Question

Two bodies of equal masses revolve in circular orbits of radii \[{R_1}\] and \[{R_2}\] with the sample period. Their centripetal forces are in the ratio.
A. \[{\left( {\dfrac{{{R_2}}}{{{R_1}}}} \right)^2}\]
B. \[\dfrac{{{R_1}}}{{{R_2}}}\]
C. \[{\left( {\dfrac{{{R_1}}}{{{R_2}}}} \right)^2}\]
D. \[\sqrt {{R_1}{R_2}} \]

Answer 1

Hint:As the given body is revolving so we use angular terms here. The force by which the body is revolving in a circular path is determined as the centripetal force. In the centripetal force formula, we use the relation between linear and angular velocity to get the ratio between the centripetal force.

Formula used:
The centripetal force is given as,
\[{F_c} = \dfrac{{m{v^2}}}{r}\]
Where m is the mass of the body, v is the velocity of the body and r is the radius of the circular path.
The relation between linear velocity and angular velocity is given as,
\[v = \omega R\]
Where \[\omega \] is the angular velocity and R is the radius.
The centripetal force is also given as
\[{F_c} = m{\omega ^2}R\]

Complete step by step solution:
Given two bodies have equal masses.
Let masses be \[{m_1} = {m_2} = m\]
As we know that centripetal force is,
\[{F_c} = \dfrac{{m{v^2}}}{r}\]
Using \[v = \omega R\]
\[{F_c} = \dfrac{m}{r} \times {\omega ^2}{R^2}\]
\[\Rightarrow {F_c} = m{\omega ^2}R\]

By this equation, we see that the centripetal force (\[{F_c}\] ) is directly proportional to the radius of the orbit(R).
So \[{F_c} \propto R\]
Thus, we can write
\[\dfrac{{{F_{c1}}}}{{{F_{c2}}}} = \dfrac{{{R_1}}}{{{R_2}}}\]
Therefore, if two bodies of equal masses revolve in circular orbits of radii \[{R_1}\] and \[{R_2}\] then their centripetal forces are in the ratio \[\dfrac{{{R_1}}}{{{R_2}}}\].

Hence option B is the correct answer.

Note: A force is necessary to move an object, and the force works differently on various objects based on the sort of motion they display. Centripetal force is defined as the force exerted on a curvilinear object that is directed towards the axis of rotation or the centre of curvature. Newton is the unit of centripetal force.