Problem $1.(20$ Points$)$

A foot pedal mechanism for a machine is crudely modeled as a pendulum connected

to a spring as illustrated in Figure. The purpose of the spring is to keep the pedal

roughly horizontal.

Derive the differential equation of this system assuming that the angular deflections

are small, such that the spring deflection can be approximated by the arc length,

that the pedal may be treated as a point mass and that pendulum rod has negligible

mass. The values in the figure are $m=0\mathrm{.}5kg,k=2100\frac{N}{m},g=9\mathrm{.}8\frac{m}{s^2},$

$l_1=0\mathrm{.}2m$ A foot pedal mechanism for a machine is crudely modeled as a pendulum connected

to a spring as illustrated in Figure. The purpose of the spring is to keep the pedal

roughly horizontal.

Derive the differential equation of this system assuming that the angular deflections

are small, such that the spring deflection can be approximated by the arc length,

that the pedal may be treated as a point mass and that pendulum rod has negligible

mass. The values in the figure are $m=0\mathrm{.}5kg,k=2100\frac{N}{m},g=9\mathrm{.}8\frac{m}{s^2},$

$l_1=0\mathrm{.}2m$ and $l_2=0\mathrm{.}3m.$ and $l_2=0\mathrm{.}3m.$