1. The problem statement, all variables and given/known data 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.] 2. Relevant equations F=m1*m2*G/R^2 3. 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.