1. Image that two particles, a proton and an electron, are confined to move on circular rings of fixed radius, r. For each particle, the kinetic energy can be written as 1 - (vx ² + v₁₂²) = = 1w² 2 m T = -77 (V₂ de where the angular velocity, w = and the moment of inertia, I = mr², arise from converting this expression from Cartesian to plane polar coordinates. dt' (a) Let the fixed radius for each particle be r = 100 pm. Suppose that both the proton and the electron can be treated classically and that they happen to have the same angular momentum, L = Iw. What would be the ratio of their angular velocities, electron? @proton (b) Now, suppose both the proton and the electron have the same angular velocity, w, and the same angular momentum, L, but are on rings of different sizes. If the proton's ring has radius r = 200 pm, what is the radius of the electron's ring (in pm)? (c) For the case examined in part (a) (same ring with same angular momentum), what is the ratio of the electron's kinetic energy to that of the proton?
1. Image that two particles, a proton and an electron, are confined to move on circular rings of fixed radius, r. For each particle, the kinetic energy can be written as 1 - (vx ² + v₁₂²) = = 1w² 2 m T = -77 (V₂ de where the angular velocity, w = and the moment of inertia, I = mr², arise from converting this expression from Cartesian to plane polar coordinates. dt' (a) Let the fixed radius for each particle be r = 100 pm. Suppose that both the proton and the electron can be treated classically and that they happen to have the same angular momentum, L = Iw. What would be the ratio of their angular velocities, electron? @proton (b) Now, suppose both the proton and the electron have the same angular velocity, w, and the same angular momentum, L, but are on rings of different sizes. If the proton's ring has radius r = 200 pm, what is the radius of the electron's ring (in pm)? (c) For the case examined in part (a) (same ring with same angular momentum), what is the ratio of the electron's kinetic energy to that of the proton?
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Transcribed Image Text:1. Image that two particles, a proton and an electron, are confined to move on circular rings of
fixed radius, r. For each particle, the kinetic energy can be written as
m
1
T = ² (vx² + v₂²) = = 1 w²
2
where the angular velocity, w = and the moment of inertia, I = mr², arise from
converting this expression from Cartesian to plane polar coordinates.
de
dt'
(a) Let the fixed radius for each particle be r= 100 pm. Suppose that both the proton and
the electron can be treated classically and that they happen to have the same angular
momentum, L = Iw. What would be the ratio of their angular velocities, electron?
@proton
(b) Now, suppose both the proton and the electron have the same angular velocity, w, and
the same angular momentum, L, but are on rings of different sizes. If the proton's ring
has radius r = 200 pm, what is the radius of the electron's ring (in pm)?
(c) For the case examined in part (a) (same ring with same angular momentum), what is the
ratio of the electron's kinetic energy to that of the proton?
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