A particle of mass m moves under the influence of a central force.The potential energy function is
V(r) = {
K (a2âr2) /2 if r ⤠a
0 if r > a
as illustrated in the figure. The initial position is (x,y) = (a/2,0) and the initial velocity is (vx,vy) = (0, v0).
[Data: m = 1.0 kg; K =17.1 N/m; a = 1.0 m; r0 = 0.5 a; v0 = 2.0m/s.]
(A) Determine the time ta when the mass crosses the boundary r =a.
(B) Determine angular position Ïa where the particle crosses theboundary r = a.
***HINT GIVEN***
Use the constants of the motion to analyze the orbit.
Write L = mr^2 dÏ/dt = the initial value.
What is the initial value?
Write E = (1/2)m (dr/dt)^2 + L^2/(2mr2) +V(r )
= the initial value. What is the
initial value?
Part A. In the energy equation,
change the variable from r to u,
where r =sqrt(u). The result can be
solved by u(t) = c1+c2 cosh(αt).
Determine c1, c2, and α.
Part B.
Ï = Integral 0 to t [L/(mr^2)] dt
