PERIODIC MOTION

I. Mass$-$spring oscillation

A spring with spring constant $k$ hangs vertically. When a mass $m$ is attached, the spring stretches a distance $y.$ The mass is initially at rest.

a$.$ Sketch a free$-$body diagram of the mass, and label all of the forces.

The mass is pulled down a short distance $y$ and released. The position of the mass at five times during one cycle of the motion is shown below.

$\backslash$table$[[,,{\displaystyle \underset{m}{\overset{?}{}}},,],[$in: lowest position just after release,t$1$: middle position, moving upivard,$t_2$ : highest position,$t_3$ : middle position, moving downivard,$4$: lowest position$]]$

b$.$ Sketch a free$-$body diagram of the mass at each of the five times. Be sure to draw all forces to scale relative to one another.

$\backslash$table$[[,,,,],[,,,,],[,,,,],[t_0,,,,],[,,,,],[t_1,,,,]]$

c$.$ Determine the direction of the net force $($if any$)$ on the block at each of the five times. Sketch an arrow to indicate the direction of the net force in the boxes below.

$\backslash$table$[[$Net force at $t_0,$Net force at $t_1,$Net force at $t_2,$Net force at $t_3,$Net force at $t_4$