Question

Write down the DE of this damped harmonic oscillator.

Answer 1

Answer:

$\frac{d^{2}x }{d^{2}t }+\frac{k}{m}x+\frac{b}{m} \frac{dx}{dt}=0$

Explanation:

The spring mass equation for the damped oscillation will be,

$F=-kx-bv$

Here, -bv is the damping term used in this b is damping constant, k is spring constant, x is elongation in the spring, F is the force.

$ma=-kx-bv\\m\frac{d^{2}x }{d^{2}t }=-kx-b\frac{dx}{dt}\\  m\frac{d^{2}x }{d^{2}t }+kx+b\frac{dx}{dt}=0\\\frac{d^{2}x }{d^{2}t }+\frac{k}{m}x+\frac{b}{m} \frac{dx}{dt}=0$

Therefore the differential equation for the damped harmonic oscillator is,

$\frac{d^{2}x }{d^{2}t }+\frac{k}{m}x+\frac{b}{m} \frac{dx}{dt}=0$