Electrical Force vs. Centripetal Force

[ilovemcat]

Assume that the electron in a deuterium atom can be viewed as orbitting the nucleus (one proton, one neutron) in uniform circular motion . If k is Coulomb's constant, e is the charge magnitude of the electron, and r is the radius of the orbit, then which one of the following expressions gives the kinetic energy of the electron?

The answer is: ke^2 / 2r

I know this is an easy question but I'd hate to miss a point because of a simple mistake. I'm a little confused with the bottom part. The way I approached this problem was by realizing that (if the electron is held in place), at distance "r" from the proton the electron spinning has an electric Potential Energy of kqq/r. All that PE is converted to KE when the object is in motion. Therefore, PE = KE. This yields: ke^2/r

The way they solved it was different: They explained that the centripetal force is provided by the electrical attraction:

Centripetal Force = Electrical Force
mv^2/r = ke^2/r^2 === multiplying both sides by "r" ==> mv^2 = ke^2/r

Then multiplying both sides by 1/2 they got:

1/2mv^2 = ke^2/2r

This is a totally different expression from what I found. So my question is ...which is correct and why?

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[Rabolisk]

All that PE is converted to KE when the object is in motion. Therefore, PE = KE. This yields: ke^2/r

Error.
Are you saying that there is no PE during centripetal motion?

 

[ilovemcat]

Error.
Are you saying that there is no PE during centripetal motion?

Oops.

 

[Rabolisk]

Basically, their method is correct, because the electrical force is the centripetal force. In other words, the centripetal acceleration and the motion is caused solely by the electrostatic interaction between the proton and the electron.

Your reasoning is wrong, because PE is not converted into KE at any point. Usually that sort of reasoning applies to problems in which an electron changes positions due to the electrical force between it and a proton. Then you can use the change in potential energy to calculate the change in kinetic energy. But in this problem, both potential and kinetic energies are constant throughout the motion.

 

[ilovemcat]

Basically, their method is correct, because the electrical force is the centripetal force. In other words, the centripetal acceleration and the motion is caused solely by the electrostatic interaction between the proton and the electron.

Your reasoning is wrong, because PE is not converted into KE at any point. Usually that sort of reasoning applies to problems in which an electron changes positions due to the electrical force between it and a proton. Then you can use the change in potential energy to calculate the change in kinetic energy. But in this problem, both potential and kinetic energies are constant throughout the motion.

Yeah, you're right. What I should of realized was that as the electron is rotating around the proton, its potential wasn't changing (it was always at a distance "r" from the proton) and because it's potential wasn't changing, the Electrical Force remained constant. This force is the centripetal force and here, velocity is constant.

I never encountered a situation where I divided the centripetal force by 1/2 to get kinetic energy. But it makes sense, since v = constant.

So then, if you took for instance the motion of a pendulum (where the velocity isn't constant). Here we can't divide the nonuniform circular motion by 1/2 to get kinetic energy because both the acceleration and velocity changes throughout its path. I know this seems fairly obvious lol, but I just want to make sure I have my understanding right