# Solving a Circular Motion Problem: Find the Center of the Path

• ubiquinone
In summary, the conversation is about a beginner in physics seeking help with a circular motion problem. The problem involves a particle moving along a circular path at constant speed over a horizontal xy coordinate system. The coordinates and velocities of the particle at two different times are given. The question is to determine the coordinates of the center of the circular path. The person asking for help has encountered three previous easy problems and wants to be able to solve more difficult problems like this one. They are seeking guidance and clarification on how to approach and solve this type of problem. The other person suggests figuring out the distance traveled and the direction of the particle between the given times to determine the size and location of the circle.
ubiquinone
Hi, I'm just beginner at physics and circular motion is really a new topic for me. I came across this problem in the opening chapter exercises. I glanced at the problems following this one and many of them are pretty much the same. I was just wondering if someone could please show me how to solve this problem, so I can use this as a guide when I solve similar problems of this chapter. Thank You.

Question: A particle moves along a circular path over a horizontal $$xy$$ coordinate system, at constant speed. At time $$t_1=4.00s$$, it is at point $$(5.00m,6.00m)$$ with velocity $$(3.00m/s)\vec{j}$$ and acceleration in the positive $$x$$ direction. At time $$t_2=10.0s$$, it has velocity $$(-3.00m/s)\vec{i}$$ and acceleration in the positive $$y$$ direction. What are the coordinates of the center of the circular path?

If you are just picking a problem before studying circular motion, this is probably not the one to start with. If you want to take a shot at it, try to figure out where the particle is at t = 10 seconds. How far does it travel between t = 4 seconds and t = 10 seconds? How big is the circle?

Hi OlderDan, thanks again for replying to my questions. No, I'm not picking random questions to do in fact this is the 4th question from my chapter exercises on circular motion. The first three were easy, straight formula calculations of solving for one of the unknows with a_c=v^2/r.
But I want to be able to solve this type of problem, as I have many similar that I need to encounter in the upcoming problems.
If you can help a little bit more, may you please offer me a hand. Thanks again for your patience.

ubiquinone said:
Hi OlderDan, thanks again for replying to my questions. No, I'm not picking random questions to do in fact this is the 4th question from my chapter exercises on circular motion. The first three were easy, straight formula calculations of solving for one of the unknows with a_c=v^2/r.
But I want to be able to solve this type of problem, as I have many similar that I need to encounter in the upcoming problems.
If you can help a little bit more, may you please offer me a hand. Thanks again for your patience.
Then you should try to answer the questions I posed. How far does it move between t = 4s and t = 10s? Knowing the direction it is moving at both times, what part of a circle is this? How big is the circle?

## 1. How do I identify the center of rotation in a circular motion problem?

The center of rotation in a circular motion problem is the point around which an object is moving in a circular path. To identify the center of rotation, you can use the following steps:
1. Find the point where the object changes direction in the circular path.
2. Draw a straight line perpendicular to the tangent of the circle at this point.
3. The center of rotation will be the intersection point of the perpendicular line with the circle's circumference.

## 2. What is the relationship between the radius and the center of rotation in a circular motion problem?

The radius and the center of rotation in a circular motion problem are directly related. The center of rotation is the point that is equidistant from all points on the circle's circumference. This means that the radius, which is the distance from the center to any point on the circle, is the same for all points. Therefore, the radius and the center of rotation are always equal in a circular motion problem.

## 3. Can the center of rotation be outside of the circular path?

No, the center of rotation must always be inside the circular path. This is because the center of rotation is the point around which the object is rotating, and if it is outside of the circular path, the object would not be rotating around it. If the center of rotation is outside of the circular path, it means that the object is moving in a different type of motion, such as an elliptical or spiral path.

## 4. How does the speed of an object affect the location of the center of rotation?

The speed of an object has no effect on the location of the center of rotation in a circular motion problem. The center of rotation is solely determined by the shape and size of the circular path, and not by the speed at which the object is moving. However, the speed of the object will affect the rate at which it moves around the circle and the centripetal force required to keep it in circular motion.

## 5. Can there be more than one center of rotation in a circular motion problem?

No, there can only be one center of rotation in a circular motion problem. This is because the center of rotation is the fixed point around which the object is rotating, and if there were multiple centers, the object would not have a consistent circular path. It is possible for an object to have multiple rotations, each with its own center, but in a single circular motion problem, there can only be one center of rotation.

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