Understanding Centripetal Force: Solving for Tension

[i.mehrzad]

I am thoroughly confused about centripetal force.

I was doing this problem myself and when looked at the solution i was fairly confused.

The problem was like this.

There is a thread which is hinged to a pole and the pole is rotating with an angular velocity of 'w'. The rope has got finite mass m. What is the tension of the rope as a function of 'r' which is the distance fro the pole aloong the rope.

In the solution i had this problem.

They considered a diffrential length dr which is situated at a distance r from the pole.

Towards the right of dr the tension is pointing to to the right, and to the left of dr the tension is pointing to the left.

I wanted to know that why is there a difference in the direction of tension. When the cause of both is the same, that is centripetal accelaration.

 

[Doc Al]

Think of the tension as being a condition of the thread, not just "a force". The bonds of the thread are being stretched. That tension will create forces on each end of a segment of the thread--often we loosely refer to the force created by tension as just "tension".

A rope or thread under tension must have an outward force exerted on each end (by whatever is pulling the rope taut), and it will exert an inward force at each end (per Newton's 3rd law) on whatever is pulling it taut. To find the force on any particular segment of rope, consider the forces acting on that segment: At the right end of the segment, the adjacent material pulls the thread to the right; at the left end, to the left.

Another way of describing it that might help: Think of the rope as being two pieces tied together. If the rope is under tension, the left half must pull the right half to the left; conversely, the right half must pull the left half to the right. Make sense?

 

[m2003]

I can understand how this problem may seem confusing at first. Centripetal force is a concept that can be difficult to grasp, but once understood, it can help explain many physical phenomena.

First, let's define centripetal force. It is the force that acts towards the center of a circular path and keeps an object moving along that path. In this case, the pole is rotating in a circular path, and the thread is attached to it, keeping it in that path.

Now, let's look at the tension in the thread. As you correctly pointed out, the tension on either side of the differential length dr is pointing in opposite directions. This is because of the nature of circular motion. In circular motion, the velocity of the object is constantly changing, as it is always changing direction. This change in velocity requires a force, which is provided by the tension in the thread. So, on one side of dr, the tension is pointing in the direction of the object's motion, and on the other side, it is pointing in the opposite direction. This is necessary to maintain the circular motion.

Furthermore, the magnitude of the tension will vary with the distance from the pole, as the distance r affects the force required to maintain the circular motion. This is why the tension is given as a function of r in the solution.

I hope this explanation helps to clarify the concept of centripetal force and how it relates to tension in this problem. Remember, in science, it is important to break down complex problems into smaller, more manageable parts, and understand the fundamental concepts behind them. With practice and a deeper understanding, you will be able to solve more complex problems involving centripetal force.