# Centripetal and tangential acceleration in rotational motion ?

centripetal and tangential acceleration in rotational motion....?

consider a stone attached to a string moving in a horizontal circle in a gravity field.
now it has two accelerations one along the $\hat{r}$ direction and the other along $\theta$$\hat{}$ (theta hat) direction
but there is only one force. the centripetal force and the only acceleration produced as a result of that is the r hat direction acceleration that is the centripetal acceleration directed inwards.
so i suppose that there must be a force in theta direction as well responsible for acceleration in that direction
if all of what i said (which i doubt) is correct then what is that theta directed force?

second question: centrifugal force exists right? so when we cut the centripetal force why does not the stone or ball whatever it is fly off in the direction opposite to centripetal force that is in direction of centrifugal force directed outwards? instead what is observed is that it goes off tangential...what is the reason for that?

last one: i got some notes from my professor about rotational motion in which a ball bearing was placed inside a hollow tube. it was not attached , just placed. so when we started rotating it it moved outwards as well as the rotated, obviously. so it was spiral motion.
if you are getting what i am saying then can you think of any force due to which it moves outwards? i mean the circular part is obvious but why move outward?
there is only one force the contact force by the tube. but that is in theta direc tion and it must be solely responsible for the circular motion. my teacher said that the outward motion part is also associated to the contact force, somehow.
i could not understand him at all on this one.
so if u can think of anything then please do .....thanks

## Answers and Replies

I don't really understand your first question, in circular motion there is always a force acting towards the center however this force can never make the object speed up or slow down, there must be at least one other force to do that; perhaps a little more context would help.

Centrifugal force doesn't really "exist" in that there is nothing you can point to and say that it causes this force. What is attributed to centrifugal force is just that object's inertia, or if it's easier to visualize for you, its momentum.

When you begin to rotate the hollow tube the side of the wall pushes on the ball bearing, giving it some motion, then in the next instant when you have rotated some small angle the ball bearing is moving in the same direction which is now pointed slightly towards the end of the tube since the tube itself has in fact moved. Again it is just the inertia of the object that makes it move to the outside while you spin the tube :)

NascentOxygen
Staff Emeritus
Science Advisor

The string is not precisely radially outwards; the string in your hand rotates a little ahead of the string on the stone. Besides, it doesn't follow a circle, more like an ellipse while you are trying to speed the stone up.

The moment the string is let go, there is no further centrifugal acceleration so the stone has nothing to oppose.

I don't really understand your first question, in circular motion there is always a force acting towards the center however this force can never make the object speed up or slow down, there must be at least one other force to do that; perhaps a little more context would help.

Centrifugal force doesn't really "exist" in that there is nothing you can point to and say that it causes this force. What is attributed to centrifugal force is just that object's inertia, or if it's easier to visualize for you, its momentum.

When you begin to rotate the hollow tube the side of the wall pushes on the ball bearing, giving it some motion, then in the next instant when you have rotated some small angle the ball bearing is moving in the same direction which is now pointed slightly towards the end of the tube since the tube itself has in fact moved. Again it is just the inertia of the object that makes it move to the outside while you spin the tube :)

i got the answer to my first question.....what i was trying to say that there are two accelerations being produced as a result of the centripetal force....now newton says that is wrong
i was refering to the centripetal acceleration and the one which i thought was the second one : the tangential acceleration
but is you compute the tangential acceleration it comes to be the same centripetal acceleration....infact that is the centripetal acceleration.

you are right in saying that centrifugal force ....but thats only in an inertial frame. if we are in a rotating frame then there is a centrifugal force does exist
so what will be the answer to the second question now?

and i could not grasp your answer to third question. can you please explain it a bit more in context to forces.?

The string is not precisely radially outwards; the string in your hand rotates a little ahead of the string on the stone. Besides, it doesn't follow a circle, more like an ellipse while you are trying to speed the stone up.

The moment the string is let go, there is no further centrifugal acceleration so the stone has nothing to oppose.

if i am getting it right then do you mean to say that as soon as the string is let go....the centripetal and centrifugal force vanish and the object moves in a tangent due to inertia?
but what i am saying is that centrifugal force is basically an inertial affect. so when i let go of the string why does not the object fly outward instead of in the tangential direction?
perhaps you could somehow explain that with respect to forces

A.T.
Science Advisor

second question: centrifugal force exists right? so when we cut the centripetal force why does not the stone or ball whatever it is fly off in the direction opposite to centripetal force that is in direction of centrifugal force directed outwards? instead what is observed is that it goes off tangential...what is the reason for that?
you are right in saying that centrifugal force ....but thats only in an inertial frame. if we are in a rotating frame then there is a centrifugal force does exist so what will be the answer to the second question now?
In the rotating frame it doesn't go off tangential.

