Problem $1$

Consider the one$-$pulley oscillating system shown in the figure below. The pulley is rigid and has

no mass. The cable hanging the weight $m$ is not stretchable and has no mass.

Indicate the origin and positive direction of the coordinate you choose.

Assume that there is a positive displacement, velocity and acceleration. Draw the free$-$body

diagram.

Derive the equation of motion. Identify the equivalent inertia and stiffness coefficients.

Write down the expression for the natural frequency ${}_n.$

Compare to the spring$-$mass oscillator without the pulley and its natural frequency ${}_n=$

$\sqrt[\phantom{2}]{\frac{k}{m}}.$ Is the natural frequency of the pulley different? Why?