H1. Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving with a velocity v = (B, az, -ay), where a and ß are positive constants. (a) What are the physical dimensions of a and B? (b) Find the general form of r(t), the position of the particle, as a function of time t, assuming the initial position of the particle is ro = (0,2,0). (hint: write v = (B, az, -ay) as a system of first order ODEs and note that the equation for x is decoupled from the others). Describe in words the motion of the particle and sketch its trajectory in R3 (you can use software packages for the plot). (c) Show that the speed of the particle is constant, but the acceleration vector a(t) is nonzero. Justify. (d) Assuming the particle has a constant mass m, use Newton's second law to show that the force acting on the particle is (as a function of the position r = r = (x, y, z)) F(x, y, z) = ma² (0, -y, —z).
H1. Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving with a velocity v = (B, az, -ay), where a and ß are positive constants. (a) What are the physical dimensions of a and B? (b) Find the general form of r(t), the position of the particle, as a function of time t, assuming the initial position of the particle is ro = (0,2,0). (hint: write v = (B, az, -ay) as a system of first order ODEs and note that the equation for x is decoupled from the others). Describe in words the motion of the particle and sketch its trajectory in R3 (you can use software packages for the plot). (c) Show that the speed of the particle is constant, but the acceleration vector a(t) is nonzero. Justify. (d) Assuming the particle has a constant mass m, use Newton's second law to show that the force acting on the particle is (as a function of the position r = r = (x, y, z)) F(x, y, z) = ma² (0, -y, —z).
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:H1. Consider a point particle with position vector r = (x, y, z) in Cartesian coordinates, moving
with a velocity v = (ß, az, -ay), where a and ß are positive constants.
(a) What are the physical dimensions of a and B?
(b) Find the general form of r(t), the position of the particle, as a function of time t, assuming
the initial position of the particle is ro = (0,2,0). (hint: write v = (B, az, -ay) as a system
of first order ODEs and note that the equation for x is decoupled from the others). Describe in
words the motion of the particle and sketch its trajectory in R3 (you can use software packages
for the plot).
(c) Show that the speed of the particle is constant, but the acceleration vector a(t) is nonzero.
Justify.
(d) Assuming the particle has a constant mass m, use Newton's second law to show that the
force acting on the particle is (as a function of the position r = = (x, y, z))
F(x, y, z) = ma² (0, -y, —z).
H2. Suppose a particle of constant mass m with position x > 0, moves in one space dimension
under the influence of the gravitational force of another point particle of constant mass M
sitting at x = 0, i.e. the attracting force is
F
(a) Using Newton's second law, show that
GmM
x²
-i.
1
d/ ( 2m²² _ GmM) =
-mx
dt
=
= 0.
(i.e., the total energy, sum of kinetic and potential energy, is conserved).
(b) Using the change of variables v dr, solve the equation of motion and determine the
velocity of the particle v(x) as a function of x assuming it starts with zero velocity at xo. Does
the particle's speed in x = 0 depend on the initial position?
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