Answer to Question #156194 in Differential Equations for Samychou

Question

A particle A moves in a resisting medium in a straight line such that its distance x from a fixed point O satisfies the equation d^2x/dt^2 + p(dx/dt) + qx = 0, where p and q are constants. Find the condition(s) on p and q such that the motion of A is
(i) simple harmonic.
(ii) damped harmonic.
In the case where the motion is damped harmonic, find
(iii) the damping factor.
(iv) the period of the motion.

Answer 1

(i) $p = { b \over m} = 0$

Where b is a constant that depends on the medium and the shape of the body

Note: "p" is the damping coefficient

$q= {k \over m}=\omega^2$

where is the elastic constant

(ii)$p = { b \over m}$ is not equal to zero

$q= {k \over m}=\omega^2$

(iii) damping ration or damping factot,$\zeta$ = ${p \over 2 \sqrt{mk}}$

Note:$2 \sqrt{mk} =$ critical damping

(iv) period,T, =${2\pi \over \omega}$