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

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.