What Is the Maximum Speed of a Mass in a Driven Damped Oscillator?

[DoctorBozon]

Homework Statement

A small block (mass 0.25 kg) attached to a spring (force constant 16 N/m) moves in one dimension on a horizontal surface. The oscillator is subject to both viscous damping and a sinusoidal drive. By varying the period of the driving force (while keeping the drive amplitude fixed), it is found that the mass oscillates with its largest steady state amplitude (0.30 m) when the driving force has a period of 1.00 second.

Find the maximum speed of the mass in this largest-amplitude steady state. Express your answer in meters per second

Homework Equations

X(steadystate) = Dcos(ωt-δ)
D = A/√((ω₀²-ω²)² + 4ω²β²)
ω₀² = k/m
ω_r² = √(ω₀²-2β²)

The Attempt at a Solution

I know that the resonance frequency ω_r = 2∏/1.00s
From this I can find my damping parameter
I know that D=0.30m

Since dX/dt = -Dωsin(ωt-δ) , where ω=ω₀ for kinetic energy resonance.
Is my maximum speed of the mass in the largest-amplitude steady state just D*ω₀

 

[anathema]

?

Yes, your maximum speed in this case would be D*ω0, where ω0 is the natural frequency of the oscillator. This is because at resonance, the amplitude of the oscillations is at its maximum and therefore the velocity is also at its maximum. So, in this case, the maximum speed would be 0.30m*ω0.