Question

A particle is moving with a uniform speed in a circular orbit of radius \[R\] in a central force inversely proportional to the \[{n^{th}}\] power of\[R\]. If the period of rotation of the particle is \[T\], then
A. \[t{{ }}\alpha {{ }}{{{R}}^{\dfrac{{^{(n + 1)}}}{2}}}\]

B. \[T{{ }}\alpha {{ }}{{{R}}^{\dfrac{n}{2}}}\] for any \[n\]

C. \[T{{ }}\alpha {{ }}{{{R}}^{\dfrac{3}{2}}}\]

D. \[T{{ }}\alpha {{ }}{{{R}}^{\dfrac{n}{2} + 1}}\]

Answer 1

Hint: Understand the concept of the circular motion, When a particle moves in circular motion, it experiences the central force acting along the radius of the orbit. Angular frequency is the frequency with which a particle moves in the circular orbit. Time taken to complete one revolution is considered as, time period.

Complete step by step answer:Understand that, the central force \[{F_c}\] is inversely proportional to the \[{n^{th}}\] power of\[R\]. Therefore it can be written as,
\[\
\Rightarrow {F_{c{{ }}}}\alpha \dfrac{1}{{{R^n}}} \\
\Rightarrow {F_c} = \dfrac{K}{{{R^n}}} \\
\ \]
Here, \[{F_c}\] is the central force, \[R\] is the radius of the orbit and \[K\] is the proportionality constant.
Substitute \[m{\omega ^2}r\] for \[{F_c}\]
\[\Rightarrow m{\omega ^2}R = \dfrac{K}{{{R^n}}}\]
Here, \[\omega \] is the angular frequency and \[m\] is the mass of the particle.

Rearrange for \[\omega \]
\[\Rightarrow\omega = \sqrt {\dfrac{K}{m}} \times \dfrac{1}{{{R^{\dfrac{{n + 1}}{2}}}}}\]

Substitute \[\dfrac{{2\pi }}{T}\] for \[\omega \]
\[\Rightarrow \dfrac{{2\pi }}{T} = \sqrt {\dfrac{K}{m}} \times \dfrac{1}{{{R^{\dfrac{{n + 1}}{2}}}}}\]
Rearrange for \[T\]
\[\Rightarrow T = 2\pi \sqrt {\dfrac{m}{K}} {R^{\dfrac{{n + 1}}{2}}}\]
Above equation can be written as,
\[\Rightarrow T{{ }}\alpha {{ }}{R^{\dfrac{{n + 1}}{2}}}\]

Therefore, the option A is the correct choice.

Additional Information:
The expression of angular frequency of the system in terms of time-period of oscillation \[\left( T \right)\] is written as,
\[\Rightarrow \omega = \dfrac{{2\pi }}{T}\]
Here, \[\omega \] is the angular frequency
The expression of frequency \[f\] of oscillation (in Hertz) is written as,
\[\Rightarrow f = \dfrac{1}{T}\]
The expression for the central force \[{F_c}\] is written as,
\[\Rightarrow{F_c} = m{\omega ^2}r\]
Here, \[\omega \] is the angular frequency and \[m\] is the mass of the particle.
The expression of the central force, angular frequency, is used to calculate the relationship between the time period and radius of the circular orbit.

Note:From Newton’s third law we know that forces always occur in pairs. The balancing force for centripetal force is centrifugal force which is a false force. Also, if at any instant of time a body is in a circular motion moving with constant speed, its velocity is not constant because velocity is a vector quantity, the direction of the motion is continuously changing.