7: Work and Kinetic Energy

*A sprinter exerts her maximum power with the greatest force in the short time her foot is in contact with the ground. This adds to her kinetic energy, preventing her from slowing down during the race. Pushing back hard on the track generates a reaction force that propels the sprinter forward to win at the finish. (credit: modification of work by Marie-Lan Nguyen)*

In this module, we discuss some basic physical concepts involved in every physical motion in the universe, going beyond the concepts of force and change in motion, which we discussed in Motion in Two and Three Dimensions and Newton’s Laws of Motion. These concepts are **work**, **kinetic energy**, and **power**. We explain how these quantities are related to one another, which will lead us to a fundamental relationship called the **work-energy theorem**. In the next module, we generalize this idea to the broader principle of conservation of energy.

Work and energy — these are two new, closely-related, concepts. Newton didn’t think about these quantities, but they are very useful!

Energy is conserved: it changes form, but the total amount (in any isolated system) never changes. This is a central idea in all of modern physics. Energy is a little tough to define, because of all the different forms it takes, e.g. kinetic (or “energy of motion”), chemical, nuclear, gravitational, thermal, … Fundamentally, energy is the ability of a system to do work. Whenever work is done, energy is transformed from one form to another, but you can never create or destroy energy, only change its form. This is of central importance in understanding a variety of phenomena (including the increasingly relevant issue of our finite supplies of energy reserves in the form of fossil fuels).

We must begin with the concept of work. Think of what it feels like to do work – plow a field, let a motor speed a car up, climb the stairs. There’s always a force involved, and it does something, moving an object.

Consistent with this, we make a formal definition of work:

The work, W, done by a constant force F moving an object through a displacement Δx is defined to be where F_{x} is the component of F in the x-direction (the direction of the displacement).

Graphically, .

From the picture (or formula), only the component of force in the direction of motion is doing work. (Only the “projection of the force along the direction of motion” does work.)

In 2- or 3-D, we write the equation as

That notation is called the “dot product” or “scalar product” (because the answer is a scalar).

When work is done, the energy of a system will change. If the work done is positive, then the energy will increase and if the work done is negative, then the energy will decrease. This leads us to one of the most useful theorems in physics: the Work-Energy Theorem.

#### 7.1 Work

- Represent the work done by any force
- Evaluate the work done for various forces

#### 7.2 Kinetic Energy

- Calculate the kinetic energy of a particle given its mass and its velocity or momentum
- Evaluate the kinetic energy of a body, relative to different frames of reference

#### 7.3 The Work-Energy Theorem

- Apply the work-energy theorem to find information about the motion of a particle, given the forces acting on it
- Use the work-energy theorem to find information about the forces acting on a particle, given information about its motion

#### 7.4 Power

- Relate the work done during a time interval to the power delivered
- Find the power expended by a force acting on a moving body