When multiple forces act on a body, the (vector) sum of those forces gives the net force, which is the force we substitute in Newton’s second law of motion to get the equation of motion of the body. If all forces sum up to zero, there will be no acceleration, and the body retains whatever velocity it had before. Statics is the study of objects that are neither currently moving nor experiencing a net force, and thus remain stationary. You might expect that this study is easier than the dynamical case when bodies do experience a net force, but that just depends on context. Imagine, for example, a jar filled with marbles: they aren’t moving, but the forces acting on the marbles are certainly not zero, and also not uniformly distributed.
Even if there is no net force, there is no guarantee that an object will exhibit no motion: if the forces are distributed unevenly along an extended object, it may start to rotate. Rotations always happen around a stationary point, known as the pivot. Only a force that has a component perpendicular to the line connecting its point of action to the pivot (the arm) can make an object rotate. The corresponding angular acceleration due to the force depends on both the magnitude of that perpendicular component and the length of the arm,and is known as the moment of the force or the torque \(\tau\). The magnitude of the torque is therefore given by \(Fr sin\theta\), where \(F\) is the magnitude of the force, r the length of the arm, and \(\theta\) the angle between the force and the arm. If we write the arm as a vector r pointing from the pivot to the point where the force acts, we find that the magnitude of the torque equals the cross product of \(r\) and \(F\):
The direction of rotation can be found by the right-hand rule from the direction of the torque: if the thumb of your right hand points along the direction of \(\tau\), then the direction in which your fingers curve will be the direction in which the object rotates due to the action of the corresponding force \(F\).
We will study rotations in detail in Chapter 5. For now, we’re interested in the case that there is no motion, neither linear nor rotational, which means that the forces and torques acting on our object must satisfy the stability condition: for an extended object to be stationary, both the sum of the forces and the sum of the torques acting on it must be zero.