The gravitational force between two objects is an attractive force, which does positive work when the objects get closer together. Gravity does positive work (20 J) when the book moves down from the shelf. There is zero difference in height for any path that begins and ends at the same place on the shelf, so W = 0.
The bottom line is that you need to analyze each particular case to determine the work done by the forces, whether positive, negative or zero. In this case, kinetic friction exerted by the rug on the object could be in the same direction as the displacement of the object, relative to the floor, and do positive work. You can pull the rug out from under an object in such a way that it slides backward relative to the rug, but forward relative to the floor. If you’re driving your car at the speed limit on a straight, level stretch of highway, the negative work done by air resistance is balanced by the positive work done by the static friction of the road on the drive wheels. Static friction can keep someone from sliding off a sled when the sled is moving and perform positive work on the person. As an external force, static friction can do work. The force of static friction does no work in the reference frame between two surfaces because there is never displacement between the surfaces. Where | l A B | | l A B | is the path length on the surface. In this case, we can factor out the force the remaining integral is just the total displacement, which only depends on the end points A and B, but not on the path between them: The simplest work to evaluate is that done by a force that is constant in magnitude and direction. Work Done by Constant Forces and Contact Forces In the English system, still used in the United States, the unit of force is the pound (lb) and the unit of distance is the foot (ft), so the unit of work is the foot-pound ( ft This combination is called a joule, for historical reasons that we will mention later, and is abbreviated as J. The units of work are units of force multiplied by units of length, which in the SI system is newtons times meters, N The maximum work is done by a given force when it is along the direction of the displacement ( cos θ = ± 1 cos θ = ± 1), and zero work is done when the force is perpendicular to the displacement ( cos θ = 0 cos θ = 0). As a result, the work done by a force can be positive, negative, or zero, depending on whether the force is generally in the direction of the displacement, generally opposite to the displacement, or perpendicular to the displacement. The components of a vector can be positive, negative, or zero, depending on whether the angle between the vector and the component-direction is between 0 ° 0 ° and 90 ° 90 ° or 90 ° 90 ° and 180 ° 180 °, or is equal to 90 ° 90 °. Recall that the magnitude of a force times the cosine of the angle the force makes with a given direction is the component of the force in the given direction. From the properties of vectors, it doesn’t matter if you take the component of the force parallel to the displacement or the component of the displacement parallel to the force-you get the same result either way. In words, you can express Equation 7.1 for the work done by a force acting over a displacement as a product of one component acting parallel to the other component.
Which choice is more convenient depends on the situation. In two dimensions, these were the x- and y-components in Cartesian coordinates, or the r- and φ φ-components in polar coordinates in three dimensions, it was just x-, y-, and z-components. We could equally well have expressed the dot product in terms of the various components introduced in Vectors. We choose to express the dot product in terms of the magnitudes of the vectors and the cosine of the angle between them, because the meaning of the dot product for work can be put into words more directly in terms of magnitudes and angles. The infinitesimal work is the dot product of these two vectors the total work is the integral of the dot product along the path. The force acting on a particle and its infinitesimal displacement are shown at one point along the path between A and B.
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