Two small insulating balls with charge +q and mass m are placed in a circular bo

Question

Two small insulating balls with charge +q and mass m are placed in a circular bowl with radius R as shown above and are free to move without friction. You may assume that the radius of the balls is negligible compared to the radius of the bowl a) If we begin with the balls at the top of the bowl (i.e. 0-90°), what is the total gravitational potential energy of both balls? You may assume that the potential energy is zero if the ball is at the bottom of the bowl. (2 points) b) Write an equation for the total gravitational potential energy of the two balls, Ug, in terms c) If the balls are initially at the top of the bowl, what is the electrostatic potential energy of d) Write an equation for the total electrostatic potential energy, UE, as a function of R, 6, and e) Combine the equations for Ue and Ug to get the total potential energy of the system, Uot ofR, , and m, and g. (2 points) the two charges? (2 points) q. (2 points) Find an equation for the equilibrium value of 0 that minimizes the potential energy in terms of the parameters that were given in the problem (R, q, m, and fundamental constants). Your answer will look similar to the solution for problem 5.68. (6 points) f) Solve for the value of q that makes the equilibrium value of 0-45 (3 points) g) If the value of is changed so that it is slightly less than 45°, will the balls move towards each other or away from each other? What about if the value of 0 is slightly greater than 45°? What does this imply about the motion of the charges about the equilibrium value of 0? (3 points)

Explanation / Answer

a)

Total gravitational PE = gravitational PE of ball 1 + gravitational PE of ball 2

b)

PEgTotal = PEg1 + PEg2 = mgRsin + mgRsin = 2mgRsin

c)

Total electric PE = (k*q1*q2)/d2=(k*q*q)/(2Rsin)2 =(kq2)/(2Rsin)2

d)

UETotal = (kq2)/(2Rsin)2

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