A particle of mass m moves in a plane with origin O, and its plane polar coordinates are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r). a. Show that the radial and transverse and transverse components after applying Newton's law reduce to: i. F-70²=-F(r), ii. r²0 = h, where h is a positive constant (and, in the usual notation, a dot over a variable denotes its time derivative). b. Make the substitution that u=1, and use (ii), iii. to show that iv. and deduce that h²(du + u) = u²²F(u¯¹). c. For the case F(r)= ur3, where is a positive constant with > h², show that dt² - a²u = 0, where a² = -1. d. Write down the general solution of the equation in (c) above.
A particle of mass m moves in a plane with origin O, and its plane polar coordinates are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r). a. Show that the radial and transverse and transverse components after applying Newton's law reduce to: i. F-70²=-F(r), ii. r²0 = h, where h is a positive constant (and, in the usual notation, a dot over a variable denotes its time derivative). b. Make the substitution that u=1, and use (ii), iii. to show that iv. and deduce that h²(du + u) = u²²F(u¯¹). c. For the case F(r)= ur3, where is a positive constant with > h², show that dt² - a²u = 0, where a² = -1. d. Write down the general solution of the equation in (c) above.
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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Transcribed Image Text:Question 3
A particle of mass m moves in a plane with origin O, and its plane polar coordinates
are (r(t), 0(t)). It is subject to a force directed at O of magnitude mF(r).
a. Show that the radial and transverse and transverse components after applying
Newton's law reduce to:
i. F-r0²-F(r),
ii. r²0 = h,
where h is a positive constant (and, in the usual notation, a dot over a variable
denotes its time derivative).
b. Make the substitution that u = 1, and use (ii),
iii. to show that
iv. and deduce that
du
+ = -hat'
h²(du + u) = u²²F(u¯¹).
c. For the case F(r) = ur-3, where is a positive constant with > h², show that
f
d'u
dt²
- a²u = 0,
where a² = -1.
d. Write down the general solution of the equation in (c) above.
e. Find the particular solution that satisfies r=a, r=0 when = 0.
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