Breakdown of Circular Motion

Hi,
Is my understanding right that when centripetal force is less than m*r*omega^2, the particle takes off at a tangent where as when the centripetal force is greater than m*r*omega^2, the particle moves in projectile motion. I am talking in the context of a string strictly, not a rod.

what is generating the force, the tension in the string?

Tension and at some points components of weight are effecting the motion as well.

Ok, can you be a bit more specific. In the horizontal plane so no gravity etc or .... It helps tp give a clear answer.

Well, its a vertical circle so gravity is playing some role.............

It does, but only if you say its vertical motion. So what is the force equation perpendicular to the tangent (along the string)? If you have circular motion, the tension is the difference between centripetal and gravity. If the string snaps, there is a parabolic motion of a projectile under gravity. if the tension goes to zero (and then negative so you don't have circular motion), its again parabolic as the string plays no role hence the circular/centripetal effect does not apply.

The force equation perpendicular to the tangent is :
m*g*sin(theta) = m*r*(d^2theta/dt^2)
Also what do you mean by the fact that tension is the difference between centripetal and gravity. Just don't get what you are saying....

Where is the tension in that equatiion?
http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/cirvert.html
Is pretty good.

Let's look at two examples: one horizontal and one vertical.

Take a mass, $m$ attatched to a string of length $r$ orbiting with angular frequency $\omega$.

Horizontal:
We know when the string snaps the mass will move off tangentially.

Vertical:
Consider the forces acting on the mass at the top of the circle, $mr\omega^2 = T + mg$ where $T$ is the tension in the string.
If the string is to remain taught, thus able to deliver force, we need $T > 0 \Leftrightarrow mr\omega^2 - mg > 0$.
If the string loses tension the mass will move as a parabola as the only force acting is gravity.
Now consider the forces acting on the mass at the bottom of the circle, $mr\omega^2 = T - mg$ so $T = mr\omega^2 + mg$.
If this is greater than the maximium tension the string can tolerate, the mass will undergo parabolic motion as the only force acting is gravity.

oh sorry the force equation perpendicular to tangent is:
T - mgcos(theta) = m*r*omega^2

and the force equation parallel to tangent is:
mgsin(theta) = m*r*(d^2theta/dt^2)

Also when you say that " If you have circular motion, the tension is the difference between centripetal and gravity. " Do you mean to say motion under gravity and motion under centripetal force (i.e. circular motion).

In vertical motion, wouldn't the force be T - mgsin(theta) = m*r*omega^2 ?

Tension is the sum / difference of centripetal and resolved gravity. If its too large the string breaks. If it goes to zero, the string goes slack. In both cases you then have parabolic (quadratic - suvat) motion.

I thought that when the string breaks in vertical motion, i.e. when centripetal acceleration is less than m*r*omega^2, the particle flies of at a tangent.

You have to consider when T<=0 or the string breaks.When positive (and not too large), the strings tension is equal to the sum / difference of centripetal & gravity and circular motion will happen.
The particle does fly off at a tangent, but this is the initial condition for parabolic motion.

what happens when the string breaks?

Please define theta before throwing it out lol
And I said to consider at the top and bottom of the circle.

Parabolic or suvat motion under gravity.

Shouldn't it be T > 0 (i.e. m*r*omega^2 - mgcos(theta) > 0 ) ?