A free body
diagram is a diagram showing all the forces and moments of forces acting on an
object. We distinguish between two types of objects:
1. Particles
that have no spatial extent and thus have no moment arm (d). An example of this
would be a satellite orbiting the earth because the spatial extent of the
satellite is very small compared to the distance from the earth to the
satellite or the radius of the earth. Particles do not have moments of forces
and thus do not rotate in response to a force.
2. Rigid bodies
that have a finite dimension and thus has a moment arm (d) associated with each
applied force. Rigid bodies have moments of forces and thus can rotate in
response to a force.
There are
several types of forces that act on particles or rigid bodies:
1. Rope, cable,
etc. – Force (tension) must be along line of action; no moment of force (1 unknown
force)
2. Rollers,
frictionless surface – Force must be perpendicular to the surface; no moment of
force (1 unknown force). There cannot be a force parallel to the surface
because the roller would start rolling! Also the force must be away from the
surface towards the roller (in other words the roller must exert a force on the
surface), otherwise the roller would lift off of the surface.
3. Frictionless
pin or hinge – Force has components both parallel and perpendicular to the line
of action; no moment of force (2 unknown forces) (note that the coordinate
system does not need to be parallel and perpendicular to either the gravity
vector or the bar)
4. Fixed support
– Force has components both parallel and perpendicular to line of action plus a
moment of force (2 unknown forces, 1 unknown moment force). Note that for our simple
statics problems with 3 degrees of freedom, if there is one fixed support then
we already have 3 unknown quantities and the rest of our free body cannot have
any unknown forces if we are to employ statics alone to determine the forces.
In other words, if the free body has any additional unknown forces the system
is statically indeterminate as will be discussed shortly.
5. Contact
friction – Force has components both parallel (F) and perpendicular (N) to
surface, which are related by F = μN, where μ is the coefficient of friction,
which is usually assigned separate values for static (no sliding) (μs)
and dynamic (sliding) (μd) friction, with the latter being lower. (2
unknown forces coupled by the relation F = μN). μ depends on both of the surfaces
in contact. Most dry materials have friction coefficients between 0.3 and 0.6
but Teflon in contact with Teflon, for example, can have a coefficient as low
as 0.04. Rubber (e.g. tires) in contact with other surfaces (e.g. asphalt) can
yield friction coefficients of almost 2.
Actually
the statement F = μsN for static friction is not correct at all,
although that’s how it’s almost always written. Consider the figure on the
right, above. If there is no applied force in the horizontal direction, there
is no need for friction to counter that force and keep the block from sliding,
so F = 0. (If F ≠ 0, then the object would start moving even though there is no
applied force!) Of course, if a force were applied (e.g. from right to left, in
the –x direction) then the friction force at the interface between the block
and the surface would counter the applied force with a force in the +x direction
so that ΣFx = 0. On the other hand, if a force were applied from
left to right, in the +x direction) then the friction force at the interface
between the block and the surface would counter the applied force with a force
in the -x direction so that ΣFx = 0. The expression F = μsN
only applies to the maximum magnitude of the static friction force. In other
words, a proper statement quantifying the friction force would be |F| ≤ μsN,
not F = μsN. If any larger force is applied then the block would
start moving and then the dynamic friction force F = μdN is the applicable
one – but even then this force must always be in the direction opposite the
motion – so |F| = μdN is an appropriate statement. Another, more
precise way of writing this would be
where
v is the velocity of the block and v is the magnitude of this velocity, thus 
is a unit vector
in the direction of motion.
Special note:
while ropes, rollers and pins do not exert a moment of force at the point of
contact, you can still sum up the moments of force acting on the free body at
that point of contact. In other words, ΣMA = 0 can be used even if
point A is a contact point with a rope, roller or pin joint, and all of the
other moments of force about point A (magnitude of force x distance from A to
the line of action of that force) are still non-zero. Keep in mind that A can
be any point, within or outside of the free body. It does not need to be a
point where a force is applied, although it is often convenient to use one of
those points as shown in the examples below.
Statically
indeterminate system
Of course, there
is no guarantee that the number of force and moment of force balance equations will
be equal to the number of unknowns. For example, in a 2D problem, a beam
supported by one pinned end and one roller end has 3 unknown forces and 3
equations of static equilibrium. However, if both ends are pinned, there are 4
unknown forces but still only 3 equations of static equilibrium. Such a system
is called statically indeterminate and requires additional information beyond the
equations of statics (e.g. material stresses and strains, discussed in the next
chapter) to determine the forces.
The Previous Article Is "Forces in the Structures" Here: Go Here
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