![]() These three variables are half the number that I would have to say if I were to state each component of the force and displacement vectors, of which there will be six if we are working in 3D. I can easily state that the “work on an object is equal to a 45 N force times 0.25 m displacement at an angle of 45. The benefit of the angular method seems to be that it is easier to speak, nothing more than that. The other method to compute work is from the magnitudes of the force and displacement multiplied by the cosine of the angle, which is the angle made from the direction of the displacement to the direction of the force. Otherwise we may have the two dimensions of a Polar Coordinate system, and. In two dimensions we will have x- and y-coordinates (typically speaking) when working from a Cartesian Coordinate System in two dimensions, or x-, y-, and z-coordinates if working in three dimensions. ![]() We can consider work in vector notation as where we will multiply each component of the displacement with the corresponding parallel component of the force. We must choose depending on what best fits the situation. Work Equation: Vectors or Angles?Īs you can see in the second set of expressions, there are two different ways we can calculate work: we can consider work from the perspective of magnitudes and angles, or from vector notation. However, the total work acting on an object between two points is the integral of the dot product at each point along the path, the sum of each individual infinitesimal work calculation, essentially. The infinitesimal work is the dot product of the force and displacement vectors. Such as the example I used in the previous article with the changing force of gravity on a rocket ship leaving our atmosphere. Which is used when the motion of the object (as a result of the force acting upon it and work being done) follows a curved trajectory, rather than a linear one, also where force is variably depending on position. To this end we will determine work using integral calculus, calculating the sum of innumerable minute displacements with a constantly varying direction between two points that we will arbitrarily call A and B. Then we will want to use a different method, calculating the sum of infinitesimal displacements and changing forces between two points. In these realistic and complex real-world situations, the forces are always changing. Or else the motion of a ship on stormy seas, rising and falling but the terrain of the waters surface between waves, the force of the waves, velocity across the surface, and the force of gravity which causes them to fall as then cut down a wave. This is the case if we are analyzing the flight of a bird for example, because the wings give a regular upward force with each down-stroke of the wings, while each upstroke gives a small downward force, with gravity causing a significant downward displacement between each down-stroke of the wings, while the force of the wind is buffeting the bird slightly one direction or another. However, what if the force acting on an object is constantly changing? In which case the direction of an object’s displacement (and the work done by a force upon it) are always changing which tells us also that the work being done on the object may be changing in both magnitude and direction from moment to moment. ![]() In both of those forms a single force with resultant work and displacement is implied. The first version of the work equation is more for calculating general average values for work measured in joules, while the second is applicable for measuring an instantaneous value for the work vector. Integral Calculus Form of the Work Equation However, if we want to calculate for instantaneous work at a given moment in time, we signify the use of the calculus operation of differentiation with the inclusion of the symbol “d” to tell us that we are computing infinitesimal work from an infinitesimal displacement as in the following expression This is the simplest form of the equation which we would in the early stages of physics development write simply as In algebraic terms work is the product of the force vector acting on the object by the displacement of the object which we might write as. The Fundamental Forms of the Work Equation Thus means that if either the force or displacement increase (or both) then the work being done on the object increases also. ![]() We also know based on the limited information I gave you above that work is directly proportional to force and displacement, because work is the product of these quantities. ![]()
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