Problem $10.(10$ points$)$ We need to select a spring for the ball feeder shown below. Each ball is $0\mathrm{.}5$ inches in diameter, made of steel, and weighs $0\mathrm{.}02$ pounds. The plunger is made up of the cap, shaft and handle. The plunger weighs $0\mathrm{.}4$ lbs $.$ The spring pushes the ball up into the recess in the slider. The slider then moves to the right and drops a ball into the slot. The slider then returns to its original position and picks up the next ball. This repeats until the ball tube is empty and the cap is even with the top of the tube. When fully loaded the ball tube holds $8$ balls. When each ball is positioned in the recess of the slider, the ball muss press upwards against the slider with no more than $1\mathrm{.}0$ lb and no less than $0\mathrm{.}6$ lb of force as shown.

a$)$ When the ball tube is fully loaded, what is the maximum allowable force in the spring?

b$)$ When the last ball is in the slider, what is the minimum allowable spring force?

c$)$ What is the maximum $k$ value that would work for the system

d$)$ If you select a spring with a k value of $0\mathrm{.}1l\frac{b}{i}n,$ what would the maximum initial length, $L_0,$ be$?$

e$)$ If you select a spring with af$($ value of $0\mathrm{.}1l\frac{b}{i}n,$ what would the minimum initial length, $L_0,$ be

a$.F_{\text{m}\text{a}\text{x}}=8(\mathrm{.}02)+\mathrm{.}4+1=1\mathrm{.}5616$

b$.F_{\text{m}\text{i}\text{n}}=1(\mathrm{.}02)+\mathrm{.}4+\mathrm{.}6=1\mathrm{.}0216$

$q,$

$q,$