# Finding Earth's Mass - Gauss Law

#### bananasplit

1. 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.

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#### chrisk

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

#### bananasplit

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

#### chrisk

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.

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