5. STATIC EQUILIBRIUM.
Key words: Static Equilibrium, First Condition of Equilibrium, Torque,
Lever Arm, and Second Condition of Equilibrium.
We studied two basic branches of the mechanics – the kinematics and
dynamics. Now we will begin to study the third basic branch of the
mechanics that is called Statics. Statics is the determination of the forces
within a structure at equilibrium. A body at rest (or in an uniform motion) is
said to be in equilibrium. These branch of the Mechanics is very important
for applications in civil engineering, architecture, medicine and so on.
5.1. First Condition of equilibrium.
The object will be in equilibrium with respect of the translational motion
when net force acting on it is zero. Actually at rest velocity of object is zero,
so the acceleration is also zero. Using the second Newton’s Law of motion,
we can get the First Condition of Equilibrium.
ΣF=0
(5-1)
We will use (5-1) in components. In 2 dimensional problems, the first
condition of equilibrium can be written as following:
Σ Fx = 0
(5-1a)
Σ Fy = 0
(5-1b)
These relationships are necessary condition for an object to be in
equilibrium. The 1st condition of equilibrium is not always the sufficient
condition. Really, consider two forces with equal magnitudes and opposite
directions, but directed not along the same straight line. The first condition
of equilibrium will be satisfied in this case. Nevertheless, the object will not
be at rest and will have a tendency to rotate. Therefore we need additional
condition of equilibrium with respect to a rotation. To introduce additional
condition we need to introduce concept of torque.
5.2. Torque.
It is understandable that to make object rotating about an axis of rotation we
need a force. But it is also important not only the magnitude of this force but
also the perpendicular distance from the axis of rotation to the line along
which a force is acting (line of action of a force). This perpendicular is
called lever arm. The rotating effect of a force also depends on the angle
between the lever arm and a force. We will take all these circumstances into
account if we introduced new physical quantity Torque that plays the same
role for rotational motion as a force plays for translational motion. The
magnitude of the torque can be determined as follows:
τ= r F sinθ
(5-2)
Where r is the distance from the axis of rotation to the point where force F
applied; F is the magnitude of the applied force; θ is the angle between the
line of action of the force and a line connecting axis of rotation and the point
at which the force is applied (see Fig. 5.1). Usually we will use the following
sign convention: a positive sign is assigned to torques that act to rotate the
object counterclockwise, and a negative sign to torques that act to rotate the
object clockwise. From relationship (5-2) we can deduce that unit of torque
in SI system is m N. There is no special name for this unit.
5.3. The second condition of equilibrium.
Now we can formulate the Second Condition of Equilibrium: The sum of
the torques acting on a body must be zero. In the 2D problems, we usually
suppose that axis of rotation is directed along Z-axis and all forces are lying
in the plane XOY. The all set of equations describing conditions of
equilibrium can be written for this case as follows:
Σ Fx = 0
(5.3a)
Σ Fy = 0
(5.3b)
Στ=0
(5.3c)
Sum of X—components and sum of Y-components of all forces acting on an
object and sum of torques of all these forces applied in the XOY plane must
be equal zero. There are some specifics related to the Second condition of
equilibrium. Torque of the force can be calculated with respect to different
axis of rotation (real or virtual). So, with respect to which axis of rotation net
torque must be zero? The answer is: in equilibrium, net torque must be zero
with respect to any axis of rotation. Therefore, solving problems, we can
choose any axis of rotation that makes our calculation easier. For example,
we can choose as axis of rotation point at which unknown force is applied.
In this case, the torque of this force will be zero. The number of unknown in
the set of equations will be decreased by this way. It does not matter is the
chosen axis real or virtual.