Couple M
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Couple M
From Wikipedia, the free encyclopedia
Pair of equal and opposite forces acting along different lines of action of force on a rigid body
Ο
=
F
d
{\displaystyle \tau =Fd}
Ο
=
|
d
Γ
F
|
.
{\displaystyle \mathbf {\tau } =|\mathbf {d} \times \mathbf {F} |.}
^ Jump up to: a b c Dynamics, Theory and Applications by T.R. Kane and D.A. Levinson, 1985, pp. 90-99: Free download
^ Physics for Engineering by Hendricks, Subramony, and Van Blerk, page 148, Web link
^ Engineering Mechanics: Equilibrium , by C. Hartsuijker, J. W. Welleman, page 64 Web link
^ Jump up to: a b c Augustus Jay Du Bois (1902). The mechanics of engineering, Volume 1 . Wiley. p.Β 186 .
^ J.L. Ericksen (1979) Timoshenko Acceptance Speech at iMechanica.org site for mechanicians
In mechanics , a couple is a system of forces with a resultant (a.k.a. net or sum) moment but no resultant force. [1]
A better term is force couple or pure moment . Its effect is to create rotation without translation . In rigid body mechanics , force couples are free vectors , meaning their effects on a body are independent of the point of application.
The resultant moment of a couple is called a torque . This is not to be confused with the term torque as it is used in physics, where it is merely a synonym of moment. [2] Instead, torque is a special case of moment. Torque has special properties that moment does not have, in particular the property of being independent of reference point, as described below.
A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment.
The simplest kind of couple consists of two equal and opposite forces whose lines of action do not coincide. This is called a "simple couple". [1] The forces have a turning effect or moment called a torque about an axis which is normal (perpendicular) to the plane of the forces. The SI unit for the torque of the couple is newton metre .
If the two forces are F and β F , then the magnitude of the torque is given by the following formula:
The magnitude of the torque is equal to F β’ d , with the direction of the torque given by the unit vector
e
^
{\displaystyle {\hat {e}}}
, which is perpendicular to the plane containing the two forces and positive being a counter-clockwise couple. When d is taken as a vector between the points of action of the forces, then the torque is the cross product of d and F , i.e.
The moment of a force is only defined with respect to a certain point P (it is said to be the "moment about P ") and, in general, when P is changed, the moment changes. However, the moment (torque) of a couple is independent of the reference point P : Any point will give the same moment. [1] In other words, a torque vector, unlike any other moment vector, is a "free vector". (This fact is called Varignon 's Second Moment Theorem .) [3]
The proof of this claim is as follows: Suppose there are a set of force vectors F 1 , F 2 , etc. that form a couple, with position vectors (about some origin P ), r 1 , r 2 , etc., respectively. The moment about P is
Now we pick a new reference point P' that differs from P by the vector r . The new moment is
However, the definition of a force couple means that
This proves that the moment is independent of reference point, which is proof that a couple is a free vector.
A force F applied to a rigid body at a distance d from the center of mass has the same effect as the same force applied directly to the center of mass and a couple Cβ = Fd . The couple produces an angular acceleration of the rigid body at right angles to the plane of the couple. [4] The force at the center of mass accelerates the body in the direction of the force without change in orientation. The general theorems are: [4]
Couples are very important in mechanical engineering and the physical sciences. A few examples are:
In a liquid crystal it is the rotation of an optic axis called the director that produces the functionality of these compounds. As Jerald Ericksen explained:
At first glance, it may seem that it is optics or electronics which is involved, rather than mechanics. Actually, the changes in optical behavior, etc. are associated with changes in orientation. In turn, these are produced by couples. Very roughly, it is similar to bending a wire, by applying couples. [5]
A special case of moments is a couple. A couple consists of two
parallel forces that are equal in magnitude, opposite in sense and do not
share a line of action. It does not produce any translation, only rotation.
The resultant force of a couple is zero. BUT, the resultant of a couple
is not zero; it is a pure moment.
For example, the forces that two hands apply to turn a steering wheel
are often (or should be) a couple. Each hand grips the wheel at points on
opposite sides of the shaft. When they apply a force that is equal in magnitude
yet opposite in direction the wheel rotates. If both hands applied a force
in the same direction, the sum of the moments created by each force would
equal zero and the wheel would not rotate. Instead of rotating around the
shaft, the shaft would be loaded with a force tending to cause a translation
with a magnitude of twice F. If the forces applied by the two hands were
unequal, there would again be an unbalanced force creating a translation
of the "system." A pure couple always consists of two forces equal
in magnitude.
The moment of a couple is the product of the magnitude of one of the
forces and the perpendicular distance between their lines of action. M
= F x d. It has the units of kip-feet, pound-inches, KN-meter, etc.
The magnitude of the moment of a couple is the same for all points in the
plane of the couple. A couple may be moved anywhere in its plane or a parallel
plane without changing its external effect. The magnitude of the couple
is independent of the reference point and its tendency to create a rotation
will remain constant. This can be illustrated with the simple illustration
of a bar with a length d that is pinned at its midpoint. Two parallel
forces of equal magnitude, opposite in sense are applied at the ends of
the bar. The magnitude of the moment generated by the couple of the forces
F , relative to the pin in the illustration, is equal to
The magnitude of the couple of the forces F relative to point
"O" is
(F)(d+x) - (F)(x)
(F)(d) + (F)(x) - (F)(x)
Again, it can be seen that the magnitude of the couple is independent
of the reference location. It is always equal to (F)(d)!
The resultant of a number of couples is their algebraic sum. A couple
CANNOT be put in equilibrium by a single force! A couple can only
be put in equilibrium by a moment or another couple of equal magnitude and
opposite direction anywhere in the same plane or in a parallel plane. If
a single force is added to the system that balances the sum of the moments,
one of the other two equations of equilibrium will not be satisfied. A couple
maintains the internal equilibrium of a simple beam or of many other simple
structural systems. The concept is very important to the further study of
structural behaviour.
There are many examples of couples in the built world. Some structures
are clear manifestations of the couple; others hide the couple inside oftheir
structural elements.
The Saltash Bridge by Brunel is a structure which allows the couple at midspan
to be clearly read. The structure illustrates the equilibrium developed
by the compressive force of the top chord and the equal and opposite tensile
force of the suspension chain. These two forces create a couple at midspan.
The magnitude of the couple would be the force F c or F t
times the distance seperating the two forces, d . This is the same
type of couple that is hidden inside of solid beams.
Would the magnitude of the couple change as a train rolled over the bridge?
Why or Why not?
Here is an example of a common street lamp. The lamp creates a moment
with the magnitude determined by the weight of the lamp times the moment
arm,d. This moment is resisted by a couple generated by the tubes of the
supporting arm. The two forces of the couple can be seen in the compression
and tension forces that are seperated by a much smaller distance. These
act to put the system back into equilibrium.
What other forces are required to keep the system in equilibrium?
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