e Given the initial conditions at t 0 determine the position of the particle on

E given the initial conditions at t 0 determine the

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(e) Given the initial conditions at t = 0, determine the position of the particle on the surface of the cylinder at any subsequent time. 0 , 0 , 0 , 0 0 0 0 0 = = = φ φ & & z z Certainly, R = ρ as the particle remains on the surface of the cylinder. Since angular momentum is conserved, ( ) ( ) t t const mR dt d 0 2 , , 0 φ φ φ φ & & & = = = . All that remains is to solve the differential equation in z. I have: kz mg z m = & & . The solution to the homogeneous equation is clearly of the form t m k B t m k A z kz z m sin cos , + = & & . A particular solution is of the form k mg z = . Solutions to the full equation are then k mg t m k B t m k A z + sin cos . Solving this for my boundary conditions, then, I have: ( ) ( ) ( ) t t R t k mg t m k k mg t z B m k B m k A k mg A k mg m k B m k A 0 cos 0 cos 0 sin 0 0 sin 0 cos 0 φ φ ρ & = = = = + = = + =
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