Question

If the radius of curvature of the path of two particles of same masses are in the ratio 1 : 2, then in order to have constant centripetal force, their velocity, should be in the ratio of
A. \[1:4\]
B. \[4:1\]
C. \[\sqrt 2 :1\]
D. \[1:\sqrt 2 \]

Answer 1

Hint: The Centripetal Force Formula is given as the product of mass which is in kg and the square of tangential velocity which is in metres per second divided by the radius which is in metres. It implies that on doubling the radius then the tangential velocity will be quadrupled. So that we can find the relation between the velocity and radius.

Formula used:
The centripetal force is given as;
\[F = \dfrac{{m{v^2}}}{r}\]
Where F is the Centripetal force, m is the mass of the object, v is the speed or velocity of the object and r is the radius.

Complete step by step solution:
Given two particles of constant masses(m) and centripetal force(F).
Ratio of the radius of curvature of two particles is, \[{r_1}:{r_2} = 1:2\]
As we know that centripetal force, \[F = \dfrac{{m{v^2}}}{r}\]
we can write this formula as, \[r = \dfrac{{m{v^2}}}{F}\]

Here mass m and force F are constant terms, so from this we get the relation between v and r as,
\[r \propto {v^2}\,{\rm{ or v}}\, \propto \sqrt r \]
Thus, by using the values, we get
\[\dfrac{{{v_1}}}{{{v_2}}} = \dfrac{{\sqrt {{r_1}} }}{{\sqrt {{r_2}} }}\]
\[\Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{{{r_1}}}{{{r_2}}}} \\
\Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \sqrt {\dfrac{1}{2}} \\
\therefore \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{1}{{\sqrt 2 }}\]
Therefore, the ratio of the radius of curvature of the path of two particles is \[1:\sqrt 2 \].

Hence option C is the correct answer.

Note: A force is required to make an object move and also the force acts differently on objects depending on which type of motion it exhibits. Centripetal force is defined as the force which is acting on an object in curvilinear motion directed towards the axis of rotation or centre of curvature. The unit of centripetal force is Newton(N). The direction of centripetal force is perpendicular to the direction of the object displacement.