An infinitely long, thin, massive cylinder exerts a gravitational force of 14OUm newtons on any particle of mass m kilograms a distance r meters from the cylinder's axis, directed towards the r In 3 cylinder. A particle is initially 729 meters from the cylinder's axis and starts moving away at velocity 140 meters/second. Let r denote the distance from the axis to the particle (in meters) and let t denote time (in seconds). Find a second-order differential equation and initial conditions that describe the particle's position. The solution is der dt2 , r(0)= ,r'(0)= %3D Use the phase plane method to find a formula relating r to the particle's velocity v=or. Solve for v. dt The solution is v= What is the furthest from the cylinder the particle gets? The solution is meters.

An infinitely long, thin, massive cylinder exerts a gravitational force of 14OUm newtons on any particle of mass m kilograms a distance r meters from the cylinder's axis, directed towards the r In 3 cylinder. A particle is initially 729 meters from the cylinder's axis and starts moving away at velocity 140 meters/second. Let r denote the distance from the axis to the particle (in meters) and let t denote time (in seconds). Find a second-order differential equation and initial conditions that describe the particle's position. The solution is der dt2 , r(0)= ,r'(0)= %3D Use the phase plane method to find a formula relating r to the particle's velocity v=or. Solve for v. dt The solution is v= What is the furthest from the cylinder the particle gets? The solution is meters.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

An infinitely long, thin, massive cylinder exerts a gravitational force of 0om newtons on any particle of mass m kilograms a distance r meters from the cylinder's axis, directed towards the
r In 3
cylinder. A particle is initially 729 meters from the cylinder's axis and starts moving away at velocity 140 meters/second.
Let r denote the distance from the axis to the particle (in meters) and let t denote time (in seconds). Find a second-order differential equation and initial conditions that describe the particle's
position.
d²r_
The solution is
,r(0)3=
r'(0)=
dt?
Use the phase plane method to find a formula relating r to the particle's velocity v=-
dr Solve for v.
dt
The solution is v=
What is the furthest from the cylinder the particle gets?
The solution is
meters.

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