# Calc-based conceptual problem with centripetal motion

• Afterthought
In summary, the conversation discusses the classic example of a ball attached to a string and moving at constant speed in a circle. The acceleration is v^2/r and always facing in the direction of the string. The conversation also delves into the use of vectors in polar coordinates and the role of infinitesimal vectors in the problem. It is concluded that infinitesimal vectors are essential in the derivation of the formula and that thinking about them is relevant to the question of whether time is continuous or discrete.
Afterthought
Note: This is more of a math question than a physics question, but I'm posting it here since it's in the context of physics.

I've been thinking about the classic example of a ball attached to a string and moving at constant speed in a circle. The acceleration is v^2/r and always facing in the direction of the string. I've derived this result in the past before by breaking the acceleration into its x and y components and integrating to find v. In these coordinates the result is intuitive to me and makes sense.

However something bothers me about thinking of the problem using vectors in polar coordinates. At any given time, there is a velocity vector tangential to the circle, and the acceleration vector normal to it. At some infinitesimal moment later, there is an infinitesimally small velocity vector pointing the same direction as the acceleration, and it adds with the normal vector to create a new velocity vector. If a = v^/r, this new vector will be tangential to the next bit of the circle; this is why the object moves in a circle. However, the thing that bothers me is, that when you add these two vectors, shouldn't the resultant velocity vector be bigger than the original velocity vector, even if only by an infinitesimal amount? And in that case, wouldn't all the infinitesimal errors eventually add up, making the velocity vector no longer constant?

The only thing I could think of as to why this thinking is wrong is because when you add the vectors, you have to use the Pythagorean Theorem to find the new vector (as the vectors are normal), and that involves the square of dt, which does not integrate into any finite number. Is this reasoning correct?

If it is, I'd like to know to what extent thinking about infinitesimal vectors being added is relevant to the question whether time is continuous or discrete. In the case of centripetal motion, If time was discrete, would the magnitude of velocity vector change in a noticeable way after a long enough time?

Thanks.

Afterthought said:
If it is, I'd like to know to what extent thinking about infinitesimal vectors being added is relevant
That's an essential part of the derivation. You start with a finite step size (taking the acceleration over finite steps) and then take the limit, as the step size approaches zero (as with all differential calculus). That gives you the well known formula. It's a standard bit of book work. You will find it in any A level Mechanics textbook.

## 1. What is centripetal motion?

Centripetal motion is the circular motion of an object around a fixed point, caused by a centripetal force acting towards the center of the circle.

## 2. How is centripetal motion different from regular motion?

Centripetal motion is different from regular motion because it involves an object moving in a circular path, rather than a straight line. Additionally, centripetal motion requires a centripetal force to maintain the circular path, while regular motion does not.

## 3. What is the formula for calculating centripetal force?

The formula for calculating centripetal force is F = mv²/r, where F is the force, m is the mass of the object, v is its velocity, and r is the radius of the circular path.

## 4. How does centripetal force affect the motion of an object?

Centripetal force is responsible for keeping an object moving in a circular path. It constantly changes the direction of the object's velocity, causing it to accelerate towards the center of the circle.

## 5. What are some real-life examples of centripetal motion?

Some real-life examples of centripetal motion include a car turning a corner, a satellite orbiting around the Earth, and a rollercoaster moving along a circular track. Other examples include the motion of a ferris wheel, a spinning top, and the swinging of a pendulum.

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