# Work-energy Theorem

JoVE Core
Physik
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JoVE Core Physik
Work-energy Theorem

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The work-energy theorem states that the total work done by all the forces on an object is equal to the change in kinetic energy of that object, where m is the mass and v1 and v2 are the initial and final velocities of the object.

For instance, when a force is applied to a box, the velocity of the box increases, which increases its kinetic energy. Hence, the work done by the force contributes to a change in the kinetic energy of the box.

According to Newton's second law of motion, force is equal to mass times acceleration.

Consider a case where the box is moving with an acceleration a along a straight line. From the third equation of motion, the relationship between velocity and acceleration can be rearranged to obtain the displacement covered by the box.

By substituting the force and displacement terms in the work formula, a relationship between work done and change in kinetic energy is obtained. This expression is called the work-energy theorem.

According to this theorem, when an applied force increases the velocity of an object, the work done is positive. If the applied force causes no change in the velocity of the object or its kinetic energy, the work done is said to be zero.

If the net force applied is used to stop the moving object, the kinetic energy decreases, and hence the work done is negative.

## Work-energy Theorem

According to Newton’s second law of motion, the sum of all the forces acting on a particle (net force) determines the rate of change in the momentum of the particle (motion). Therefore, we should consider the work done by all forces acting on a particle, or the net work, to see its effect on the particle’s motion.

The work-energy theorem equates work done by all the forces on an object to the change in its kinetic energy. The theorem can be used to calculate work done by a force when acceleration is not constant, provided we know the change in velocity. When approaching a problem using the work-energy theorem, the following steps should be considered.

1. Identify the given quantities and draw a free-body diagram based on the forces acting on the object. Calculate the work done by the known forces, keeping the sign of work done in mind. Add the total amount of work done by each force.
2. Use the work-energy theorem and equate the total work to the change in kinetic energy. Substitute the known quantities and solve for the unknown.
3. Evaluate the answer. If the object is traveling at a constant speed, there is no change in kinetic energy; therefore, the total work done should be zero. If the object has accelerated or has increased kinetic energy, the total work done is positive. If the object has slowed down, the total work done is negative.

This text is adapted from Openstax, University Physics Volume 1, Section 7.3: Work-Energy Theorem.