Translational equilibrium exists only when the vector sum of all of the forces acting on an object is zero. This is called the first condition of equilibrium, and it is merely a reiteration of Newton’s first law. Translational motion occurs when forces cause an object to move without any rotation. The simplest pathways may be linear, such as when a skier slides down a snowy hill on a sled, or parabolic, as in the case of a cannonball shot out of a cannon.4 When the resultant force upon an object is zero, the object will not accelerate. This may mean that the object is stationary, but it could just as well mean that the object is moving with a constant nonzero velocity. Thus, an object experiencing translational equilibrium will have a constant velocity.
Rotational motion occurs when forces are applied against an object in such a way as to cause the object to rotate around a fixed pivot point, also known as the fulcrum. Application of force at some distance from the fulcrum generates torque (τ) or the moment of force. The distance between the applied force and the fulcrum is termed the lever arm. It is the torque that generates rotational motion, not the mere application of the force itself. This is because torque depends not only on the magnitude of the force but also on the length of the lever arm and the angle at which the force is applied.4 The equation for torque is a cross product:
τ = r × F
τ = rF sinθ
where r is the length of the lever arm, F is the magnitude of the force, and θ is the angle between the lever arm and force vectors. Since sin 90° = 1, this means that torque is greatest when the force applied is 90 degrees (perpendicular) to the lever arm. Knowing that sin 0° = 0 tells us that there is no torque when the force applied is parallel to the lever arm. Rotational equilibrium exists only when the vector sum of all the torques acting on an object is zero. This is called the second condition of equilibrium.5 Torques that generate clockwise rotation are considered negative, while torques that generate counter-clockwise rotation are positive. Thus, in rotational equilibrium, it must be that all of the positive torques exactly cancel out all of the negative torques. Similar to the behaviour defined by translational equilibrium, there are two possibilities of motion in the case of rotational equilibrium. Either the object is not rotating at all (that is, it is stationary), or it is rotating with a constant angular velocity.
1) Kane, Thomas R.; Levinson, David A. (1996), Dynamics Online, Sunnyvale, California: Online Dynamics. vectors
2) Resnick; Halliday; Krane (1992). Physics, Volume 1 (4th ed.). p. 83.
3) Beatty, Millard F. (2006), Principles of Engineering Mechanics, Volume 2: Dynamics—The Analysis of Motion, Mathematical Concepts and Methods in Science and Engineering, 33, Springer
4) Louis Bloomfield, Professor of Physics at the University of Virginia, How Everything Works: Making Physics Out of the Ordinary, John Wiley & Sons (2007)
5) Adkins, C.J. (1968/1983). Equilibrium Thermodynamics, third edition, McGraw-Hill, London
6) Jammer, Max (1957). Concepts of Force. Dover Publications, Inc. p. 167; footnote 14 work
7) C Hellingman (1992). “Newton’s third law revisited”. Phys. Educ. 27 (2): 112–115. Bibcode:1992 PhyEd..27.
8) Resnick & Halliday (1977). Physics (Third ed.). John Wiley & Sons. pp. 78–79. Any single force is only one aspect of a mutual interaction between two bodies.
9) Jain, Mahesh C. (2009). Textbook of Engineering Physics (Part I). p. 9. Chapter 1, p. 9