A force may be thought of as a push or
pull in a specific direction. When a force is applied to an object,
the resulting motion of the object depends on where the
force is applied and how the object is confined.
If the object is unconfined and the force is applied through the
center of gravity,
the object moves in pure
translation,
as described by Newton's
laws of motion.
If the object is confined (or pinned) at some location called a
pivot, the object rotates
about the pivot, but does not translate.
The force is transmitted through the pivot
and the details of the rotation depend on the distance from
the applied force to the pivot.
If the object is unconfined and the force is applied at some
distance from the center of gravity, the object both translates and
rotates
about the center of gravity.
The details of the rotation depend on the distance from the
applied force to the center of gravity.
The motion of flying objects and spacecraft is
described
by this third type of motion; a combination of translation and rotation.
A force F is a
vector quantity,
which means that it has both a magnitude and
a direction associated with it. The
direction of the force
is important because the translation of the object
is in the same direction as the force.
The product of the force and the perpendicular distance to the
center of gravity for an unconfined object,
or to the pivot for a confined object, is
called the torque or the moment.
A torque is also a vector quantity and produces a rotation
in the same way that a force produces a translation. Namely, an object at
rest, or rotating at a constant angular velocity, will continue to do so
until it is subject to an external torque. A torque produces a change
in angular velocity which is called an angular acceleration.
On the figure, we show four examples of torques to illustrate the basic
principles governing torques.
In the examples, we will study the action of torques on a confined
object. If the object were unconfined, the pivot p would be replaced
by the center of gravity cg.
In each example a blue weight W generates a force F on a red bar,
which is called an arm.
The distance L used to determine the torque T is the distance from the
pivot to the force, but measured perpendicular to the
direction of the force.
In Example 1, the force is applied perpendicular
to the arm. In this case, the perpendicular distance is the length of the
bar and the torque is equal to the product of the length and the force.
T = F * L
In Example 2, the same force is applied to the arm,
but the force now acts right through the
pivot. In this case, the distance from the pivot perpendicular to the force
is zero. So, in this case:
T = 0
Think of a hinged door. If you push on
the edge of the door, towards the hinge, the door doesn't move
because the torque is zero.
Example 3 is the general case in which the force is applied
at some angle a to
the arm. The perpendicular distance is given by
trigonometry
as the length of the arm (L) times the
cosine (cos)
of the angle.
The torque is then given by:
T = F * L * cos(a)
Examples 1 and 2 can be derived from this general formula,
since the
cosine of
0 degrees is 1.0 (Example 1), and the cosine of 90 degrees is 0.0 (Example 2).
In Example 4, the pivot has been moved from the end of the bar to
a location near the middle of the bar. Weights are added to both sides
of the pivot.
To the right a single weight W produces a force F1 acting
at a distance L1 from the pivot. This creates a torque T1 equal to the
product of the force and the distance.
T1 = F1 * L1
To the left of the
pivot two weights W produce a force F2 at a distance L2.
This produces
a torque T2 in a direction opposite from T1 because the distance
is in the opposite direction.
T2 = F2 * L2
If the system were in equilibrium,
or balanced, the torques would be equal and no net torque would act on the system.
T1 = T2 or T1  T2 = 0
F1 * L1 = F2 * L2
If the system is not in equilibrium, or unbalanced, the bar rotates
about the pivot in the direction of the higher torque.
If F2 = 2 * F1,
what is the relation between L1 and L2 to balance the system? If F2 = 2 * F1,
and L1 = L2, in which direction would the system rotate?
Aerospace engineers use torques
to stabilize and control flying objects.
On airplanes, the control surfaces produce
aerodynamic forces
which are applied at some distance from the
center of gravity
and therefore
cause the aircraft to rotate. The same effect
is used on rockets flying within the atmosphere. The
aerodynamic forces
act through the
center of pressure
to provide
stability
during powered flight.
On most full scale rockets, the
thrust
of the nozzle is
gimbaled,
or angled, to produce a torque to maneuver the rocket
in flight. In orbit, small thrusters can torque the
spacecraft to point in a desired direction.
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