Driven harmonic oscillator problem

[Taylor Grubbs]

Homework Statement

A mass m sits on a horizontal frictionless surface and is attached to a wall by means of a spring having force constant k. The mass is now subjected to an additional force of the form.
F(t) = Acosbt
(a) Write the equation of motion for this mass.(b) What is the solution to this equation which satisfies the conditions x(t==0) = 0; and v(t=0) = 0?(c) Assume now that the driving frequency is twice the natural frequency of the oscillator. What is
the period of the motion?(d) Write an expression for the power delivered to the mass by the external force.(e) Integrate your expression for the power over one cycle of the motion. Is this the result youwould expect? Explain.

Homework Equations

So mx''= -kx + Acos(bt)
ω₀

The Attempt at a Solution

Solving for the initial conditions I found that

x = [A/m(ω²-ω₀²)](cos(bt) - cos(ωt))

Is this the correct form to find the period of motion?

 

[OldEngr63]

You have not yet identified the natural frequency and incorporated that into the solution for both the homogeneous and particular solutions.

 

[rude man]

Solving for the initial conditions I found that

x = [A/m(ω²-ω₀²)](cos(bt) - cos(ωt))

Is this the correct form to find the period of motion?

On the right track, but you have more varialbles than there should be: b, ω and ω₀. Fix this, find the resonant frequency as oldengr63 suggests, then decide on what b should be for part (c).

 

[BvU]

Hi Taylor,
Could you please show how you find your expression ? It's easy to point you to the right answer, but as a learning experience that scales lower than if you get some small help while working it out yourself.