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## 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 m

^{2}/kg

^{2}.

(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 10

^{4}kg/m

^{3}, B = 1.16 x 10

^{4}kg/m

^{3}, and Earth’s radius R = 6.370 x 10

^{6}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)r

^{2}dr.

(b) Determine the gravitational field inside Earth.

(c) Using the result of part b, determine the gravitational field magnitude at Earth’s surface.

## Homework Equations

## 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 +U

_{s}+ U

_{e})

_{i}= (K+ U

_{s}+U

_{e})

_{f}

0+0+0=0+(1/2)kx

^{2}

_{max}−QEx

_{max}so x

_{max}= (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.