$($For the following, when entering a mathematical expression, do not substitute numerical values; use variables only.$)($a$)$ Calculate the speed of the block as it passes through the equilibrium position x$=0$ if the surface is frictionless.

SOLUTION

Conceptualize This situation has been discussed before, and it is easy to visualize the block being pushed to the right by the spring and moving with some speed at $x=0.$

Categorize We identify the system as the block and model the block as a nonisolated $$ system.

Analyze In this situation, the block starts with $v_i=0$ at $x_i=-4\mathrm{.}0cm,$ and we want to find $v_f$ at $x_f=0.$

Use the following equation to find the work done by the spring with $x_{\text{m}\text{a}\text{x}}=x_i$ :

$w_s=\frac{1}{2}k{x}_{\text{m}\text{a}\text{x}}^2$

Work is done on the block, and its speed changes. The conservation of energy equation reduces to the work$-$kinetic energy theorem. Use that theorem to find the speed at $x=0($Use the following as necessary: $k,x_{\text{m}\text{a}\text{x}\text{'}}$ and $m)$:

$W_s=\frac{1}{2}m{v}_f^2-\frac{1}{2}m{v}_i^2$

$v_f=\sqrt[\phantom{2}]{v_i^2+\frac{2}{m}{w}_s}=\sqrt[\phantom{2}]{v_i^2+\frac{2}{m}(\frac{1}{2}k{x}_{\text{m}\text{a}\text{x}}^2)}$

Substitute numerical values to calculate the speed $($in $\frac{m}{s})$:

$v_f=,\frac{m}{s}$