# The change in magnitude of centripetal acceleration

• Frawx
In summary: The only difference is that with a changing speed, you need to take into account the change in tangential velocity as well.
Frawx
When an object (e.g. racecar) moves around in circles with constant tangential velocity, constant centripetal acceleration is present.

What happens to the centripetal acceleration when the racecar is at rest, then increases its speed? I know that the tangential velocity increases due to the tangential acceleration, but what about the centripetal acceleration?

Since centripetal acceleration is tangential velocity squared divided by the radius, and the tangential velocity is increasing from rest, the centripetal acceleration must then be increasing as well.

How do you calculate the values for centripetal acceleration if it is changing? There doesn't seem to be a formula for it. And it seems that centripetal acceleration is changing, is there a term for the rate of change of it?

If the object is moving in a circle, the centripetal acceleration must be ##v^2/r## - ie all that matters is the instantaneous linear speed and distance from the centre of the circle. If the object is not moving in a circle (eg ellipse or more complex shape) it becomes more complicated.

andrewkirk said:
If the object is moving in a circle, the centripetal acceleration must be ##v^2/r## - ie all that matters is the instantaneous linear speed and distance from the centre of the circle. If the object is not moving in a circle (eg ellipse or more complex shape) it becomes more complicated.

What if the instantaneous linear speed is increasing? How would I be able to find out the centripetal acceleration's increase as the instantaneous linear speed increases? The formula only applies to an instantaneous point in time.

Additionally, what happens when the object is not moving in a circle (e.g. ellipse)? Would you assume the ellipse to be almost equal to a circle? If it is a complex shape, do you draw many different circles?

Frawx said:
What if the instantaneous linear speed is increasing? How would I be able to find out the centripetal acceleration's increase as the instantaneous linear speed increases? The formula only applies to an instantaneous point in time.
If you know a formula for the linear speed as a function of time then you automatically have a formula for the centripetal accel as a function of time. Just square the first formula and divide by ##r##. That's all there is to it.

Perhaps what you are wondering about is what would happen with an object that is constrained to move on a circular track, to which a constant circumferential acceleration is applied?

The "v" in v^2/r is the instantaneous tangential velocity.

So if v was changing with some complicated function of time like v= 3t^3 + t^2 then you just substitite it so..

A = v^2/r
Becomes
A =(3t^3 + t^2)^2/r

Note that A is just the centripetal acceleration NOT the net acceleration. If you want to calculate the net acceleration you must do the vector sum of the tangential acceleration and the centripetal acceleration. That's not difficult because they are at 90 degrees to each other so Pythagoras can help.

Frawx said:
If it is a complex shape, do you draw many different circles?
Conceptually yes. You use the local radius & center of curvature to determine the magnitude and direction of the centripetal acceleration.

https://en.wikipedia.org/wiki/Curvature

Last edited:
Thanks for the responses! Turns out that the answer to this question was simpler than I had thought.

The formula for centripetal acceleration at an instantaneous point in time is ac=v2/r.

The formula for centripetal acceleration for a changing tangential velocity over time is ac=(v(t))2/r.

Frawx said:
Thanks for the responses! Turns out that the answer to this question was simpler than I had thought.

The formula for centripetal acceleration at an instantaneous point in time is ac=v2/r.

The formula for centripetal acceleration for a changing tangential velocity over time is ac=(v(t))2/r.

Well, it's the same formula, regardless if the speed changes or not.

## What is centripetal acceleration?

Centripetal acceleration is the acceleration that an object experiences when moving in a circular path. It is always directed towards the center of the circle and its magnitude depends on the object's speed and the radius of the circle.

## How does the magnitude of centripetal acceleration change?

The magnitude of centripetal acceleration changes as the object's speed or the radius of the circle changes. If the speed increases, the magnitude of centripetal acceleration also increases. If the radius of the circle increases, the magnitude of centripetal acceleration decreases.

## What factors affect the magnitude of centripetal acceleration?

The magnitude of centripetal acceleration is affected by the object's speed, the radius of the circle, and the mass of the object. Higher speeds and smaller radii result in larger magnitudes of centripetal acceleration, while larger masses result in smaller magnitudes of centripetal acceleration.

## Why is centripetal acceleration important?

Centripetal acceleration is important because it is responsible for keeping objects moving in a circular path. It is also a key concept in understanding circular motion and is used in various real-life applications such as amusement park rides, car racing, and satellite motion.

## How is centripetal acceleration calculated?

The magnitude of centripetal acceleration can be calculated using the formula a = v^2/r, where a is the centripetal acceleration, v is the speed of the object, and r is the radius of the circle. This formula can also be rearranged to solve for other variables if needed.

• Mechanics
Replies
6
Views
1K
• Introductory Physics Homework Help
Replies
5
Views
2K
• Mechanics
Replies
15
Views
2K
• Mechanics
Replies
21
Views
1K
• Introductory Physics Homework Help
Replies
55
Views
1K
• Mechanics
Replies
21
Views
7K
• Introductory Physics Homework Help
Replies
7
Views
2K
• Mechanics
Replies
24
Views
2K
• Mechanics
Replies
37
Views
3K
• Classical Physics
Replies
8
Views
1K