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 isa)Proportional to r2b)Independent of rc)Proportional to rd)Proportional to 1/rCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Hiii....so let the force be some constant divided by r and this will equal to the centripetal force i. e mv^2/r this cancels r from both side and velcity becomes independant of r....

Answer

It could be taken similar to uniform circular motion in which central attractive force (centripetal force) is mv2/r i.e. inversely proportional to r, but the speed doesnot depend on r.

Answer

The central attractive force acting on the particle is inversely proportional to the distance r. In other words, as the distance between the particle and the center decreases, the force becomes stronger.

To understand why the speed of the particle is independent of r, let's consider the basic principles of circular motion and centripetal force.

1. Circular Motion:
When an object moves in a circular path, it experiences a centripetal acceleration towards the center of the circle. This acceleration is constantly changing the direction of the object's velocity, keeping it moving in a circular orbit.

2. Centripetal Force:
The centripetal force is the force required to keep an object moving in a circular path. It always acts towards the center of the circle and is responsible for the centripetal acceleration.

Now let's analyze the situation in more detail:

- As the particle moves in the circular orbit, the central attractive force acts as the centripetal force, keeping the particle in its orbit.

- According to Newton's second law of motion, the centripetal force is equal to the mass of the particle multiplied by its centripetal acceleration.

- Since the centripetal acceleration is always directed towards the center of the circle, it is perpendicular to the velocity of the particle.

- The centripetal force is given by the equation F = m * a_c = m * (v^2 / r), where m is the mass of the particle, v is its velocity, and r is the radius of the circular orbit.

- Rearranging the equation to solve for v, we get v = √(F * r / m).

Now, let's consider the effect of the central attractive force being inversely proportional to the distance r:

- If the force is inversely proportional to r, we can write F = k / r, where k is a constant.

- Substituting this expression into the equation for velocity, we get v = √((k / r) * r / m) = √(k / m).

From the above equation, we can clearly see that the velocity of the particle (v) is independent of the radius of the circular orbit (r). It only depends on the constant k and the mass of the particle (m).

Therefore, the correct answer is option 'B': The speed of the particle is independent of r.

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