Question

Two particles of equal masses are revolving in circular paths of radii r1 and r2 respectively with the same period. The ratio of their centripetal force is
A. \[\dfrac{{{r}_{1}}}{{{r}_{2}}} \\ \]
B. \[\sqrt{\dfrac{{{r}_{2}}}{{{r}_{1}}}} \\ \]
C. \[{{\left( \dfrac{{{r}_{1}}}{{{r}_{2}}} \right)}^{2}} \\ \]
D. \[{{\left( \dfrac{{{r}_{2}}}{{{r}_{1}}} \right)}^{2}}\]

Answer 1

Hint: Here the time period of both the particles is same, so that means they will have the same angular velocity as well. (As time per is inversely proportional to angular velocity). By formula it is clear that force is directly dependent on mass, radius and square of angular velocity. Time period is inversely dependent on angular velocity.

Formula used:
The expression of centripetal force is,
\[F=mr{{w}^{2}}\]
The expression of time period is,
\[T=\dfrac{2\pi }{w}\]
Here, $m$ is the mass, $w$ is the angular velocity and $r$ is the radius of curvature.

Complete step by step solution:
As both the bodies have equal mass and equal angular velocity {as they have same time period} so which means centripetal force directly proportional to radius of their circular path. Here \[\left( F \right)\] represents centripetal force. The force which is responsible for circular motion of the body, if centripetal force becomes zero the body will not move in a circular path.

\[\left( m \right)\] represent mass of body, \[\left( w \right)\] represent angular velocity (note that angular velocity and velocity are two different things). Angular velocity is an axial vector that is calculated by right hand thumb rule.

\[\left( r \right)\] represent the radius of a circular path and \[\left( T \right)\] represent the time period of the body (time period means the time required by the body to complete one revolution or to complete one single circle).
\[F\infty r\]
Therefore we can write,
\[{{F}_{1}}\infty {{r}_{1}}\] and \[{{F}_{2}}\infty {{r}_{2}}\]
\[\left( {{F}_{1}} \right)\] represent centripetal force on first particle and \[\left( {{F}_{2}} \right)\] represent centripetal force on second particle. \[\left( {{r}_{1}} \right)\] represent radius of circular path of first particle and \[\left( {{r}_{2}} \right)\] represent radius of circular path of second particle.
So, \[\dfrac{{{F}_{1}}}{{{F}_{2}}}=\dfrac{{{r}_{1}}}{{{r}_{2}}}\]

Therefore option A is correct.

Note: Here we need to connect the relation between centripetal force and time period which can only be linked by angular velocity here. As mass is already equal and relation between force and time can be easily drawn by the formula of time period