

Lesson 1: Basic Terminology and ConceptsWork: Definition and Mathematics of Work 
Lesson 1: Basic Terminology and ConceptsDefinition and Mathematics of WorkIn the first three units of The Physics Classroom, we utilized Newton's laws to analyze the motion of objects. Force and mass information were used to determine the acceleration of an object. Acceleration information was subsequently used to determine information about the velocity or displacement of an object after a given period of time. In this manner, Newton's laws serve as a useful model for analyzing motion and making predictions about the final state of an object's motion. In this unit, an entirely different model will be used to analyze the motion of objects. Motion will be approached from the perspective of work and energy. The affect that work has upon the energy of an object (or system of objects) will be investigated; the resulting velocity and/or height of the object can then be predicted from energy information. In order to understand this workenergy approach to the analysis of motion, it is important to first have a solid understanding of a few basic terms. Thus, Lesson 1 of this unit will focus on the definitions and meanings of such terms as work, mechanical energy, potential energy, kinetic energy, and power.
When a force acts upon an object to cause a displacement of the object, it is said that work was done upon the object. There are three key ingredients to work  force, displacement, and cause. In order for a force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement. There are several good examples of work which can be observed in everyday life  a horse pulling a plow through the field, a father pushing a grocery cart down the aisle of a grocery store, a freshman lifting a backpack full of books upon her shoulder, a weightlifter lifting a barbell above his head, an Olympian launching the shotput, etc. In each case described here there is a force exerted upon an object to cause that object to be displaced.
Read the following five statements and determine whether or not they represent examples of work. Then click on the See Answer button to view the answer.
Mathematically, work can be expressed by the following equation. where F is the force, d is the displacement, and the angle (theta) is defined as the angle between the force and the displacement vector. Perhaps the most difficult aspect of the above equation is the angle "theta." The angle is not just any 'ole angle, but rather a very specific angle. The angle measure is defined as the angle between the force and the displacement. To gather an idea of its meaning, consider the following three scenarios.
To Do Work, Forces Must Cause DisplacementsLet's consider Scenario C above in more detail. Scenario C involves a situation similar to the waiter who carried a tray full of meals above his head by one arm straight across the room at constant speed. It was mentioned earlier that the waiter does not do work upon the tray as he carries it across the room. The force supplied by the waiter on the tray is an upward force and the displacement of the tray is a horizontal displacement. As such, the angle between the force and the displacement is 90 degrees. If the work done by the waiter on the tray were to be calculated, then the results would be 0. Regardless of the magnitude of the force and displacement, F*d*cosine 90 degrees is 0 (since the cosine of 90 degrees is 0). A vertical force can never cause a horizontal displacement; thus, a vertical force does not do work on a horizontally displaced object!! It can be accurately noted that the waiter's hand did push forward on the tray for a brief period of time to accelerate it from rest to a final walking speed. But once up to speed, the tray will stay in its straightline motion at a constant speed without a forward force. And if the only force exerted upon the tray during the constant speed stage of its motion is upward, then no work is done upon the tray. Again, a vertical force does not do work on a horizontally displaced object. The
equation for work lists three variables  each variable is
associated with one of the three key words mentioned in the
definition of work (force, displacement,
and cause). The angle theta in the equation is associated
with the amount of force which causes a displacement. As
mentioned in a previous
unit, when a force is exerted on an object at an angle
to the horizontal, only a part of the force contributes to
(or causes) a horizontal displacement. Let's consider the
force of a chain pulling upwards and rightwards upon Fido in
order to drag Fido to the right. It is only the horizontal
component of the tension force in the chain which causes
Fido to be displaced to the right. The horizontal component
is found by multiplying the force F by the cosine of the
angle between F and d. In this sense, the cosine theta in
the work equation relates to the cause factor  it
selects the portion of the force which actually
causes a displacement.
The Meaning of ThetaWhen determining the measure of the angle in the work equation, it is important to recognize that the angle has a precise definition  it is the angle between the force and the displacement vector. Be sure to avoid mindlessly using any 'ole angle in the equation. A common physics lab involves applying a force to displace a cart up a ramp to the top of a chair or box. A force is applied to a cart to displace it up the incline at constant speed. Several incline angles are typically used; yet, the force is always applied parallel to the incline. The displacement of the cart is also parallel to the incline. Since F and d are in the same direction, the angle theta in the work equation is 0 degrees. Nevertheless, most students experienced the strong temptation to measure the angle of incline and use it in the equation. Don't forget: the angle in the equation is not just any 'ole angle. It is defined as the angle between the force and the displacement vector.
The Meaning of Negative WorkOn occasion, a force acts upon a moving object to hinder a displacement. Examples might include a car skidding to a stop on a roadway surface or a baseball runner sliding to a stop on the infield dirt. In such instances, the force acts in the direction opposite the objects motion in order to slow it down. The force doesn't cause the displacement but rather hinders it. These situations involve what is commonly called negative work. The negative of negative work refers to the numerical value which results when values of F, d and theta are substituted into the work equation. Since the force vector is directly opposite the displacement vector, theta is 180 degrees. The cosine(180 degrees) is 1 and so a negative value results for the amount of work done upon the object. Negative work will become important (and more meaningful) in Lesson 2 as we begin to discuss the relationship between work and energy.
Whenever a new quantity is introduced in physics, the standard metric units associated with that quantity are discussed. In the case of work (and also energy), the standard metric unit is the Joule (abbreviated J). One Joule is equivalent to one Newton of force causing a displacement of one meter. In other words,


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