A doorknob is located as far as possible from the door’s hinge line for a good reason. If you want to open a heavy door, you must certainly apply a force; that alone, however, is not enough. Where you apply that force and in what direction you push are also important. If you apply your force nearer to the hinge line than the knob, or at any angle other than 90° to the plane of the door, you must use a greater force to move the door than if you apply the force at the knob and perpendicular to the door’s plane.
Fig. 10-16 (a) A force 
acts at point P on a rigid body that is free to rotate about an axis through O; the axis is perpendicular to the plane of the cross section shown here. (b) The torque due to this force is (r)(F sin 
). We can also write it as rFt, where Ft is the tangential component of . (c) The torque can also be written as r⊥F, where r⊥ is the moment arm of .
Figure 10-16a shows a cross section of a body that is free to rotate about an axis passing through O and perpendicular to the cross section. A force is applied at point P, whose position relative to O is defined by a position vector 
. The directions of vectors and make an angle with each other. (For simplicity, we consider only forces that have no component parallel to the rotation axis; thus, is in the plane of the page.)
To determine how results in a rotation of the body around the rotation axis, we resolve into two components (Fig. 10-16b). One component, called the radial component Fr, points along . This component does not cause rotation, because it acts along a line that extends through O. (If you pull on a door parallel to the plane of the door, you do not rotate the door.) The other component of , called the tangential component Ft, is perpendicular to and has magnitude Ft = F sin . This component does cause rotation. (If you pull on a door perpendicular to its plane, you can rotate the door.)
The ability of to rotate the body depends not only on the magnitude of its tangential component Ft, but also on just how far from O the force is applied. To include both these factors, we define a quantity called torque τ as the product of the two factors and write it as
Two equivalent ways of computing the torque are
and
where r⊥ is the perpendicular distance between the rotation axis at O and an extended line running through the vector (Fig. 10-16c). This extended line is called the line of action of , and r⊥ is called the moment arm of . Figure 10-16b shows that we can describe r, the magnitude of , as being the moment arm of the force component Ft.
Torque, which comes from the Latin word meaning “to twist,” may be loosely identified as the turning or twisting action of the force . When you apply a force to an object—such as a screwdriver or torque wrench—with the purpose of turning that object, you are applying a torque. The SI unit of torque is the newton-meter (N · m). Caution: The newton-meter is also the unit of work. Torque and work, however, are quite different quantities and must not be confused. Work is often expressed in joules (1 J = 1 N · m), but torque never is.
In the next chapter we shall discuss torque in a general way as being a vector quantity. Here, however, because we consider only rotation around a single axis, we do not need vector notation. Instead, a torque has either a positive or negative value depending on the direction of rotation it would give a body initially at rest: If the body would rotate counterclockwise, the torque is positive. If the object would rotate clockwise, the torque is negative. (The phrase “clocks are negative” from Section 10-2 still works.)
Torques obey the superposition principle that we discussed in Chapter 5 for forces: When several torques act on a body, the net torque (or resultant torque) is the sum of the individual torques. The symbol for net torque is τnet.
CHECK POINT 6 The figure shows an overhead view of a meter stick that can pivot about the dot at the position marked 20 (for 20 cm). All five forces on the stick are horizontal and have the same magnitude. Rank the forces according to the magnitude of the torque they produce, greatest first.
Leave a Reply