A particle of mass is moving around the origin with a constant force pulling it towards the origin. If Bohr model is used to describe its motion, the radius of the orbit and the particle’s speed in the orbit depend on as [2025]

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Q.
A particle of mass $m$ is moving around the origin with a constant force $F$ pulling it towards the origin. If Bohr model is used to describe its motion, the radius $r$ of the $n^{th}$ orbit and the particle’s speed $v$ in the orbit depend on $n$ as [2025]

1 $r \propto n^{2 / 3}; v \propto n^{1 / 3}$

2 $r \propto n^{4 / 3}; v \propto n^{- 1 / 3}$

3 $r \propto n^{1 / 3}; v \propto n^{1 / 3}$

4 $r \propto n^{1 / 3}; v \propto n^{2 / 3}$

Ans.
(1)

As force (F) is constant, so

$F = \frac{m v^{2}}{r}$

We can write, $F = \frac{m^{2} v^{2} r^{2}}{m r^{3}} = \frac{L^{2}}{m r^{3}} (As, L = m v r)$

As force is constant, so $L^{2} \propto m r^{3}$

Using, $L = \frac{n h}{2 π},$ we get $\frac{n^{2} h^{2}}{4 π^{2}} = m r^{3} \Rightarrow n^{2} \propto r^{3} or r \propto n^{2 / 3}$

Also, $m v r = \frac{n h}{2 π}$

As $r \propto n^{2 / 3},$ so $m v (n^{2 / 3}) \propto \frac{n h}{2 π}$

or $v \propto n^{1 / 3}$

Solution

Q.
A particle of mass $m$ is moving around the origin with a constant force $F$ pulling it towards the origin. If Bohr model is used to describe its motion, the radius $r$ of the $n^{th}$ orbit and the particle’s speed $v$ in the orbit depend on $n$ as [2025]

1 $r \propto n^{2 / 3}; v \propto n^{1 / 3}$

2 $r \propto n^{4 / 3}; v \propto n^{- 1 / 3}$

3 $r \propto n^{1 / 3}; v \propto n^{1 / 3}$

4 $r \propto n^{1 / 3}; v \propto n^{2 / 3}$

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Solution

Q. A particle of mass m is moving around the origin with a constant force F pulling it towards the origin. If Bohr model is used to describe its motion, the radius r of the nth orbit and the particle’s speed v in the orbit depend on n as [2025]

1 r∝n2/3; v∝n1/3

2 r∝n4/3; v∝n-1/3

3 r∝n1/3; v∝n1/3

4 r∝n1/3; v∝n2/3

Ans. (1) As force (F) is constant, so F=mv2r We can write, F=m2v2r2mr3=L2mr3 (As, L=mvr) As force is constant, so L2∝mr3 Using, L=nh2π, we get n2h24π2=mr3⇒n2∝r3 or r∝n2/3 Also, mvr=nh2π As r∝n2/3, so mv(n2/3)∝nh2π or v∝n1/3

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Q.
A particle of mass $m$ is moving around the origin with a constant force $F$ pulling it towards the origin. If Bohr model is used to describe its motion, the radius $r$ of the $n^{th}$ orbit and the particle’s speed $v$ in the orbit depend on $n$ as [2025]

1 $r \propto n^{2 / 3}; v \propto n^{1 / 3}$

2 $r \propto n^{4 / 3}; v \propto n^{- 1 / 3}$

3 $r \propto n^{1 / 3}; v \propto n^{1 / 3}$

4 $r \propto n^{1 / 3}; v \propto n^{2 / 3}$