(c) Find the complementary function (CF) and the particular integral (PI) of the differential equation (11). (d) Hence or otherwise, show that the general solution can be written as r(t) = L cosh wt + M sinh wt + sin wt (12) (e) Identify the initial conditions from the description in part (b) and use them to find L and M. Hence show that r(t) = a cosh wt + -) sinh ut - (13) sin wt 2w
(c) Find the complementary function (CF) and the particular integral (PI) of the differential equation (11). (d) Hence or otherwise, show that the general solution can be written as r(t) = L cosh wt + M sinh wt + sin wt (12) (e) Identify the initial conditions from the description in part (b) and use them to find L and M. Hence show that r(t) = a cosh wt + -) sinh ut - (13) sin wt 2w
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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Hello I have done A and B. Please help me solving C, D and E. Please also take 3 questions from me as well. But DON'T SOLVE a and b.
![3. Rotating Tube
(a) In a Cartesian plane, if the position vector of a moving particle is given
by ř = rỉ + yj, then the r and y components of velocity is given by #
and # respectively. On the other hand, the polar components, i.e., the r
and ö components of velocity is given by and r. Using these, prove
that the r component of acceleration is
dr
de
f.
dt?
(10)
(b) A smooth straight thin tube revolves with uniform angular velocity w in
a vertical plane about one end which is fixed. A particle inside the tube is
sliding along the tube with a constant velocity v. At time t = 0, the tube
was horizontal and the particle was at a distance a from the fixed end.
Show that the motion of the particle can be described by the differential
equation
dr
- rw? = -g sin wt
(11)
dt?
[Hint: Use formula (10)|
(c) Find the complementary function (CF) and the particular integral (PI)
of the differential equation (11).
(d) Hence or otherwise, show that the general solution can be written as
r(t) = L cosh wt + M sinh wt +
sin wt
(12)
%3D
(e) Identify the initial conditions from the description in part (b) and use
them to find L and M. Hence show that
r(t) = a cosh wt + (-
sinh wt + sin wt
2w
(13)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44430f18-ea34-4e6b-a9f4-280941c4b5be%2F6641839e-5180-4b2d-b17f-3acef7f33efa%2Fzoon4c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Rotating Tube
(a) In a Cartesian plane, if the position vector of a moving particle is given
by ř = rỉ + yj, then the r and y components of velocity is given by #
and # respectively. On the other hand, the polar components, i.e., the r
and ö components of velocity is given by and r. Using these, prove
that the r component of acceleration is
dr
de
f.
dt?
(10)
(b) A smooth straight thin tube revolves with uniform angular velocity w in
a vertical plane about one end which is fixed. A particle inside the tube is
sliding along the tube with a constant velocity v. At time t = 0, the tube
was horizontal and the particle was at a distance a from the fixed end.
Show that the motion of the particle can be described by the differential
equation
dr
- rw? = -g sin wt
(11)
dt?
[Hint: Use formula (10)|
(c) Find the complementary function (CF) and the particular integral (PI)
of the differential equation (11).
(d) Hence or otherwise, show that the general solution can be written as
r(t) = L cosh wt + M sinh wt +
sin wt
(12)
%3D
(e) Identify the initial conditions from the description in part (b) and use
them to find L and M. Hence show that
r(t) = a cosh wt + (-
sinh wt + sin wt
2w
(13)
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