Work is done when a force is applied, at least partially, in the direction of the displacement of the object. If that force is constant then the work done by the force is the dot product of the force with the displacement:
The net work done on an object (the work of the net force) is equal to the energy added to the object. This is the reason for both the unit of work being the Joule, , and the work-kinetic energy theorem:
Anne spends most of her day napping on a futon. In the reference frame of the Earth, what is the work done to Anne by the futon in one hour?
In the reference frame of the Earth, Anne is sitting still, so her displacement is 0 in any amount of time. Hence the work done in one day is
Newton's laws provide a method for turning force into acceleration, but this only provides the acceleration at an instant. In order to affect the motion of an object, the force must be applied over some time (impulse) or over a distance (work). For an object with displacement, , only in the x-direction being acted on only by a force in the x direction, the work is simply defined in terms of the magnitudes as
If there are also some y and z components to the force, the work of each of those forces is
The net work on an object moving through space is just the sum of these individual components.
Since the right side is the definition of the dot product, the left side must be as well.
This dot-product realization also produces another valuable form of the work definition.
A force causes a block to undergo a displacement Find the work done.
Since the force and displacement are given in terms of their components, use the component-wise definition of work.
The Earth takes 1 year to complete its orbit around the Sun. Treating the orbit as perfectly circular, find the net work done on the Earth by the Sun in of a year. Give your answer to three significant figures.
Mass of the Earth =
Distance from Sun to Earth =
1 Earth year = 365 Earth days
1 Earth day = 24 hours
Combining Newton's second law, with the definition of work in one-dimension gives
The product shows up in one dimensional kinematics.
Divide both sides by 2 to solve for the desired product on the left.
Finally, multiply both sides by mass to get work on the left.
Notice, while the left is the definition of work, the right also happens to be the change in kinetic energy.
Combined with the conservation of energy, provides the following relation.
Work of a conservative force
The kinetic energy of a car decreases from to over a distance of 10 by a braking force. Find the magnitude of that braking force.
The negative sign indicates the force is antiparallel to the displacement.