Chapter 2 Review of Forces and Moments - Brown University
Chapter 2 Review of Forces and Moments - Brown University
Chapter 2 Review of Forces and Moments - Brown University
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2.3 Force Couples, Pure <strong>Moments</strong>, Couples <strong>and</strong> Torques<br />
We have seen that a force acting on a rigid body has two effects: (i) it tends to move the body; <strong>and</strong> (ii) it<br />
tends to rotate the body.<br />
A natural question arises – is there a way to rotate a body without moving it? And is there a kind <strong>of</strong> force<br />
that causes only rotation without translation?<br />
The answer to both questions is yes.<br />
2.3.1 Force couples<br />
A system <strong>of</strong> forces that exerts a resultant moment, but no resultant force, is<br />
called a force couple.<br />
The simplest example <strong>of</strong> a force couple consists <strong>of</strong> two equal <strong>and</strong> opposite<br />
forces +F <strong>and</strong> −F acting some distance apart. Suppose that the force −F<br />
acts at position r −<br />
while the force +F acts at position r +<br />
The resultant<br />
moment is<br />
M = r+ × F+ r−× ( −F)<br />
= ( r+ − r−)<br />
× F<br />
Of course, the vector r+<br />
− r<br />
−<br />
is just the vector from the point where −F acts<br />
to the point where +F acts. This gives a quick way to calculate the moment<br />
induced by a force couple:<br />
j<br />
r_<br />
-F<br />
r +<br />
r + - r_<br />
F<br />
i<br />
Two equal <strong>and</strong> opposite<br />
forces exert a purely<br />
rotational force.<br />
The moment induced by two equal <strong>and</strong> opposite forces is equal to the moment <strong>of</strong> one force about the point<br />
<strong>of</strong> action <strong>of</strong> the other. It doesn’t matter which force you use to do this calculation.<br />
Note that a force couple<br />
(i) Has zero resultant force<br />
(ii) Exerts the same resultant moment about all points.<br />
Its effect is to induce rotation without translation.<br />
The effect <strong>of</strong> a force couple can therefore be characterized by a single vector moment M. The physical<br />
significance <strong>of</strong> M is equivalent to the physical significance <strong>of</strong> the moment <strong>of</strong> a force about some point.<br />
The direction <strong>of</strong> M specifies the axis associated with the rotational force. The magnitude <strong>of</strong> M specifies<br />
the intensity <strong>of</strong> the rotational force.<br />
There are many practical examples <strong>of</strong> force systems that are best thought <strong>of</strong> as force-couple systems.<br />
They include<br />
1. The forces exerted by your h<strong>and</strong> on a screw-driver<br />
2. The forces exerted by the tip <strong>of</strong> a screw-driver on the head <strong>of</strong> a screw<br />
3. The forces exerted by one part <strong>of</strong> a constant velocity joint on another<br />
4. Drag forces acting on a spinning propeller