is released from rest at a moment when the spring is unstretched (that is, when x = 0).
(a) By what maximum amount does the spring expand?
(b) What is the equilibrium position of the block?

Problem 2. Consider a closed surface S in a region of gravitational field g. Gauss’s law for gravitation tells us that the gravitational flux through surface S is linearly proportional to the total mass min occupying the volume contained by S. More specifically, Gauss’s law states that
(closed integral)g x da = -4Gmin :
Note that g here is the total electric field, due to mass sources both inside and outside S. The value of G, the gravitational constant, is about 6.673 x10-11 N m2/kg2.
(a) Earth’s volume mass density, at any distance r from its center, is given approximately by the function p = A-Br=R, where A = 1.42 x 104 kg/m3, B = 1.16 x 104 kg/m3, and Earth’s radius R = 6.370 x 106 m. Calculate the numerical value of Earth’s mass M. Hint: The volume of a
spherical shell, lying between radii r and r + dr, is dv = 4(pie)r2dr.
(b) Determine the gravitational field inside Earth.
(c) Using the result of part b, determine the gravitational field magnitude at Earth’s surface.

2. Homework Equations

3. The Attempt at a Solution
1.Arbitrarily choose V = 0 at 0. Then at other points
V= −Ex and Ue =QV=−QEx.
Between the endpoints of the motion,
(K +Us+ Ue)i = (K+ Us +Ue)f
0+0+0=0+(1/2)kx2max −QExmax so xmax = (2QE)/k

At equilibrium,
ΣFx= −Fs+Fe= 0 or kx =QE .
So the equilibrium position is at x = QE/k

Problem 2. I have no clue at where to begin, or what equations to use. Any help is appreciated.

Problem 2. The density function does not make sense. There are two equal signs and the units do not match. Check the function again.

Im sorry, I wrote it in correctly the density function should be roe=A-Br/R.

Problem 2. Using Gauss's Law for gravitation gives

$$\int\vec{g}\bullet d\vec{S}=-4\pi GM$$

where M is the total mass enclosed within the surface S. Then,

$$\int\vec{g}\bullet d\vec{S}=-4\pi G\int dm\mbox{ where }dm=\rho dv$$

Use the hint for dv.