Torque
The
second condition necessary to achieve equilibrium involves avoiding
accelerated rotation (maintaining a constant angular velocity. A
rotating body or system can be in equilibrium if its rate of rotation is
constant and remains unchanged by the forces acting on it. To
understand what factors affect rotation, let us think about what happens
when you open an ordinary door by rotating it on its hinges.
Torque is the turning or twisting effectiveness of a force…
The magnitude,
direction, and point of application of the force are incorporated into
the definition of the physical quantity called torque. Torque
is the rotational equivalent of a force. It is a measure of the
effectiveness of a force in changing or accelerating a rotation
(changing the angular velocity over a period of time). In equation form,
the magnitude of torque is defined to be
where τ (the Greek letter tau) is the symbol for torque, r is the distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force and the vector directed from the point of application to the pivot point, as seen in Figure 9.7 and Figure 9.8. An alternative expression for torque is given in terms of the perpendicular lever arm r⊥ as shown in Figure 9.7 and Figure 9.8, which is defined as
so that
The perpendicular lever arm r⊥ is the shortest distance from the pivot point to the line along which F acts; it is shown as a dashed line in Figure 9.7 and Figure 9.8. Note that the line segment that defines the distance r⊥ is perpendicular to F, as its name implies. It is sometimes easier to find or visualize r⊥ than to find both r and θ. In such cases, it may be more convenient to use τ = r⊥ F rather than τ = rF sin θ for torque, but both are equally valid.
The SI unit of torque
is newtons times meters, usually written as N · m. For example, if you
push perpendicular to the door with a force of 40 N at a distance of
0.800 m from the hinges, you exert a torque of 32 N · m(0.800 m × 40 N ×
sin 90°) relative to the hinges. If you reduce the force to 20 N, the
torque is reduced to 16 N · m, and so on.
The torque is
always calculated with reference to some chosen pivot point. For the
same applied force, a different choice for the location of the pivot
will give you a different value for the torque, since both r and θ
depend on the location of the pivot. Any point in any object can be
chosen to calculate the torque about that point. The object may not
actually pivot about the chosen “pivot point.”
Note that for
rotation in a plane, torque has two possible directions. Torque is
either clockwise or counterclockwise relative to the chosen pivot point,
as illustrated for points B and A, respectively, in Figure 9.8.
If the object can rotate about point A, it will rotate
counterclockwise, which means that the torque for the force is shown as
counterclockwise relative to A. But if the object can rotate about point
B, it will rotate clockwise, which means the torque for the force shown
is clockwise relative to B. Also, the magnitude of the torque is
greater when the lever arm is longer.
Now, the second condition necessary to achieve equilibrium is that the net external torque on a system must be zero.
An external torque is one that is created by an external force. You can
choose the point around which the torque is calculated. The point can
be the physical pivot point of a system or any other point in space—but
it must be the same point for all torques. If the second condition (net
external torque on a system is zero) is satisfied for one choice of
pivot point, it will also hold true for any other choice of pivot point
in or out of the system of interest. (This is true only in an inertial
frame of reference.) The second condition necessary to achieve
equilibrium is stated in equation form as
where
net means total. Torques, which are in opposite directions are assigned
opposite signs. A common convention is to call counterclockwise (ccw)
torques positive and clockwise (cw) torques negative.
When two children balance a seesaw as shown in Figure 9.9,
they satisfy the two conditions for equilibrium. Most people have
perfect intuition about seesaws, knowing that the lighter child must sit
farther from the pivot and that a heavier child can keep a lighter one
off the ground indefinitely.
