WORK & FORCE

Work and Energy: Definitions and Physics Concepts

The Physics Definition of Work

Just like force, the words "work" and "energy" have a variety of connotations in common language. When we say work, sometimes we mean exercise, and other times we mean a place we go every day to earn money. However, in physics, we must allow rigorous definitions to replace colloquial connotations. Work is defined as action done on an object whereby an applied force causes a displacement of that object.

A basic example would involve pushing a box 10 meters across the floor. Work is equal to the magnitude of the applied force times the displacement of the object ($W = Fd$). Consequently, it has units of force times distance. If you push with a constant force of 100 Newtons over a 10-meter distance, you would do 1,000 Newton-meters of work. These units are equivalent to another SI unit called Joules.

The Necessity of Displacement and Angles

Work can only be done when displacement occurs. If you push a brick wall with the same force for the same amount of time as the box, you may get very tired, but technically, you have done zero work because the wall did not move. Additionally, only the components of the applied force that are parallel to the displacement are considered to contribute to work.

If you push the box with your arms parallel to the floor, all of your effort contributes to the work done. However, if you are pushing at a downward angle, only the x-component contributes to work, while the y-component does not. In such a scenario, we calculate work using the following formula:

$$W = Fd \cos \theta$$

This equation gives us the portion of the applied force operating in the horizontal direction. If $\theta$ is zero (meaning the applied force is horizontal), $\cos \theta$ will equal 1, leaving us with $W = Fd$. If you were to push straight down on the box, $\theta$ would be 90 degrees, and $\cos \theta$ would be zero—meaning no work is done on the box, and it will not move.

Positive vs. Negative Work

We must bear in mind that work is a scalar, meaning it has magnitude but not direction, though it can be positive or negative.

• Positive Work: Occurs when the applied force is in the same direction as the object's motion (e.g., pushing the box forward).

• Negative Work: Occurs if the applied force is in the direction opposite to the object's motion (e.g., someone pushing the box the other way without enough force to stop it).

The sign on work is crucial because it indicates if the speed of an object will increase or decrease as a result of the work applied.

The Work-Energy Theorem

Energy is quite simply the capacity to do work. This relationship is summarized in the Work-Energy Theorem, which states that when work is done on an object, there will be a change in that object's kinetic energy (the energy of motion). This implies that an object in motion is also able to do work on another object.

We must purge ourselves of prior connotations of the word "energy" and understand that it merely describes a property of an object or system that can be transferred into another object or system—often in the form of work. There are different kinds of energy, including kinetic, potential, thermal, and chemical energy. The SI unit for energy is also the Joule, defined as a Newton-meter, even though energy is not always transferred in the form of work.