Question

This question assesses the gravitational attraction due to a cylinder of mass. Consider a cylinder with radius R equal to the mean radius of the Earth, length L and a mass M equal to the mass of the Earth. The specific task is to determine the escape velocity from the end of the cylinder. First, though, consider a different but related problem: a particle with mass mp placed a distance r from the center of a ring of mass M, along the line through the center of the ring and perpendicular to its plane. (Ignore the ring's thickness.) Ci (a) (2 points) Consider a small mass element of the ring, dm. Find the gravitational potential energy dUG associated with that mass dm and the particle mp (b) (3 points) Integrate your answer to (a) to find the gravitational potential energy Uring of the ring-particle system as a function of r. (Take U ing-0 very far away from the ring.) Now consider a hollow cylinder of radius R, mass M, and length L (c) (4 points) Think about the cylinder as a stack of thin rings, each of mass dm. Integrate your answer to (b) to find the gravitational potential energy Uhe of the hollow-cylinder-and-particle system, as a function of the distance from the end of the cylinder to point w in the diagram. (Again, take Uaisk - 0 very far away from the end of the cylinder.) Last, imagine a solid cylinder of tadius R, mass M, and length L. x-h A(x) (d) (3 points) Integrate your answer to (c) to find the gravitational potential energy Ucyt of the cylinder-particle system, as a function of distance away from the point marked a - h in the diagram. (Again, take Ueyl-0 very far away from the end of the cylinder.) (e) (3 points) Use F.- and your answer to (d) to find the force on the particle (f) (3 points) Show that your result for (d) reduces to the correct expression for x → (g) (4 points) Use your answer to (d) to find the escape velocity from the flat surface actual Earth? (If the answer changes depending on the value of L, discuss how.) mp as a function of distance away from the end of the cylinder r. 00. of the cylinder. Is it greater than, less than, or equal to the escape velocity from

Answer 1

(dm) m - 2-1て

In tuts cnse M m A Mm 2FRL

We ean csidert the volid clind e as shown belous using Eq⑩ cwe have in tuis ease 2.

ラ fR" [ LE(: リ두妇2.lar TCR2 L deo TR' L

(5 -70 we have 卜.sj veifea) tem enertoy Con s have