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Atoms and Nuclei question

2024 · Shift 1 · Q39
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  5. /2024 · Shift 1 · Q39

Atoms and Nuclei question

2024 · Shift 1 · Q39

JEE AdvancedPhysicsAtoms and NucleiMultiple correct+4 / −2
A particle of mass mmm is moving in a circular orbit under the influence of the central force F(r)=−krF(r)=-k rF(r)=−kr, corresponding to the potential energy V(r)=kr2/2V(r)=k r^2 / 2V(r)=kr2/2, where kkk is a positive force constant and rrr is the radial distance from the origin. According to the Bohr's quantization rule, the angular momentum of the particle is given by L=nℏL=n \hbarL=nℏ, where ℏ=h/(2π),h\hbar=h /(2 \pi), hℏ=h/(2π),h is the Planck's constant, and nnn a positive integer. If vvv and EEE are the speed and total energy of the particle, respectively, then which of the following expression(s) is(are) correct?
  1. A
    r2=nℏ1mkr^2=n \hbar \sqrt{\frac{1}{m k}}r2=nℏmk1​​
  2. B
    v2=nℏkm3v^2=n \hbar \sqrt{\frac{k}{m^3}}v2=nℏm3k​​
  3. C
    Lmr2=km\frac{L}{m r^2}=\sqrt{\frac{k}{m}}mr2L​=mk​​
  4. D
    E=nℏ2kmE=\frac{n \hbar}{2} \sqrt{\frac{k}{m}}E=2nℏ​mk​​
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