What Causes the Counterbalancing Force in a Frictionless, Rotating System?

[Bashyboy]

Hello, the problem I am working on is:

"A puck of mass $m = 51.0 g$is attached to a taut cord passing through a small hole in a frictionless, horizontal surface (see figure below). The puck is initially orbiting with speed $v_i = 1.30 m/s$ in a circle of radius $r_i = 0.310 m$. The cord is then slowly pulled from below, decreasing the radius of the circle to $r = 0.120 m$."

In the problem, it states that the surface is friction-less, so what would counterbalance the tension force in the string, so that the puck wouldn't be pulled completely towards the hole in the center of the table? Is it the centrifugal force (or, inertial force)? If so, could someone give me a descriptive answer as to why it is the centrifugal acting.

 

[mfb]

Centrifugal force, in a rotating coordinate system.
Centripetal force, in an inertial system.

The puck is changing its direction of motion all the time - and this acceleration is always "towards the center".

It is the same concept as in planetary orbits: They are accelerated towards the sun all the time, which leads to a circular motion around the sun.
Here, you change the force a bit, this changes the radius.

 

[Bashyboy]

Yes, I know the acceleration is directed towards the center. But what keeps the puck from moving towards the center of the table under the tension force?

 

[mfb]

The force is not sufficient to pull it in (it would have to be more than the centripetal force, but that is not limited). It moves closer to the center, this increases centripetal force until the system is in equilibrium again.

 

[Bashyboy]

Is it not sufficient because of the puck's inertia?

 

[mfb]

The puck's inertia is one of the concepts used to describe the system. It does not exist in a meaningful way with a theory what inertia does, and that is classical mechanics.

 

[rcgldr]

what would counterbalance the tension force in the string?

At the end of the string that is attached to the puck, you have a pair of Newton third law forces, the inwards force that the string exerts onto the puck and the outwards reaction force (due to acceleration) that the puck exerts onto the string. If the string is not being pulled inwards or released outwards, so that the radius of the pucks path is constant (circular motion), then the inwards force is centripetal, and the outwards force is sometimes called "reactive centrifugal force." Wiki article:

http://en.wikipedia.org/wiki/Reactive_centrifugal_force

During the time the string is being pulled inwards, the path of the puck is a spiral, and a component of the tension in the string is in the direction of the inwards spiraling puck, increasing it's speed. Angular momentum will be conserved, while kinetic energy will be increased due to the work done by pulling the string inwards.

 

[Bashyboy]

rcgldr, so are you, and the wikipedia article, essentially saying that though the centrifugal force isn't real, its effects are?

 

[haruspex]

rcgldr, so are you, and the wikipedia article, essentially saying that though the centrifugal force isn't real, its effects are?

The effect of a (unopposed) force is acceleration. If centrifugal force were real it would result in an acceleration, and it would be away from the axis. It all comes down to reference frame. In an inertial frame, there is acceleration towards the axis provided by centripetal force. In the frame of reference of the orbiting body (which is not an inertial frame, and so can lead to confusion), there is no acceleration. To explain that, given the experienced centripetal force, you have to invent an equal and opposite centrifugal force.

 

[rcgldr]

rcgldr, so are you, and the wikipedia article, essentially saying that though the centrifugal force isn't real, its effects are?

The normal usage (at least by physicists) of the term centrifugal force is the fictitious force used in a rotating frame of reference, along with the fictitious coriolis force (and the Euler force if there is angular acceleration):

http://en.wikipedia.org/wiki/Rotating_reference_frame

wiki_centrifugal_force_(rotating_reference_frame)

Note that all of these rotating frame fictitious forces appear to be forces exerted onto an object within the rotating frame. For your example, all of these rotating frarme fictitious forces would appear to act on the puck.

This is different than the term "reactive centrifugal force". For your example, the "reactive centrifugal force" is exerted onto the string (not the puck). There is a net centripetal force exerted by the string onto the puck that accelerates the puck so that it follows a circular path. The puck reacts to this acceleration by exerting an outwards "reactive centrifugal force" onto the string. This "reactive centrifugal force" would correspond to the tension in the string.

http://en.wikipedia.org/wiki/Reactive_centrifugal_force

The wiki article on centrifugal force includes a section that explains the difference between fictitious and reactive forces:

wiki_fictitious_vs_reactive_force.htm