Question:
Despite the very long description - question 3 by itself is trivial. In some sense, it is just a hint for the upcoming questions. To a bare minimum, the question listed a bunch of quantity and asked which one has to unit of energy. So all we have to do is to match dimensions.
Solution:
Energy unit
= Force times Distance
= Mass times Acceleration times Distance
= M(LT-2)(L)
= ML2T-2
kr = Energy
Therefore k has a unit of Force MLT-2
Planck's constant has unit of Joule Second = Energy Second = ML2T-1
Finally $ \mu $ has unit of mass.
So $ \frac{\hbar^2 k^2}{\mu} $ has unit $ \frac{(ml^2t^{-1})^2(mlt^{-2})^2}{m} = m^3l^6t^{-6} $
Therefore we see $ \left(\frac{\hbar^2 k^2}{\mu}\right)^{1/3} $ has the unit of energy.
Despite the very long description - question 3 by itself is trivial. In some sense, it is just a hint for the upcoming questions. To a bare minimum, the question listed a bunch of quantity and asked which one has to unit of energy. So all we have to do is to match dimensions.
Solution:
Energy unit
= Force times Distance
= Mass times Acceleration times Distance
= M(LT-2)(L)
= ML2T-2
kr = Energy
Therefore k has a unit of Force MLT-2
Planck's constant has unit of Joule Second = Energy Second = ML2T-1
Finally $ \mu $ has unit of mass.
So $ \frac{\hbar^2 k^2}{\mu} $ has unit $ \frac{(ml^2t^{-1})^2(mlt^{-2})^2}{m} = m^3l^6t^{-6} $
Therefore we see $ \left(\frac{\hbar^2 k^2}{\mu}\right)^{1/3} $ has the unit of energy.
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