Question content area top part 1 for a mass spring...

Question content area top Part 1 For a mass-spring oscillator, Newton's second law implies that the position y(t) of the mass is governed by the second-order differential equation m y double prime left parenthesis t right parenthesis plus b y prime left parenthesis t right parenthesis plus ky left parenthesis t right parenthesis equals 0.

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00:01 In this problem we are going to solve the problem of undamped oscillator with a periodic force using the laplace transform.

00:11 So the force is f of t equal f0, sine omega t.

00:23 And with that we write the differential equation using the newton low as m times x double dot plus k times x equal to the force let's first write like this and then we have f0 sine mega t and we are given the initial conditions x0 equal to zero and x dot 0 equal to zero so we will solve this differential equation using the laplace transform okay let us indicate the laplace transform of our function x by capital x then using the derivative rule we have the laplace transform of x dot equal to s times x capital x minus x 0 and for a second derivative we have s squared capital x minus x0 minus x dot zero and we also know that the laplace transform of is or sine omega t is omega divided by s squared plus omega squared.

01:48 Now let us put everything into the differential equation.

01:53 But first, i want to rewrite this equation as follows.

02:01 X double dot plus omega zero squared x equal to small f0, times sine omega t so here i have defined omega zero to be k over m under the square root and f0 equal to capital f or m now let us everything let us put everything into this equation and simplify it we obtain s squared plus omega zero squared times capital x minus s times x0 plus x stud zero this is equal to f0 omega over s squared plus omega squared so here we have a huge simplification these terms are equal to zero so we obtain capital x equal to f0 omega divided by s squared plus omega 0 squared times s squared plus omega squared now we need to take the inverse laplace transform of this guy and to do that i'm going to first perform a partial fraction decomposition so i will assume capital x to be equal to a s plus b over s square plus omega 0 squared plus c s plus d over plus omega squared...

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