# Solving for Constant Centripetal Acceleration: Understanding Spiral Motion

• Ale98
In summary, the conversation discusses an object accelerating in a circular motion and the need for a spiral path to maintain constant centripetal acceleration while the object is also accelerating forward. The equation for this spiral is discussed, but the formula to keep centripetal acceleration constant is still unknown.
Ale98
Assume an object accelerating at a certain value dV/dt. If this object was traveling in a circular motion the centripetal force would increase as the object moves faster.
To maintain centripetal acceleration constant while the object is accelerating (in its forward motion dV/dt) I think it would need to follow some sort of spiral path so that after a certain time the object would be traveling faster however the radius also increases.

Does anybody know what type of spiral would keep centripetal acceleration constant? What formula would this spiral have?

Thanks in advance for the help.

The differential equation should be

$\dot v = a_t \cdot \frac{v}{{\left| v \right|}} + a_r \cdot \frac{{n \times v}}{{\left| {n \times v} \right|}}$

but I'm too lazy to solve it.

edit: equation amended for a spiral in a plane with the normal n

Just to make sure i understood your notation

The acceleration ##\dot v## is the sum of tangential ##a_t \cdot \frac{v}{{\left| v \right|}}## (= your dV/dt) and the radial acceleration ##a_r \cdot \frac{{n \times v}}{{\left| {n \times v} \right|}}## (= centripetal acceleration).

Okay, but I am confused on how I can find the formula for a spiral to keep centripetal acceleration constant after I solve the integral.

Ale98 said:
Okay, but I am confused on how I can find the formula for a spiral to keep centripetal acceleration constant after I solve the integral.

Just keep ##a_r## constant.

## 1. What is constant centripetal acceleration?

Constant centripetal acceleration refers to the acceleration of an object moving in a circular path at a constant speed. This acceleration is always directed towards the center of the circle and is responsible for keeping the object in its circular motion.

## 2. How is constant centripetal acceleration calculated?

The formula for calculating constant centripetal acceleration is a = v^2/r, where a is the acceleration, v is the velocity, and r is the radius of the circular path. This means that the acceleration is directly proportional to the square of the velocity and inversely proportional to the radius.

## 3. What is the difference between constant centripetal acceleration and constant tangential acceleration?

Constant centripetal acceleration is always directed towards the center of the circle and is responsible for maintaining the circular motion, while constant tangential acceleration is directed tangentially to the circle and is responsible for changing the speed of the object.

## 4. How does understanding spiral motion relate to constant centripetal acceleration?

Spiral motion is a type of circular motion where the radius of the circle gradually changes over time, resulting in a spiral path. In order for an object to move in a spiral path, it must maintain a constant centripetal acceleration to stay in its circular motion, while also experiencing a changing tangential acceleration to create the spiral shape.

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

Some common examples of constant centripetal acceleration include a car moving around a circular track, a satellite orbiting the Earth, and a roller coaster looping around a track. Essentially, any object that moves in a circular path at a constant speed experiences constant centripetal acceleration.

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