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# What is Work-Energy Theorem?

Jay Gregorio, Elizabeth Friedl
• Author
Jay Gregorio

Jay Gregorio has taught Physics and STEM courses for over 11 years. He has a degree in Education with specialization in Physics from the Philippines and a STEM leadership certificate from Columbia University, New York. He is currently taking his doctorate degree in Organizational Development at the Southeast Asia Interdisciplinary Development Institute.

• Instructor
Elizabeth Friedl

Elizabeth, a Licensed Massage Therapist, has a Master's in Zoology from North Carolina State, one in GIS from Florida State University, and a Bachelor's in Biology from Eastern Michigan University. She has taught college level Physical Science and Biology.

Learn to define work and energy. Discover the work energy theorem and the work energy theorem equation. See the relationship between work and energy with examples. Updated: 02/23/2022

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## Work Energy

The word "work" is used in many contexts, but in physics, work is done when a force causes an object to have a displacement. There are certain requirements that must be satisfied for work to be done on an object â€“ force, displacement, cause. A force is a push or a pull, while a displacement is a distance traveled by an object in a specific direction. When I push a grocery cart down the aisle, my hands and arms exert force on the handle. This force causes the wheels of the cart to turn to allow it to move in the direction of my push, hence displacement. Since there is a force, displacement, and cause, work is done on the grocery cart. Consider another situation when a heavy book is picked up from the floor. A force is needed to lift the book, which causes it to be displaced from the floor. Therefore, work is done on the book. In these situations, the direction of the force and displacement is parallel.

Energy is the capacity to do work, and this work-energy relationship is shown in pushing the grocery cart, picking up a book, and many other situations. When work is done and the object moves, we say that the object has kinetic energy (KE) or the energy due to motion. The grocery cart and the book have kinetic energy because they move as a result of the forces acting on them. In addition, objects can have energy due to their position called potential energy (PE), also called stored energy. When a ball is placed on the top edge of an incline, it possesses potential energy because it has the potential or tendency to slide down the incline. In other words, it is called potential or stored energy because it has the potential to do work. Once the ball is released, this potential energy is converted into kinetic energy, and work is done.

It is important to note that there are situations when work is not done. That is, when the force applied is perpendicular to the object's direction of motion. When a restaurant server carries a tray and walks on a straight line, the force applied on the tray is upward while the displacement of the tray is horizontal. The upward direction of the force on the tray does not cause the tray to move along the horizontal direction; therefore, no work is done. For work to be done, the tray can be lifted from the table, which makes the direction of the force and its displacement parallel. In the same manner, no work is done when an object does not move, even when force is applied. Pushing a wall with so much force and not causing it to move results in no work done since there is no change in its energy.

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## What is the Work-Energy Theorem?

The work-energy principle or work-energy theorem relates the work done by all forces acting on an object to its energy. It states that the total amount of work done on an object is equal to the object's change in kinetic energy (final kinetic energy minus initial kinetic energy). This principle applies when multiple forces act on an object that changes its state of rest or motion. Therefore, it is an important principle to understand how forces do work which eventually allows us to predict whether the object will speed up or slow down. When the object speeds up, its energy increases and the energy decreases when the object slows down. In the language of physics, "positive work" and "negative work" are sometimes being used. A positive work (+W) means that the object's kinetic energy increases, which means it is speeding up because of the force applied. If the kinetic energy decreases, negative work (-W) is done, and the object slows down.

An example would be a biker moving at a constant speed on a flat, horizontal surface. When another person pushes the biker from behind, this applied force can increase the kinetic energy of the biker and speed up its motion (+W). Likewise, when a force is applied in the opposite direction of the biker, the applied force can decrease the kinetic energy slowing it down (-W).

## Work-Energy Theorem Equation

The relationship between work done and kinetic energy can be expressed by the work-energy theorem formula expressed as:

{eq}W=?KE {/eq}

where W is the work done, {eq}? {/eq} is delta which means change, and KE is kinetic energy. This equation can be read as "the work done is equal to the change in kinetic energy." Note that {eq}?KE {/eq} is the difference between the final kinetic energy and the initial kinetic energy expressed as KEf - KEi. If the equation for kinetic energy can be written as {eq}KE = 1/2mv^2%%(vertical-align: super)%% {/eq}, then

{eq}W=?KE {/eq}

W = KEf - KEi

{eq}W=1/2 mv_f^2 - 1/2 mv_i^2 {/eq}

where

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Frequently Asked Questions

#### What is the work-energy theorem, and why is it important?

The work-energy theorem states that the total work done by all of the forces on the object is equal to the change in kinetic energy of the object. This theorem is important because it allows us to predict the direction, speed, or energy of a moving object.

#### What is an example of work energy?

The work-energy theorem is applied when a grocery cart is pushed in the aisle of a grocery store. The amount of force applied to the cart causes it to move and cover a horizontal distance, thereby increasing its kinetic energy.

#### What is the work-energy theorem equation?

The work-energy theorem equation is W = ?KE where W is work and ?KE is change in kinetic energy or the difference between the final kinetic energy and the initial kinetic energy of the object. Since kinetic energy is 1/2 mv^2, the work-energy theorem equation becomes W = 1/2 mvf^2 - 1/2 mvi^2.

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