Problem #$1$: $(35$ points$)$ A slider of mass $m=0\mathrm{.}6kg$ is attached to a spring of stiffness $k=120$

$\frac{N}{m}$ and slides down a smooth parabolic guide, as shown in Figure $1.$ The guide lies in a vertical

plane $($in which gravity acts downward$)$ and the slider is released from rest. The unstretched length

of the spring is $0\mathrm{.}15$ m $.$ The shape of the parabolic guide is given by the equation $y=2x^2$ with

both $x$ and $y$ expressed in meters $($in an $x-y$ coordinate system $-$ not shown on the figure $-$ in which

the $x-$coordinate measures positive horizontal distance to the right of the origin at point $B$ and the

$y-$coordinate measures positive vertical distance above the origin at point $B.)$

a$)$ What is the potential energy in the spring when the slider is at point $A,$ and what is the

potential energy in the spring when the slider is at point $B?$

b$)$ Determine the speed $v_B$ of the slider when it reaches point $B.$

c$)$ Find the normal force $N$ exerted by the guide on the slider when it reaches point $B.$

d$)$ Find the initial speed $v_A($at point $A)$ that the slider would need in order to acquire a speed at

$B$ equal to $1\mathrm{.}5$ times that found in part b $).$

e$)$ If the guide were frictional, the slider was released from rest, and the speed of the slider at $B$

were half that found in part b$),$ find the work done by friction.