2. As we will study in the coming weeks, the one-dimensional motion of a mass attached to a spring without friction is governed by the linear second-order equation with constant coeffi- cients (1) d²x dt² +w²x = 0, where w2 is a positive parameter. (a) Find the general solution to equation (1). You should have found a periodic solution: can you say what the period is? (b) Find the particular solution assuming that the mass is initially still in position xo at t = 0, i.e., using the initial conditions (0) = ₁ and d (0) = 0. (c) With friction, the motion is described by the equation d²x +w²x = 0, (2) dt² where both f and w² are positive parameters. Find the general solution to equation (2) and sketch the behaviour of the solution in time (remember to discuss all the possible cases!). Is this a periodic function? If so, can you find the period? + f dx dt (If you're interested in the physical modelling details: equation (1) is the fundamental equa- tion of the simple harmonic motion; the parameter ² is a compound parameter defined as w² = k/m, where m is the mass of the particle and k a parameter characteristic of the spring. Equation (2) is the fundamental equation of the damped harmonic motion and f describes the friction intensity. Later in the semester we will discuss these applications in more detail.)
2. As we will study in the coming weeks, the one-dimensional motion of a mass attached to a spring without friction is governed by the linear second-order equation with constant coeffi- cients (1) d²x dt² +w²x = 0, where w2 is a positive parameter. (a) Find the general solution to equation (1). You should have found a periodic solution: can you say what the period is? (b) Find the particular solution assuming that the mass is initially still in position xo at t = 0, i.e., using the initial conditions (0) = ₁ and d (0) = 0. (c) With friction, the motion is described by the equation d²x +w²x = 0, (2) dt² where both f and w² are positive parameters. Find the general solution to equation (2) and sketch the behaviour of the solution in time (remember to discuss all the possible cases!). Is this a periodic function? If so, can you find the period? + f dx dt (If you're interested in the physical modelling details: equation (1) is the fundamental equa- tion of the simple harmonic motion; the parameter ² is a compound parameter defined as w² = k/m, where m is the mass of the particle and k a parameter characteristic of the spring. Equation (2) is the fundamental equation of the damped harmonic motion and f describes the friction intensity. Later in the semester we will discuss these applications in more detail.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Transcribed Image Text:2. As we will study in the coming weeks, the one-dimensional motion of a mass attached to a
spring without friction is governed by the linear second-order equation with constant coeffi-
cients
(1)
d²x
dt²
+w²x =
where w² is a positive parameter.
(a) Find the general solution to equation (1). You should have found a periodic solution:
can you say what the period is?
d²x
dt²
(b) Find the particular solution assuming that the mass is initially still in position xo at
t = 0, i.e., using the initial conditions x(0) = xo and d (0) = 0.
(c) With friction, the motion is described by the equation
OOOO
= 0,
dx
dt
+w²x = 0,
(2)
where both fand w² are positive parameters. Find the general solution to equation (2) and
sketch the behaviour of the solution in time (remember to discuss all the possible cases!). Is
this a periodic function? If so, can you find the period?
+ f
(If you're interested in the physical modelling details: equation (1) is the fundamental equa-
tion of the simple harmonic motion; the parameter ² is a compound parameter defined as
w² = k/m, where m is the mass of the particle and k a parameter characteristic of the spring.
Equation (2) is the fundamental equation of the damped harmonic motion and f describes
the friction intensity. Later in the semester we will discuss these applications in more detail.)
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VIEWStep 2: Solve a second order differential equation representing simple harmonic motion
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