Uniform Circular motion and Equilibrium

Question

Is a body in uniform circular motion in dynamic equilibrium? My book says that it is true. But my argument is: a body in uniform circular motion has a force vector that is radially inwards, but for a system to be in equilibrium, there should be another force vector that is radially outward, so to make the algebraic sum of all force vectors equal to zero. If that force vector exists, then it shouldn't be in uniform circular motion in the first place.

Answer 1

Is a body in uniform circular motion in dynamic equilibrium?

As far as I am aware, dynamic equilibrium requires a body to be moving at constant velocity, that is, moving at constant speed in a straight line. A body in uniform circular motion is moving at constant speed but not in a straight line, thus its velocity is not constant. There is a net force (centripetal force) acting toward the center of rotation. Consequently, a body in uniform circular motion would not be in dynamic equilibrium.

On the other hand, if viewed from the non-inertial (rotating) frame of the body in circular motion, it could be argued that the body appears to be in dynamic equilibrium. That's because in the non inertial frame the centripetal force is balanced by the centrifugal force, which is a pseudo or fictitious force, for a net force of zero.

Hope this helps.

Answer 2

We have this compelling intuition that uniform circular motion involves some form of equilibrium.

There is in fact a good motivation for that intuition. Before I get to that I want to emphasize that uniform circular motion is not a case of force equilibrium.

Example of dynamic equilibrium: A parachute jumber has just jumped out of the plane, and gravity is accelerating him. As his velocity increases the air drag increases. At the point where the air drag matches the force of gravity the acceleration levels out, and from that point on the velocity of the jumper is roughly constant.

Generally things start out in a configuration without force equilibrium, and that unbalanced configuration then changes. At some point the forces do balance out, and from there on the configuration is in dynamic equilibrium.

Circular motion:
In order to sustain circumnavigating motion a force is required: something has to provide required centripetal force.

A common demonstration of uniform circular motion is to have some dish with a fluid in it on a turntable. When the turntable is spun up there is a redistribution of the fluid; towards the rim of the dish. When the dish is rotating at a constant angular velocity the surface of the fluid has a corresponding shape; the cross section of that shape is a parabola.

This rotation with a constant shape of the fluid is called 'solid body rotation'.

Solid body rotation is a state of motion with the following property: there is no opportunity to dissipate energy. There is kinetic energy and potential energy, and they are both constant.

If you would impart a sloshing motion to that fluid then the system has more energy than in the state of solid body rotation. Sloshing motion means that the parts of the fluid are "rubbing" along each other. That relative motion gives opportunity for energy dissipation.

Once the fluid is back to solid body rotation there is still the energy of the uniform circular motion, but no opportunity to dissipate that energy.

We know the amount of required centripetal force in order to maintain the state of solid body rotation.

With $\omega$ for the angular velocity:

$$ F_{centripetal} = m \omega^2 r $$

In order to maintain solid body rotation the centripetal force must be proportional to the distance $r$ to the central axis of rotation.

The potential energy at a point at a distance $r$ to the central axis of rotation is given by the integral of force over distance. Hence: in the case of solid body rotation the potential energy increases with the square of the distance to the central axis of rotation.

As we know, the kinetic energy is proportional to the square of velocity, hence in the state of solid body rotation the kinetic energy is proportional to the square of the distance to the central axis of rotation.

So we have that the state of solid body rotation is a state of energy equilibrium.

I believe that that is the origin of our intuition that uniform circular motion involves some form of equilibrium. Even if we cannot pinpoint why, our physical motor skills allow us to feel that.

To your question specifically: You inform about a single body in uniform circular motion. In order to maintain that circular motion a corresponding force must be provided.

One way to provide that required centripetal force is to have a second body present, and an attracting force between them. That force can be provided by a string, or some elastic material. And in the case of celestial bodies the attracting force is provided by gravity.

Pluto and Charon are a two body system. Pluto is larger than Charon, but (if memory serves me) the common center of mass of Pluto and Charon is located somewhere in the space between them.

If the required centripetal force is provided by a another body then that other body is at the diametrically opposite end of the circular motion. Of course, if that other body is much larger than the first one the motion of the larger body is imperceptible. The point is: to maintain circular motion a centripetal force must be provided, there is no escaping that.

Sustained circular motion is inherently not a state of force equilibrium.

Of course, if two bodies are circumnavigating their common center of mass then the two centripetal forces are opposite and equal in magnitude. But those two centripetal forces are exerted at two different locations: on the first body and the on the second body.

Force equilibrium refers to equilibrium of all the forces that are exerted upon one particular body.