<p> The most common dissipation process is friction, but you cannot obtain
ID: 2156635 • Letter: #
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<p> The most common dissipation process is friction, but you cannot obtain an analytic solution for the harmonic oscillator with frictional loss. However, consider the case for which the frictional loss/period is small <span class="formula"> <span class="math_text_before"> </span><span class="jax_wrap finalized_jax" data-jax="(Delta E/E << 1)">((Delta E/E << 1)) <sup href="#" class="formula_delete" data-mce-href="#">X</sup><sub href="#" class="formula_edit" data-mce-href="#">Edit</sub></span><span class="math_text_after"> </span></span>. Obtain an approximate solution for this case. Sketch your solution E(t) and compare the result with the usual damped oscillator solution. The oscillator has mass m, spring k, and constant friction force Fo. (You may find it helpful to write a differential equation for the energy E(t) of the oscillator.)</p>Explanation / Answer
since the oscillator is always in motion, so the frictional force on it is a constant but directed against the direction of motion.
to calculate the work done by the frictional force, we need (W=-Fs)
where s is the distance covered by the particle. There is no role of scalar product in this formula as the force is always directed against the velocity and thus (dW= ec{F}.d ec{s} = -Fds)
In a single period the distance covered is (s=4A)
where A is the amplitude of oscillations,
so total energy loss in one period is (Delta E=-4FA)
thus is a time dt assuming E is small, the energy loss is (dE=Delta E*dt/T = -4FA dt/T)
thus, ( rac{dE}{dt}=- rac{4FA}{T})
also, energy of a harmonic oscillator is (E= rac{1}{2}momega^2 A^2)
where is angular frequency of the oscillator.
differentiating last gives ( rac{dE}{dt}=momega^2 A rac{dA}{dt})
Thus, (momega^2 A rac{dA}{dt}=-4FA/T)
Also, (T = 2pi/omega)
so, ( rac{dA}{dt}=- rac{2 F}{pi momega})
which has the solution (A=A_0- rac{2F}{pi momega}t)
This is different from the damped harmonic oscillator case where amplitude decays exponentially.
But here amplitude decays linearly.
Also, (E(t)= rac{1}{2}momega^2 (A_0- rac{2F}{pi momega}t)^2)
which is parabolic.
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