a particle of mass m is contrained to lie on along a frictionless,horizontal plane subject to a force given by the expression F(x)=-kx. It is projected from x=0 to the right along the positive x direction with initial kinetic energy T(o)=1/2kA^2, k and A are positive constants. Find
a) The potential energy function V(x) for this force.
b)The kinetic energy
c) The total energy of the particle as a function of its position.
d) Find the turning points of the motion
integral of F(x) from x to x0=T-T(o)
integreal of dx/+- sqrt(2/m(E-V(x))) from X to Xo=t-t0
The Attempt at a Solution
ok part A,B,C i think i got, but i would want someone to double check for me just in case i made mistakes....As for D, my book doesn't really explain the turning points to well so im not exactly sure about how to find it...
i just integrate both sides here from x to x0 to get
im guessing x0 is = 0 so
B) integral of F(x) dx= T-T(o)
integral from x to x(o) of -kx dx=-1/2kx^2=T-T(o)
now im a little bit confused here,is this supposed to mean that x=a making T=0? how am i supposed to know that?
C) Assuming i did the other 2 parts properly for the total energy i get:
D) This part I have no idea how to do....i dont understand exactly what this turning point is really, i see in the book it shows a graph of V(x) vs x and theres a region in the middle called the allowed region,the two points of the curve where they are both exactly equal to E are said to be the turning points,but how am i supposde to find them?
if i use velocity=+- sqrt(2/m(E-V(x))) E cant be smaller than V(x) and what about if they are both euqal? the velocity would be 0...so is it like the point in which E is slightly bigger than V(x) how do i figure this out??