 #  Consider a particle of mass m moving in a plane, under the infl...

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 Consider a particle of mass m moving in a plane, under the influence of a central potential V(r) = ma, for some constant A Assume the particle is initially at a distance To from the origim and has initial speed to = VA Assume that to = not is tangential and TO = for is radial. (a) Find the force excerted om the particle. Express it im polar coordinates and in unit vector (i, 0) motation (b) Find the total energy of the particle. (Write it im terms of m, & and ro.) (c) Write down the F = ma equations for the system in polar coordinates. (See the note on the first page.) (d) The magnitude of the angular momentum is L = mr². (L points in the direction perpendicular to the plame of motiom.) Using part (c), show that L is conserved. (e) Suppose that i is a constant (i = a <0, where a is some comstamt), and that the initial angular speed is - (Since Do and To are perpendicular, - is just Then use (c) to find the angular speed -(t) and angle 0(1) as functions of time. The answers will depend on p, - and To- Assume the initial angle is 62 =0. (Hint: if you don't know how to solve the resulting differential equation, look back to how we treated velocity-dependent force problemas in chapter 2.)

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