Conceptual Physics

All the force need not be in the direction of the displacement. When the force and displacement vectors are not in the same direction, only the component of the force in the direction of the displacement contributes to work. Consider the woman pulling the crate at an angle with a handle, as shown in Concept 2. Again, the crate slides along the ground. The component of the force perpendicular to the displacement contributes nothing to the work because there is no motion up or down. Perhaps subconsciously, you may have applied this concept. When you push on a heavy object that is low to the floor, like a sofa, it is difficult to slide it if you are mostly pushing down on top of it. Instead, you bend low so that more of your force is horizontal, parallel to the desired motion. The equation on the right is used to calculate how much work is done by a force. It has notation that may be new to you. The equation states that the work done equals the “dot product” of the force and displacement vectors (the name comes from the dot between the F and the ǻx). The equation is also expressed in a fashion that you will find useful: (F cosș)ǻx. The angle ș is the angle between the force and displacement vectors when they are placed tail to tail. The vectors and the angle are shown in Equation 1.

By multiplying the amount of force by cos ș, you calculate the component of the force parallel to the displacement. You may recall other cases in which you used the cosine or sine of an angle to calculate a component of a vector. In this case, you are calculating the component of one vector that is parallel to another.

The equation to the right is for a constant or average force. If the force varies as the motion occurs, then you have to break the motion into smaller intervals within which the force is constant in order to calculate the total work. The everyday use of the word “work” can lead you astray. In physics, if there is no displacement, there is no work. Suppose the woman on the right huffed and puffed and pushed the crate as hard as she could for ten minutes, but it did not move. She would certainly believe she had done work. She would be exhausted. But a physicist would say she has done zero work on the crate because it did not move. No displacement means no work, regardless of how much force is exerted. Work can be a positive or negative value. Positive work occurs when the force and displacement vectors point in the same direction. Negative work occurs when the force and displacement vectors point in opposite directions. If you kick a stationary soccer ball, propelling it downfield, you have done positive work on the ball because the force and the displacement are in the same direction. When a goalie catches a kicked ball, negative work is done by the force from the goalie’s hands on the ball. The force on the ball is in the opposite direction of the ball’s displacement, with the result that the ball slows down.

The sign of work can be calculated with the equation to the right. When force and displacement point in the same direction, the angle between them is 0°, and the cosine of0° is positive one. When force and displacement point in opposite directions, the angle between the vectors is 180°, and the cosine of 180° is negative one. This mathematically confirms the points made above: Force in the direction of motion results in positive work; force opposing the motion results in negative work. Work is a scalar quantity, which means it has magnitude but no direction. The joule is the unit for work. The units that make up the joule are kg·m^2 /s^2 and come from multiplying the unit for force (kg·m/s^2 ) by the unit for displacement (m). If several forces act on an object, each of them can do work on the object. You can calculate the net work done on the object by all the forces by calculating the net force and using the equation in Equation 1.

Force at angle to displacement

Only force component along displacement contributes to work

W = F ·ǻx = (F cos ș)ǻx

W = work

F = force

ǻx= displacement

ș = angle between force and

displacement

Unit: joule (J)

How much work does the woman

do on the crate?

W = (F cos ș)ǻx

W = (120 N)(cos 0°)(3.0 m)

W = (120 N)(1)(3.0 m) = 360 J