Condition for absolute equilibrium
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.
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.
A standard definition of mechanical equilibrium for a particle is:
The necessary and sufficient conditions for a particle to be in mechanical equilibrium
are that the net force acting upon the particle is zero.
If an object is stationary all the forces acting on the object must cancel out. We say
that the object is in equilibrium.
The necessary conditions for mechanical equilibrium for a system of particles are:
(i)The vector sum of all external forces is zero;
(ii) The sum of the moments of all external forces about any line is zero.
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.
A rigid body in mechanical equilibrium is undergoing neither linear nor rotational
acceleration; however it could be translating or rotating at a constant 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.
.
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. A child sliding down a slide at constant speed would be
in mechanical equilibrium, but not in static equilibrium.
If an object is in equilibrium, 2 things must be true:
1 there is no net force acting in any direction, and
2 there is no net turning effect about any point
The size of the turning effect produced by a force is called the moment of the force.
A resultant force would cause a stationary object to start moving or an object moving
with a given velocity to speed up or slow down or change direction such that the
velocity of the object changes.
In other words, for stationary objects or objects moving with constant velocity, the
resultant force acting on the object is zero. The object is said to be in equilibrium.
If a resultant force acts on an object then that object can be brought into equilibrium
by applying an additional force that exactly balances this resultant. Such a force is
called the equilibrant and is equal in magnitude but opposite in direction to the
original resultant force acting on the object.
Consider the two objects pictured in the force diagram shown below. Note that the
two objects are at equilibrium because the forces which act upon them are balanced;
however, the individual forces are not equal to each other. The 50 N force is not equal
to the 30 N force.
An object at equilibrium is either at rest and staying at rest , or in motion and
continuing in motion with the same speed and direction. This too extends from
Newton's first law of motion.
Components of a Force
Force is a vector quantity. This means that it has its full effect in a
particular direction but that it also has reduced effects in other
directions.
The effect of a force in a direction not along its own line of action is
called a component of the force.
The process of finding the magnitudes of the components of a force is
called resolving the force into its components.
Resolving a force into vertical and horizontal components
The force,
, in the diagram can be considered to be the sum of two
forces as shown below. These forces are the vertical and horizontal
components of
.
The magnitude of the vertical component of
is given by
Fv = Fcos
The magnitude of the horizontal component of
Fh = Fcos
is given by
The magnitude of the component of a force in a direction at 90° to its own
line of action is therefore always equal to zero. This is why it is often
useful to resolve a force into its vertical and horizontal components:
these two components can then be considered as two independent
forces.
Conditions for the Equilibrium of Three Non-Parallel Forces
If we say that an object is under the influence of forces which are in
equilibrium, we mean that the object is not accelerating.
The following rules help to solve problems in which a body is acted on
by three forces.
i)
The lines of action of the three forces must all pass through the same
point.
ii) The principle of moments: the sum of all the clock-wise moments
about any point must have the same magnitude as the sum of all the
anti-clockwise moments about the same point.
iii) a) The sum of all the forces acting vertically upwards must have the
same magnitude as the sum of all the forces acting vertically
downwards
b) The sum of all the forces acting horizontally to the right must have
the same magnitude as the sum of all the forces acting horizontally
to the left.
In practice, when using condition iii), we will usually be considering the
vertical and horizontal components of forces.