The stepped disk has a mass $m=20kg.$ The spring constant is $180\frac{N}{m},$ the damping constant $c=200N*\frac{s}{m}$ and the disk radius is one metre. The moment of inertia of the disk as a function of its mass and radius is $3m{R}^2.$ The disk is released from rest with the spring unstretched. $($a$)$ Determine the equation of motion of the disk in the vertical direction. $($b$)$ Determine the position of the centre of the disk as a function of time.

$**($c$)$ Using Matlab, numerical integrate the equation of motion to find the position $x(t)$ and compare it against the analytical equation derived in $($b$).$

Answer. $($a$)x^{}+\frac{9c}{4m}{x}^{}+\frac{k}{m}x=0$;

$($b$)x(t)=0\mathrm{.}2776e^{-0\mathrm{.}407t}-0\mathrm{.}0051e^{-22\mathrm{.}0926t}$

The stepped disk has a mass $m=20kg.$ The spring constant is $180\frac{N}{m},$ the damping constant $c=200N*\frac{s}{m}$ and the disk radius is one metre. The moment of inertia of the disk as a function of its mass and radius is $3m{R}^2.$ The disk is released from rest with the spring unstretched. $($a$)$ Determine the equation of motion of the disk in the vertical direction. $($b$)$ Determine the position of the centre of the disk as a function of time.

$**($c$)$ Using Matlab, numerical integrate the equation of motion to find the position $x(t)$ and compare it against the analytical equation derived in $($b$).$

Answer. $($a$)x^{}+\frac{9c}{4m}{x}^{}+\frac{k}{m}x=0$;

$($b$)x(t)=0\mathrm{.}2776e^{-0\mathrm{.}407t}-0\mathrm{.}0051e^{-22\mathrm{.}0926t}$