Question

Centripetal and tangential acceleration

When I introduced non-uniform circular motion, I said and I quote "The centripetal acceleration can only change the direction of velocity but not the speed, and thus if we now wish to change the speed, we need an acceleration along the tangent, called tangential acceleration"

Answer 1

Now I am pretty sure there is nothign wrong with this, however one kid kept asking me about centripetal acceleration, and only now do I understand what he was really asking.

So what he is saying is , imagine if we take a stone tie to a string and whirl it in uniform circular motion, suddenly if I increase the pull (the centripetal pull increasing the centripetal acceleration) the stone goes faster..

so doesn't increasing centripetal acceleration increase the speed?
also

a = v^2/R supports that equation?

Now I know something is definitely wrong with this

My answer is, when we do it, we ACTUALLY provide a tangential acceleration which increases the speed, and to maintain this increased speed, we need to increase the pull as well.

But suppose if we JUST pulled it in.. (increased the centripetal pull) what would happen in that case? ( i think the stone would no longer be in a circular motion, it would do something weird)

So a = v^2/R is quite a special equation, in that, you cannot just say lets double the 'a' and expect to see a doubled 'v' right?

its more like, you first change 'v' (somehow) and then you would calculate the required 'a' to maintain that circular motion!

So I just need someone to tell me if I am wrong somewhere

Answer 2

@Vincent-Lyon.Fr

Answer 3

As far as i understand, when you pull the sting, the radius of the circular path is shortened very little in a very small amount of time but the center that is seen by stone remains as it was before again in a short time. then, conservation of angular momentum makes stone faster.. But i am not sure that why the center of rotation does not change. Just a thought.. i should think about it.

Answer 4

The stone having been released from its circular motion does NOT go faster. its velocity now remains constant since there is no forces acting on it. The speed being the same as it was in the circular motion.

Answer 5

Now in the actual process of hurling a stone as with a sling. The thrower does exert and addition force just before the release by moving the center of rotation which you can feel as it is released. The motion should be pure rotation until just before the release and then at the release the arm moves forward providing a pulling force momentarily.

Answer 6

The centripetal acceleration is a response to the speed. To increase the acceleration you must increase the speed by providing a tangential force. Note in increasing the speed of a stone on a string the rhythmical pumping of the arm required to build up the speed.

Answer 7

So gleem

what would happen if imagine you pulled in the string .. (not the pull that we do, but pull in to make the radius shorter?) what would happen then to the speed of the particle?
speed can't change right?

@Oksuz
i don't understand what you said :P

Answer 8

@gleem

The centripetal force is therefore something that we cannot pre - decide right?

Its not like.. Ok ll put 20 N of centripetal force.
Depending upon what the rotation speed is, automatically a part of your force becomes centripetal, and the rest basically accelerates the whole system?

Answer 9

Right, centripetal force a result of the constraining string holding it in a circular path.

Now we are back to the skater thing, If you pull in the string the circulating stone will increase in speed to conserve angular momentum since no torque is being appilied. the constraining force is a central force so rxp =0.

Answer 10

Thank you :)

Answer 11

Now again, if you JUST concentrate on the particle.. the particles speed has increased, so there must be an acceleration and thus a force (tangential) right?
So where is this force coming from?

I understand from the energy point of view, work done increased its speed. But from dynamics point of view? what supplied the necessary tangential force to accelerate the particle?

Answer 12

I think that there is an (understandable) confusion between normal acceleration and radial (aka centripetal) acceleration if you stay in the restricted frame of the circular motion.

In a curvilinear motion (elliptic for instance), acceleration is centripetal, but it changes the speed of the body, except at 2 point on the path of the body.

It is not the centripetal, but the normal acceleration that bears no influence on the speed of the body. A better example of pure normal acceleration is the motion of a particle in a B-field.

Answer 13

In your example, if you pull harder, the body leaves the circular path and the tension in the string, although still centripetal, does not act normal to the trajectory anymore, hence increasing the speed of the particle.

Answer 14

@Vincent-Lyon.Fr

Yes yes.. i just thought of that.. thanks for clearing that out.

But lets talk about torques and angular acceleration.
clearly the particle experiences an angular acceleration (about the Sun ) right? or it doesn't?
Cause there is no torque though.
so how to explain that?

Answer 15

Careful, orthoradial acceleration is zero, but tangential acceleration is not.

\(r \ddot {\theta}+2\;\dot r \dot \theta=0\)
but
\(\dot v \neq 0\)