Vector fields and Line integrals, Part 2

Vector Fields and Line integrals

Part 2: The Concept of Work in Physics

One of the fundamental precepts of Newtonian physics is that the total energy of any system is conserved. A popular statement of this idea is, ''Energy is neither created nor destroyed, it is simply transferred into different forms.''

For example, in Newton's apocryphal story of the apple falling from a tree, the apple might be thought of as gaining energy because it goes from stationary (i.e. hanging motionless on a tree) to moving (i.e. falling rapidly towards Newton's head). The apple appears to have gained energy of motion (or kinetic energy). But according to the Newtonian view, the apple has not gained energy. Rather, the apple had energy all along -- gravitational potential energy. The apple is not creating energy out of nothing as it's speed (and kinetic energy) increase. Instead, the apple's potential energy is being converted into kinetic energy.

The mechanism for this energy exchange is the physical idea of work. What is causing the apple to fall, thereby gaining speed (and kinetic energy), while it loses height (and potential energy)? Newton's answer was the force of gravity. Gravity is "doing work" on the apple by exerting a force.

The falling apple illustrates a situation in which the force doing work acts in the same direction as the direction of motion. However, a force might be exerted in a different direction from the direction of motion, as, for example, when a tow truck tows a disabled car see left figure below). Symbolically (right figure below), the amount of work done when a force $\vec{F}$ causes an object to be displaced by a direction vector $\vec{D}$ is the dot product of the two vectors: $\begin{displaymath}{\em Work} = \vec{F} \cdot \vec{D} \; . \end{displaymath}$

Photo Credit: James Marshall, from
a trip to Yosemite National Park, 1998

A constant force is described by the equation $\begin{displaymath}\vec{F} = 2\vec{i} + 3\vec{j} + \vec{k} \; . \end{displaymath}$ This force acts on an object and moves it from the point (1,3,4) to the point (2,7,9). (See the graph in your worksheet.) How much work is done?
The figure below shows a force field $\vec{F}$ and a displacement vector $\vec{D}$ . If an object is moved along $\vec{D}$ from the upper left hand corner to the lower left hand corner, how much work would the force $\vec{F}$ do? Explain why your answer is reasonable.
The following figure shows a force field $\vec{F}$ . Describe the paths that an object could move along that would require no work to be done by $\vec{F}$ .
The next figure shows a force field, $\begin{displaymath}\vec{F} = \vec{i} + \vec{j} \; , \end{displaymath}$ and a path, .

Suppose that an object moves around the path in the direction shown. Your worksheet has been set up to calculate the work done by $\vec{F}$ along each straight line segment of the path individually, and then to evaluate the total work done by $\vec{F}$ when it acts on an object all of the way around . How is the total amount of work related to the amount of work done by $\vec{F}$ along the straight line segments?
Based on your answer to step 4, make a conjecture describing how the total work along a path made up of straight lines is related to the amount of work done by the force $\vec{F}$ along the straight line segments.
Test your conjecture:
- Devise a path made up of straight line segments.
- Calculate both the total work done.
- Calculate the work done along each individual line segment.

Do the results fit with your conjecture? Explain.

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