NascentOxygen
Staff Emeritus
Science Advisor

if i am getting it right then do you mean to say that as soon as the string is let go....the centripetal and centrifugal force vanish and the object moves in a tangent due to inertia?
Yes.
but what i am saying is that centrifugal force is basically an inertial affect. so when i let go of the string why does not the object fly outward instead of in the tangential direction?
perhaps you could somehow explain that with respect to forces
The string exerts a force that continuously causes the stone to deviate from the straight line path the stone's inertia would cause it to follow. When the string ceases to exert that force, the stone continues on along the tangential path that it currently is following.

If the stone were to suddenly stop and change direction and hurtle away radially outwards, where would the force be coming from to cause it to perform this spectacular feat? Nowhere. There is no such force.

A.T.
Science Advisor

The string exerts a force that continuously causes the stone to deviate from the straight line path the stone's inertia would cause it to follow. When the string ceases to exert that force, the stone continues on along the tangential path that it currently is following.
That is true for the inertial frame, but he asks about the rotating frame.
If the stone were to suddenly stop and change direction and hurtle away radially outwards, where would the force be coming from to cause it to perform this spectacular feat? Nowhere. There is no such force.
In the co-rotating frame there is an inertial centrifugal force on the stone pointing radially outwards. So when you cut the string, the initially static stone seen from the co-rotating frame will accelerate radially outwards. Then as soon it moves the inertial Coriolis force will bend its path, so it follows a spiral in the co-rotating frame. In the inertial frame where no forces are acting it moves linearly, as you said.

yes that what i wanted to know about....its motion as observed from a rotating frame....
so the stone does accelerate outward...right? but how do you justify it....i mean which force causes that acceleration? is not centrifugal opposite to centripetal force and exists as long as the centripetal exists?
what is this coriolis force and spiral motion?
can you give me a good source where all this is explained thoroughly? i could not find a good one.
i am new to this rotating frame thing and dont know much about it....thx A.T.

A.T.
Science Advisor

yes that what i wanted to know about....its motion as observed from a rotating frame....
so the stone does accelerate outward...right? but how do you justify it....i mean which force causes that acceleration?
Inertial centrifugal force
is not centrifugal opposite to centripetal force and exists as long as the centripetal exists?
No. The inertial centrifugal force exists as long as your frame rotates:
http://en.wikipedia.org/wiki/Rotating_reference_frame

what is this coriolis force
http://en.wikipedia.org/wiki/Coriolis_effect#Formula

and spiral motion?
http://en.wikipedia.org/wiki/Spiral

BruceW
Homework Helper

The position, velocity and acceleration in 2d polar coordinates:
$$\vec{r}=r \hat{r}$$
$$\dot{\vec{r}}= \dot{r} \hat{r} + r \dot{\theta} \hat{\theta}$$
$$\ddot{\vec{r}}=(\ddot{r} - r \dot{\theta}^2) \hat{r} + (2 \dot{r} \dot{\theta} + r \ddot{\theta}) \hat{\theta}$$

Where theta is the angle, and r is the radius. The hat over theta and r indicate the unit vectors of this coordinate system. Be careful with differentiation, because the unit vectors are time dependent! The last equation is for the acceleration of the object, so if you multiply by its mass, then you can equate to the total forces on the object, to find out what its motion will be.

These equations are very general, so you can use them in many 2d situations. For example, if the object has a radial force inwards which is equal to $- m r \dot{\theta}^2 \hat{r}$, then $\ddot{r}$ must be equal to zero, meaning that the radius of the object must change at a constant rate (or be zero). This is interpreted as the effect of a centripetal force. And if we happen to have $\dot{r}=0,$ then the object will have circular motion.

EDIT: Also, I would recommend getting hold of a textbook for learning about rotating reference frames, etc, because I haven't found many places on the web which explain it well. There are probably good sites out there though.

rcgldr
Homework Helper

consider a stone attached to a string moving in a horizontal circle in a gravity field. ... but there is only one force.
There are two forces acting upon the stone, a tension force from the string, and a downwards force from gravity. You should be able to create an equation relating angle (from horizontal) of string to radius and speed of the stone.

second question: centrifugal force exists right?
Only in a rotating frame, unless you want to consider the horizontally outwards component of force exerted by the stone on the string as a reactive centrifugal force, but many physicists do not like the term reactive centrifugal force.

ball bearing was placed inside a (rotating) hollow tube.
As mentioned previously, what is a tangental force at one moment in time results in radial velocity at a later moment in time.