# Work

Energy is transferred through the process of work when something exerts forces on or against something else.6 This is expressed mathematically by the equation

## W = Fd cos θ

where W is work, F is the magnitude of the applied force, d is the magnitude of the displacement through which the force is applied, and θ is the angle between the applied force vector and the displacement vector. The SI unit for work is the joule. However, work and energy are not the same thing despite the similar SI unit. Work is the process by which a quantity of energy is moved from one system to another.6 Work is a dot product and as such, it is a function of the cosine of the angle between the vectors. This also means that only forces (or components of forces) parallel or antiparallel to the displacement vector will do work (that is, transfer energy). In the first example above, the force applied is vertical but the displacement is horizontal. A vertical force cannot affect horizontal motion. Vertical forces affect vertical motion and horizontal forces affect horizontal motion. Remember, work is done whenever a force or a component of a force results in a displacement. No component of the force is acting in the direction of motion when the book is moved horizontally with a constant velocity. The force and the displacement are independent, therefore, no work is done by the hand on the book.

Sloping inclines, such as hillsides and ramps, make it easier to lift objects because they distribute the required work over a larger distance, decreasing the required force. For a given quantity of work, any device that allows for work to be accomplished through a smaller applied force is thus said to provide mechanical advantage. Mechanical advantage is the ratio of magnitudes of the force exerted on an object by a simple machine (Fout) to the force actually applied on the simple machine (Fin):

## Mechanical advantage = Fout/ Fin

The mechanical advantage, because it is a ratio, is dimensionless. Reducing the force needed to accomplish a given amount of work does have a cost associated with it, as the distance through which the smaller force must be applied in order to do the work must be increased.

The work–energy theorem is a powerful expression of the relationship between work and energy. In its mechanical applications, it offers a direct relationship between the work done by all the forces acting on an object and the change in kinetic energy of that object. The net work done by forces acting on an object will result in an equal change in the object’s kinetic energy.6 In other words,

## Wnet = ΔK = Kf – Ki

This relationship is important to understand, as it allows one to calculate work without knowing the magnitude of the forces acting on an object or the displacement through which the forces act. If one calculates the change in kinetic energy experienced by an object, then—by definition—the net work done on or by an object is the same.

References

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

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