# I have a big doubt. Is centripetal acceleration linear acceleration?

1. Dec 19, 2013

### abhijithmatt

It is generally said that centripetal acceleration is directed towards the center of the circular path along the radius. So can we say that centripetal acceleration is a linear acceleration. Is it or can it be represented along a straight line. I had a question in my exams. A kid is rotating a stone tied to a rope in a constant speed and question is 'Is there linear acceleration, and why?' So can centripetal acceleration precisely be described as a linear acceleration. What is the correct answer.

Also if there is change in the speed in which child rotating it what will be the direction of the net acceleration?

2. Dec 19, 2013

### gmax137

I guess I am not clear on the definition of "linear acceleration." What does it mean to you?

ps - welcome to PF !

3. Dec 19, 2013

### rcgldr

Linear acceleration normally means an acceleration that changes the speed (magnitude) of an object, and is normally used in contrast to angular acceleration, which means an acceleration than changes the rate of rotation of an object.

Acceleration of an object following some type of curved path can be separated into two components: tangental acceleration is in the direction of velocity and changes the speed of an object without changing it's direction, while centripetal acceleration is perpendicular to the direction of velocity and changes the direction of the object without changing it's speed.

4. Dec 19, 2013

### tiny-tim

hi abhijithmatt! welcome to pf!
acceleration that isn't linear is rotational

rotational acceleration is angle per time squared

so if you have two stones on the same string, at different distances, they have the same rotational acceleration (zero,if the rotation is constant! )

they obviously don't have the same "ordinary" acceleration, which is distance per time squared

personally, i would say that all "ordinary" acceleration is linear acceleration

[and that rotational acceleration means acceleration about the centre of mass]

but i can imagine that some people might divide it into tangential acceleration (in the instantaneous direction of motion), and centripetal acceleration (towards the instnantaneous centre of curvature of the path) …

and if they then call tangential acceleration linear acceleration, then they would say the stone has no linear acceleration
accelerations are vectors, and add like vectors, so you just vector-add the centripetal acceleration to the tangential acceleration

Last edited: Dec 19, 2013
5. Dec 19, 2013

### A.T.

I think you are confusing linear vs. angular with tangential vs. centripetal. To me centripetal acceleration is linear acceleration, even though it doesn't change the speed. Linear acceleration is the time derivative of linear velocity. Angular acceleration is the time derivative of angular velocity.

6. Dec 19, 2013

### gmax137

So, the answer to the OP's question depends on how "linear acceleration" was defined in his course. Once we get that straight, we can talk about how it should be defined :tongue2:

7. Dec 19, 2013

### AlephZero

+1 I hope everybody agrees that
$$\text{Acceleration} = \frac{d\, \text{Velocity}}{d\,\text{Time}}$$
but whether or not you should call it "linear," if it acts on the center of mass of an object but the direction changes with time, I have no idea.

8. Dec 19, 2013

### Andrew Mason

It would suggest that it is not a good idea to use the term linear acceleration for the reason you allude to. Centripetal acceleration is just acceleration, which can be thought of as the time rate of change of momentum per unit mass and has units of distance x time^-2. Angular acceleration has dimensions of just time^-2 and it is not the time rate of change of angular momentum per unit mass.

One also has to be clear on how centripetal acceleration is defined. Whether centripetal acceleration is the component of the time rate of change of velocity that is perpendicular to the instantaneous velocity (the common usage) or the component of the time rate of change velocity in the radial direction ie directed to a fixed central point (which is how Newton used it) depends on how you want to define it. They are the same for uniform circular motion.

AM

9. Dec 28, 2013

### abhijithmatt

So final question. Is centripetal acceleration a linear acceleration. or is it a radial aceleration. Is it along a line?

10. Dec 28, 2013

### jbriggs444

The answer depends on what you mean by linear acceleration. So... what do you mean by linear acceleration?

My personal interpretation would match Andrew Mason's, that a "linear" acceleration is one that changes a body's velocity (either magnitude or direction), as opposed to an "angular" acceleration that changes its rotation rate.

A "centripetal" acceleration would then be that component of a linear acceleration which is directed toward a defined center point. There is a connotation that this component is positive, i.e. that the force is directed toward the center point rather than away.

A "radial" acceleration would also be that component of a linear acceleration which is directed toward a defined center point. But there is no connotation that this component is positive. A radial acceleration could be either toward the center point or away.

You also ask: "Is it along a line?". I am not sure what it means to you to say that an acceleration is "along a line". Without further context, what it might mean to me is that the momentary acceleration is parallel to the momentary velocity as an object moves along a path. That is, it would be a synonym for "tangential".

11. Dec 28, 2013

### Andrew Mason

Acceleration is the time rate of change of velocity. Its direction is whatever the change in velocity is. If the acceleration is constant in the direction of velocity, you could say the acceleration is linear: its direction does not change, so the body prescribes a straight line path. If the acceleration is constant and is always perpendicular to velocity, the body will prescribe uniform circular motion. In that case, the acceleration is entirely centripetal, which is radial - toward a constant central point.

But those are not the only two possibilities. Acceleration can be variable and in any direction relative to the body's velocity.

AM