# Circular motion from a central force

• steven09
In summary, the conversation discusses a particle moving under a central force and the given initial conditions. The homework questions include determining the potential energy function, finding the velocity for the particle to move on a circle, and calculating the time it takes for the particle to reach the origin. The student is unsure about how to solve for the velocity and time, but mentions considering the equation for a circle and using the equation ma=− α r^−3.
steven09
1. Homework Statement

A particle moves under the influence of a central force, F_r(r) = − α r^−3. At time t = 0, the radius = r0 and the velocity = v0 as shown in the figure.
(A) Determine the potential energy function U(r).
(B) Determine the velocity vc such that the particle will move on the circle.
(C) Now suppose v0 = vc /2. The particle will spiral into the origin. Calculate the time it will take the particle to reach r = 0.

2. Homework Equations
U(r) = - integral F(r) dr

3. The Attempt at a Solution
I got U(r).

For my Part B) I don't know how to get velocity and Part C) how to get time. I think I might have to set ma=− α r^−3 but that doesn't seem right

. For Part C I'm not sure if I have to use the equation for a circle and solve for velocity and then use that to get the time. Any help is appreciated.

## 1. What is circular motion from a central force?

Circular motion from a central force refers to the movement of an object around a fixed point, known as the center, due to the influence of a force that acts towards the center of the circle.

## 2. What is the relationship between circular motion and centripetal force?

In circular motion, the centripetal force is responsible for keeping the object moving in a circular path by continuously pulling it towards the center. This force is always directed towards the center of the circle and is perpendicular to the object's velocity.

## 3. How does the speed of an object in circular motion relate to the centripetal force?

The speed of an object in circular motion is directly proportional to the centripetal force. This means that as the force increases, the speed of the object also increases, and vice versa.

## 4. What is the difference between circular motion from a central force and uniform circular motion?

The main difference between these two types of motion is the presence of a central force. In circular motion from a central force, the object is constantly being pulled towards the center, while in uniform circular motion, the object moves at a constant speed around the circle without any external force acting on it.

## 5. How is circular motion from a central force related to Newton's laws of motion?

Circular motion from a central force can be explained by Newton's laws of motion, specifically the first and second laws. The first law states that an object will continue moving in a straight line at a constant speed unless acted upon by an external force. The second law explains how an external force, in this case the centripetal force, can change the direction of motion and cause an object to move in a circular path.

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