A 10.6 cm radius thin ring carries a uniformlydistributed 16.2 C charge. A small

Question

A 10.6 cm radius thin ring carries a uniformlydistributed 16.2 C charge. A small 7.8 g sphere witha charge of 4.2 C is placed exactly at the center of thering and given a very small push so it moves along the ring axis(+x axis). How fast will the sphere be moving when it is2.80 m from the center of the ring (ignoregravity)? A 10.6 cm radius thin ring carries a uniformlydistributed 16.2 C charge. A small 7.8 g sphere witha charge of 4.2 C is placed exactly at the center of thering and given a very small push so it moves along the ring axis(+x axis). How fast will the sphere be moving when it is2.80 m from the center of the ring (ignoregravity)? How fast will the sphere be moving when it is2.80 m from the center of the ring (ignoregravity)?

Explanation / Answer

This is a conservation of energy problem. You have: .     initial PE = final PE + final KE . Calculate the initial and final potential energy, and then youhave the final KE and can find the final speed. . The potential energy of the charges, i.e. the ring with thesphere, is given by .     initial PE = k Q q /r   = 8.99 x 109 * 16.2 x10-6 * 4.2 x 10-6 / 0.106 = .                   =   5.77056 Joules . When the sphere reaches   2.80 m from thering, the distance from the sphere and each bit of charge can befound by using the pythagorean theorem: .            d = (2.802 + 0.1062)1/2 =    2.8020 . So the final PE is... .           final PE =    k Q q /r   = 8.99 x 109 * 16.2 x10-6 * 4.2 x 10-6 / 2.8020 = .                   =   0.2183 Joules . Now... .     initial PE = final PE + final KE .        5.77056 =   0.2183   +   final KE . KE = 5.552 Joules . We know that KE is   (1/2) mv2    so .    KE = (1/2) mv2    .    v = [ 2 K / m ]1/2 = [ 2 * 5.552 / 0.0078 ]1/2=    37.73 m/s   is the final speed of the sphere .                   =   0.2183 Joules . Now... .     initial PE = final PE + final KE .        5.77056 =   0.2183   +   final KE . KE = 5.552 Joules . We know that KE is   (1/2) mv2    so .    KE = (1/2) mv2    .    v = [ 2 K / m ]1/2 = [ 2 * 5.552 / 0.0078 ]1/2=    37.73 m/s   is the final speed of the sphere

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