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oddjobmj
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Homework Statement
A particle with mass=m moves in the xy plane. It is under the influence of a repulsive central force described by:
F(r)={A[itex]\hat{r}[/itex] if r<R0 and 0 if r>R0}
[itex]\hat{r}[/itex] is the unit radial vector and R0 is the range of the force
The initial conditions are x=-3 R0, y=0.5 R0, Vx=w, vy=0
(A) Calculate energy and angular momentum in terms of the parameters A, R0, and w.
(B) Calculate the approximate distance of closest approach to the origin in terms of A, R0, and w, accurate to the order A assuming A is small.
Homework Equations
Energy=U+K=[itex]\frac{1}{2}[/itex]mv2+Force*Distance
Angular momentum=mvrsin[itex]\theta[/itex]
The Attempt at a Solution
I am having a hard time visualizing this problem. My current take is represented in the linked image:
http://imgur.com/8btZdls (I forgot the negative on -3R0)
Does that look correct?
(A)
You could represent the energy of the particle by [itex]\frac{1}{2}[/itex]mv2+A(R0-r) if the particle is within the range of the force (r<R0) but it starts outside that range meaning it only has kinetic energy. Am I to answer with regard to its current energy or generally?
Currently energy = [itex]\frac{1}{2}[/itex]mw2
Angular Momentum=mwrsin(theta) but I am asked to represent this value in terms of the parameters A, R0, and w. I can write in terms of R0 by providing a variable multiple of R0 as r where R0*[itex]\alpha[/itex]=r thus:
mw[itex]\alpha[/itex]R0sin[itex]\theta[/itex]
Does that seem like the answer I am looking for?
(B)
Is this asking, given the initial velocity w and its initial position, how close would the particle come to the origin? i.e. it travels in the positive x direction until reaching the range of affect of the force A and then takes a curved path away from the origin. Find the nearest distance along that path to the origin?
Thank you for your help!