- #1

Benzoate

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## Homework Statement

Show that the gravitational force exerted on a particle inside a hollow symmetric sphere is zero. [Hint. The proof is the same as for a particle outside a symmetric sphere, except in one detail.]

## Homework Equations

F=m1*m2*G/R^2

## The Attempt at a Solution

If a particle is going to interact with a sphere, My polar coordinates should be 3-dimensional.

variables will be theta, phi and r.

r hat will be in the direction of the r vector

theta ; rotation in the r-hat, k-hat plane , perpendicular to r-hat.

theta is the angle between the r vector and k vector.

phi-hat rotates about k-hat, in the i hat , j-hat plane

phi is the angle between i-hat and projection of the r vector in the i hat , j hat , plane r*sin(theta)*dphi.

dV= dr*(r*theta)*(r*sin(theta)*d-phi= lim(dr=>0) r^2*dr*(sin(theta)*dtheta)*dphi

sin(theta)*dtheta I think is the solid angle.

Since the horizontal components are i-hat and j-hat, by symmetry, the only component left is the k-hat component.

F=integral(from 0 to a) dr*integral(from 0 to pi) dtheta* integral(0 to pi)dphi*(-G*m(particle)*r^2*sin(theta)/R^2 * (b-r*cos(theta)/R)

R=sqrt((b-r*cos(Theta)^2+(r*sin(theta))^2)

Since symmetry is only around k axis, there isn't any dependence on phi

F=-2*pi*G*m(particle)k-hat((integral(from 0 to a)dr*integral(frome 0 to pi) dtheta integral(no upper or lower limits) r^2*sin(theta)/(R^3) *(b-cos(theta))

F=-2*pi*G*m(particle)k-hat((integral(from 0 to a)r^2*dr*integral(frome 0 to pi)*(b-r*cos(theta)*sin(theta)/(R^3)

R^2=(b-r*cos(theta))^2+(r*sin(theta)^2=b^2-2*b*r+r^2

2*R*dR=2*b*r*sin(theta)*dtheta

(b-r*cos(theta))=2b^2-2*b*r*cos(theta)=R^2+(b^2-r^2)/(2*b)

F=-(2*m*G*m(p)*rho/2b^2) integral(from 0 to a) r^2*dr

F=-(G*4*pi*a^3) *rho*m(particle)/(b^2) k-hat

=-(G*M*m(particle)/b^2 k-hat

Not correct.

I think my R is wrong.

e particle is inside the sphere not outside; Does it make a difference whether or not the sphere is hollow or solid?

Sorry for writing the integrals out in text form. I didn't know how to write out the integrals , whhere you would write out the integral and it would look like an integral you see in a math book.