$1\mathrm{.}4$ Theory Questions

It is important to recognize that the spring force depends on the displacement from equilibrium. A more general expression for Hooke's law is

$F_s=-kx$

where $x=x-x_0$ is the amount by which the spring is stretched. We typically define coordinates $($the measurement system$)$ such that the equilibrium position is $x_0=0$ to obtain the form $F_s=-kx.$ The diagram below shows three cases of an identical spring suspending different masses. The masses are stationary with displacements indicated on the $x-$axis. The position $x=0$ corresponds to the end of the unstretched spring.

a$.(1$ pt$)$ A free$-$body diagram is a tool to help you write an equation for $F_{\text{n}\text{e}\text{t}}.$ Draw a free$-$body diagram showing the forces acting on an arbitrary mass, $m.$

b$.(1$ pt$)$ Write down an equation for net force acting on a mass. With this, solve for the acceleration, $a,$ in terms of the spring constant $k,$ the displacement $x,$ the gravitational acceleration $g,$ and the mass.

c$.(2$ pt$)$ What is the acceleration when the mass is at rest in the equilibrium position? Use this fact and your result from the previous question to obtain an equation that gives the displacement in terms of $g,k,$ and $m.$

d$.(1$ pt$)$ Use your equation to predict the displacement you would measure in a lab if you hung a $1$ kg mass from a spring with $k=7\frac{N}{m}.$ Use $g=9\mathrm{.}8\frac{m}{s^2}$ as the gravitational acceleration.

e$.(1$ pt $)$ Sketch a graph of the displacement vs$.$ mass. Indicate the three points $(($:$m,x_1$