Question

A particle moves in a circular orbit under the action of a central attractive force inversely proportional to the distance 'r'. The speed of the particle is
(A) proportional to $\mathrm{r}^{3}$
(B) independent of $\mathrm{r}$
(C) proportional to $\mathrm{r}^{2}$
(D) proportional to $1 / \mathrm{r}$

Answer 1

We know that in an inertial frame, there is no outward acceleration since the system is not rotating. Since Earth rotates around a fixed axis, the direction of centrifugal force is always outward away from the axis. Thus, it is opposite to the direction of gravity at the equator; at Earth's poles it is zero. The concept of centrifugal force can be applied in rotating devices, such as centrifuges, centrifugal pumps, centrifugal governors, and centrifugal clutches, and in centrifugal railways, planetary orbits and banked curves, when they are analysed in a rotating coordinate system.

Complete step by step answer
We know that centripetal force is the name given to any force which causes a change in direction of velocity toward the center of the circular motion. The component of the force which is perpendicular to the velocity is the part resulting in the centripetal force.
As the centripetal force acts upon an object moving in a circle at constant speed, the force always acts inward as the velocity of the object is directed tangent to the circle. Thus, the work done by the centripetal force in the case of uniform circular motion is 0 Joules.
We know that Centripetal force, $\mathrm{F}=\dfrac{\mathrm{mv}^{2}}{\mathrm{r}} \mathrm{so}$
$\Rightarrow \text{k}/\text{r}=\dfrac{\text{m}{{\text{v}}^{2}}}{\text{r}}$
$\mathrm{Thus}, \mathrm{mv}^{2}=\mathrm{k}$
So, v is independent of r.

So, the correct answer is option B.

Note: We know that centripetal force is always considered a real force. A centripetal force can be measured even in an inertial frame. Now according to most formal definitions, neither centrifugal nor Coriolis forces are real. Centripetal force is defined as, the force that is necessary to keep an object moving in a curved path and that is directed inward toward the center of rotation, while centrifugal force is defined as the apparent force that is felt by an object moving in a curved path that acts outwardly away from the center of attraction. Spinning a ball on a string or twirling a lasso: Here the centripetal force is provided by the force of tension on the rope pulling the object in toward the centre. Turning a car: Here the centripetal force is provided by the frictional force between the ground and the wheels.