# Damped harmonic oscillator

Damping force and total energy. See attached file for full problem description.

© BrainMass Inc. brainmass.com October 24, 2018, 8:41 pm ad1c9bdddfhttps://brainmass.com/physics/periodic-motion/damping-force-total-energy-104548

#### Solution Preview

See the attachments.

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begin{document}

title{Damped Oscillator}

date{}

author{}

maketitle

Newton's second law implies that:

begin{equation}label{diff}

begin{split}

&mfrac{d^{2}x}{dt^{2}}=-kx-2beta m frac{dx}{dt}Longrightarrow

&frac{d^{2}x}{dt^{2}} + 2beta frac{dx}{dt} + frac{k}{m}x =0

end{split}

end{equation}

If you insert a trial ...

#### Solution Summary

A detailed solution is given that discusses the damping force and total energy.

Problems with a Damped Harmonic Oscillator

(A) A damped oscillator is described by the equation:

m [(x)ddot] = ?b [(x)dot] ? kx

What is the condition for critical damping? Assume this condition is satisfied.

(B) For t < 0 the mass is at rest at x = 0. The mass is set in motion by a sharp impulsive force at t = 0, so that the velocity is v0 at time t = 0. Determine the position x(t) for t > 0.

(C) Determine the maximum displacement of the mass for t > 0.

(D) Suppose m = 1 kg and squareroot{k/m} = 2(pi)rad/s. Calculate the maximum displacement for t > 0, as a function of v0. Hand in an accurate graph of xmax versus v0.

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