Mechanical equilibrium

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Force diagram showing the forces acting on an object at rest on a surface. Notice that the amount of force that the table is pushing upward on the object (the N vector) is equal to the downward force of the object's weight (shown here as mg, as weight is equal to the object's mass multiplied with the acceleration due to gravity): because these forces are equal, the object is in a state of equilibrium (all the forces acting on it balance to zero).

A standard definition of static equilibrium is:

A system of particles is in static equilibrium when all the particles of the system are at rest and the total force on each particle is permanently zero.[1]

This is a strict definition, and often the term static equilibrium is used in a more relaxed manner interchangeably with mechanical equilibrium, as defined next.[2]

Pendulum in a stable equilibrium (left) and unstable equilibrium (right)

A standard definition of mechanical equilibrium for a particle is:

The net force (sum of all external forces) acting upon the particle is zero.[3] This implies the object may be motionless or moving at constant speed in a straight line.
R = ∑ F = 0[4]

The necessary conditions for mechanical equilibrium for a system of particles are:

(i) the vector sum of all external forces is zero and
(ii) the sum of the moments of all external forces about any line is zero.[3]

As applied to a rigid body, the necessary and sufficient conditions become:

A rigid body is in mechanical equilibrium when the sum of all forces on all particles of the system is zero, and also the sum of all torques on all particles of the system is zero.[5][6]
R = ∑ F = 0
MRO = ∑ MO = ∑ (r x F) = 0[4]

A rigid body in mechanical equilibrium is undergoing neither linear nor rotational acceleration; however it could be translating constant velocity in a straight line or rotating with constant angular momentum but not necessarily at constant angular velocity.

However, this definition is of little use in continuum mechanics, for which the idea of a particle is foreign. In addition, this definition gives no information as to one of the most important and interesting aspects of equilibrium states—their stability.

An alternative definition of equilibrium that applies to conservative systems and often proves more useful is:[7]

A system is in mechanical equilibrium if its position in configuration space is a point at which the gradient with respect to the generalized coordinates of the potential energy is zero.

Because of the fundamental relationship between force and energy, this definition is equivalent to the first definition. However, the definition involving energy can be readily extended to yield information about the stability of the equilibrium state.

For example, from elementary calculus, we know that a necessary condition for a local minimum or a maximum of a differentiable function is a vanishing first derivative (that is, the first derivative is becoming zero). To determine whether a point is a minimum or maximum, one may be able to use the second derivative test. The consequences to the stability of the equilibrium state are as follows:

Unstable equilibria
  • Second derivative < 0: The potential energy is at a local maximum, which means that the system is in an unstable equilibrium state. If the system is displaced an arbitrarily small distance from the equilibrium state, the forces of the system cause it to move even farther away.
Stable equilibria
  • Second derivative > 0: The potential energy is at a local minimum. This is a stable equilibrium. The response to a small perturbation is forces that tend to restore the equilibrium. If more than one stable equilibrium state is possible for a system, any equilibria whose potential energy is higher than the absolute minimum represent metastable states.
Neutral equilibria
  • Second derivative = 0 or does not exist: The second derivative test fails, and one must typically resort to using the first derivative test. Both of the previous results are still possible, as is a third: this could be a region in which the energy does not vary, in which case the equilibrium is called neutral or indifferent or marginally stable. To lowest order, if the system is displaced a small amount, it will stay in the new state.

In more than one dimension, it is possible to get different results in different directions, for example stability with respect to displacements in the x-direction but instability in the y-direction, a case known as a saddle point. Without further qualification, an equilibrium is stable only if it is stable in all directions.

The special case of mechanical equilibrium of a stationary object is static equilibrium. A paperweight on a desk would be in static equilibrium. The minimal number of static equilibria of homogeneous, convex bodies (when resting under gravity on a horizontal surface) is of special interest. In the planar case, the minimal number is 4, while in three dimensions one can build an object with just one stable and one unstable balance point, this is called gomboc. A child sliding down a slide at constant speed would be in mechanical equilibrium, but not in static equilibrium.

An example of mechanical equilibrium is a person trying to press a spring. He or she can push it up to a point after which it reaches a state where the force trying to compress it and the resistive force from the spring are equal, so the person cannot further press it. At this state the system will be in mechanical equilibrium. When the pressing force is removed the spring attains its original state.

See also

Notes and references

  1. ↑ Herbert Charles Corben & Philip Stehle (1994). Classical Mechanics (Reprint of 1960 second ed.). Courier Dover Publications. p. 113. ISBN 0-486-68063-0. 
  2. ↑ Lakshmana C. Rao, J. Lakshminarasimhan, Raju Sethuraman, Srinivasan M. Sivakumar (2004). Engineering Mechanics. PHI Learning Pvt. Ltd. p. 6. ISBN 81-203-2189-8. 
  3. ↑ 3.0 3.1 John L Synge & Byron A Griffith (1949). Principles of Mechanics (2nd ed.). McGraw-Hill. p. 45–46. 
  4. ↑ 4.0 4.1 Beer FP, Johnston ER, Mazurek DF, Cornell PJ, and Eisenberg, ER. (2009) Vector Mechanics for Engineers: Statics and Dymanics. 9th ed. McGraw-Hill. p 158.
  5. ↑ Mechanical Equilibrium
  6. ↑ The torque is taken with respect to some reference point. Because the sum of the forces is zero the total torque is independent of the choice of this point.
  7. ↑ Herbert Goldstein (1950). Classical Mechanics (1st ed.). Addison-Wesley. p. 318. ISBN 0-201-02918-9. 

Further reading

  • Marion JB and Thornton ST. (1995) Classical Dynamics of Particles and Systems. Fourth Edition, Harcourt Brace & Company.
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