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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.)

Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Topics In Analytic Geometry
Section6.2: Introduction To Conics: parabolas
Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.
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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.)
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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