Question

The gravitational force between two objects is proportional to \[\dfrac{1}{R}\] (and not as \[\dfrac{1}{{{R^2}}}\]) where\[R\] is separation between them, then a particle in a circular orbit under such a force would have its orbital speed \[v\] proportional to
A. \[\dfrac{1}{{{R^2}}}\]
B. \[{R^0}\]
C. \[{R^1}\]
D. \[\dfrac{1}{R}\]

Answer 1

Hint:Gravitational force is a force exerted by the earth on all objects on it. The SI unit of force is \[Newton\] and denoted by \[N\] and If an object is moving in a circle and experiences an outward force then this force is called centrifugal force.

Formula Used:
\[F = \dfrac{{GMm}}{{{R^2}}}\]
Where, \[F\] is force of gravity, \[G\] is gravitational constant,\[M\] is mass of one object,\[m\] is the mass of another object and\[R\] is the distance between two objects.
\[{F_c}\]= \[\dfrac{{m{v^2}}}{R}\]
Where, \[{F_c}\] is centrifugal force, \[v\] is velocity and \[R\] is the distance.

Complete step by step answer:
As we know that, force is inversely proportional to the distance.
 \[F\alpha \dfrac{1}{R}\]
Here, \[F\] is the force of gravity and\[R\] is distance.
\[F\alpha \dfrac{K}{R}\] Here, \[K\] is any constant.
We also know that centrifugal force directly depends on the mass velocity and inversely depends on the distance, \[{F_c}\]= \[\dfrac{{m{v^2}}}{R}\]
Now, compare these two equations-
\[\dfrac{{m{v^2}}}{R} = \dfrac{K}{R}\]
\[\Rightarrow v = \sqrt {\dfrac{K}{m}} \]
\[\therefore v\alpha \]\[{R^0}\]

So, option B is correct.

Additional information:
Gravitation or just gravity is the force of attraction between any two bodies. All the objects in the universe attract each other with a certain amount of force, but in most of the cases, the force is too weak to be observed due to the very large distance of separation. Besides, gravity’s range is infinite but the effect becomes weaker as objects move away.This force of attraction was first observed by Sir Isaac Newton and was presented as Newton’s law of gravitation in the year 1680.

Note:Newton discovered the law stating the centrifugal force. A bike making a turn, Equatorial railway, clothes are dried automatically in washing machines are some applications of centrifugal force. Centrifugal force is equal in magnitude and opposite in direction to centripetal force. It is observed from a non-inertial frame of reference.