Q7.1 Kinds of equilibrium (T)

Suggest locations for a marble which exemplify stable, unstable and neutral mechanical equilibrium.

Can you think of situations which do not fall into one of these three categories?

Graphic - No title - formsofequum

The figure shows a graph of the potential energy of a particle as a function of some coordinate x. Sketch the associated force F(x). Identify the positions of equilibrium and classify them.

The particle is released from rest (a) close to xA and (b) close to xB. Compare the resulting behaviour.

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Hint

You will need to recall the relationship between force and potential energy discussed in section S3.5.

In the last part, the key word is curvature.

 

Solution

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Solution

  1. When displaced from a position of stable equilibrium a particle will experience a force that drives it back towards that equilibrium position. Example: marble at bottom of a bowl.
  2. When displaced from a position of unstable equilibrium a particle will experience a force that drives it further away from that equilibrium position. Example: marble on top of a football.
  3. When displaced from a position of neutral equilibrium a particle will continue to experience no net force. Example: marble on a flat table.

Not all situations fall cleanly into one or other of these categories. Thus, for example, a system may be stable against displacements in one direction but unstable against displacements in another. Example: marble on a saddle.

The force F(x) associated with a potential energy U(x) is
\[ F(x)= -\frac{dU(x)}{dx} \]
The sketch shows (roughly!) the force implied by the form of U(x) given.
  1. The points xA and xB are positions of stable equilibrium.
  2. The point xC is a position of unstable equilibrium.
  3. Points in the region to the left of xD are positions of neutral equilibrium.
Graphic - No title - formsofequumans

If a particle is released from either xA or xB (positions of stable equilibrium) it will experience a restoring force and therefore oscillate. In case A the curvature of the potential is smaller than in case B. If one thinks of a marble rolling back and forth in the hollow one knows immediately that the frequency of oscillation will be lower in case A than in case B. The force curve confirms this picture: the restoring force for a given displacement from equilibrium is smaller in case A than in case B.