A particle of mass m moves around the origin in a potential 1 2 m… - Vidyadip

MCQ

A particle of mass $m$ moves around the origin in a potential $\frac{1}{2} m \omega^{2} r^{2}$, where $r$ is the distance from the origin. Applying the Bohr's model in this case, the radius of the particle in its $n$th orbit in terms of $a=\sqrt{h /(2 \pi m \omega)}$ is

✓
$a \sqrt{n}$
B
$a n$
C
$a n^{2}$
D
$a n \sqrt{n}$

✓

Answer

Correct option: A.

$a \sqrt{n}$

a
$(a)$ Energy of particle is

$\frac{1}{2} m \omega^{2} r^{2}=\frac{1}{2} m v^{2}$

where, $v=$ velocity of particle around the path.

$\Rightarrow \quad v=r \omega$

Now, angular momentum of particle will be

$L=m v r=m r^{2} \omega$

By Bohr's model, we have

$L=\frac{n h}{2 \pi}$

$\Rightarrow \quad m r^{2} \omega=\frac{n h}{2 \pi} \Rightarrow r^{2}=\frac{n h}{2 \pi m \omega}$

or $r=\sqrt{\frac{h}{2 \pi m \omega}} \times \sqrt{n}$

$\Rightarrow \quad r=a \sqrt{n}$

$\left[\because\right.$ given, $\left.\sqrt{\frac{h}{2 \pi m \omega}}=a\right]$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 7. gravitation→7. gravitation — all question types→Physics STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

A block $A$ kept on an inclined surface just begins to slide if the inclination is $30^\circ$ . The block is replaced by another block $B$ and it is found that it just begins to slide if the inclination is $40^\circ :$

A box of mass $m\, kg$ is placed on the rear side of an open truck accelerating at $4\, m/s^2$. The coefficient of friction between the box and the surface below it is $0.4$. The net acceleration of the box with respect to the truck is zero. The value of $m$ is :- $[g = 10\,m/s^2]$

A tyre filled with air $({27^o}C,$ and $2$ atm) bursts, then what is temperature of air ....... $^oC$ $(\gamma = 1.5)$

An air column in a pipe, which is closed at one end, will be in resonance wtih a vibrating tuning fork of frequency $264\, Hz$ if the length of the column in $cm$ is (velocity of sound $= 330\, m/s$)

The force $F$ on a sphere of radius $'a'$ moving in a medium with velocity $'v'$ is given by $F = 6\pi \eta av$. The dimensions of $\eta $ are

For a gas $\gamma = 7/5.$ The gas may probably be

Two discs having masses in the ratio $1: 2$ and radii in the ratio $1: 8$ roll down without slipping one by one from an inclined plane of height $h$. The ratio of their linear velocities on reaching the ground is ........

The maximum velocity of a simple harmonic motion represented by $y = 3\sin \,\left( {100\,t + \frac{\pi }{6}} \right)$is given by

$T_1$ and $T_2$ in the given figure are

A solid sphere of mass $500\ gm$ and radius $10\ cm$ rolls without slipping with the velocity $20\ cm/s$. The total kinetic energy of the sphere will be ........ $J$