To solve the problem of two charged balls connected by a thread, we need to analyze the forces acting on them and how their motion evolves over time. The setup involves two balls of mass m, each carrying a charge q, connected by a non-conducting thread of length 2l. When the middle of the thread starts moving with a constant velocity v, we want to find the minimum distance between the two balls as they move.
Understanding the Forces and Motion
Initially, the two balls are at rest, and the only forces acting on them are the electrostatic repulsion due to their charges and the tension in the thread. When the middle of the thread moves, it causes the balls to move outward due to the tension and the repulsive force between them.
Analyzing the Geometry of Motion
As the middle of the thread moves horizontally, the two balls will also move outward. Let's denote the distance between the two balls as d. The length of the thread remains constant at 2l, so we can express the relationship between the distances:
- The horizontal distance from the center of the thread to each ball is l.
- The vertical distance from the center of the thread to the line connecting the two balls is h.
Using the Pythagorean theorem, we can express the distance d between the two balls as:
d = 2 * √(l² + h²)
Determining the Motion of the Balls
As the middle of the thread moves with velocity v, the horizontal component of the motion of each ball can be described as:
h = vt
Substituting this into our equation for d gives:
d = 2 * √(l² + (vt)²)
Finding the Minimum Distance
To find the minimum distance between the two balls, we need to minimize the expression for d. Notice that as time t increases, the term (vt)² will dominate, and thus d will increase. Therefore, the minimum distance occurs at t = 0, when the balls are still at their initial positions.
At t = 0, the distance d is simply:
d_min = 2l
Conclusion
In summary, the minimum distance between the two balls occurs when the middle of the thread starts moving, and it is equal to the original length of the thread connecting the two balls, which is 2l. As time progresses, the distance will increase due to the motion of the balls moving outward. This analysis highlights the interplay between electrostatic forces and motion in a dynamic system.
