In 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 scheme for analyzing motion and making predictions
about the final state of an object's motion. In this
unit, an entirely different scheme will be utilized to
analyze the motion of objects. Motion will be approached
from the perspective of work and energy. The effect 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
work-energy 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.
In physics, work is
defined as a force acting upon an object to
cause a displacement. There are three key
words in this definition - 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
fields, 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 her head, a shot-put launching the shot, 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
depress the mouse upon the pop-up menu to view the
answers.
Statement
Answer with
Explanation
A teacher applies a force to a wall and
becomes exhausted.
A book falls off a table and free falls to
the ground.
A waiter carries a tray full
of meals above his head by one arm across the
room. (Careful! This is a very difficult
question which will be discussed in more detail
later.)
A rocket accelerates through space.
Mathematically,
work can be expressed by the following equation.
where F = force, d = 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.
Scenario A: A force acts rightward upon an object
as it is displaced rightward. In such an instance, the
force vector and the displacement vector are in the
same direction. Thus, the angle between F and d is 0
degrees.
Scenario B: A force acts leftward upon an object
which is displaced rightward. In such an instance, the
force vector and the displacement vector are in the
opposite direction. Thus, the angle between F and d is
180 degrees.
Scenario C: A force acts upward
upon an object as it is displaced rightward. In such
an instance, the force vector and the displacement
vector are at right angles to each other. Thus, the
angle between F and d is 90 degrees.
Let'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 across the room. 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!!
The
equation for work lists three variables - each variable
is associated with the 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 tensional
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.
When
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. For
instance, consider the activity performed in the "It's
All Uphill" lab. A force was applied to a cart to pull it
up an incline at constant speed. Several incline angles
were used; yet, the force was always applied parallel to
the incline. The displacement of the cart was also
parallel to the incline. Since F and d were in the same
direction, the angle was 0 degrees. Nonetheless, 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.
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,
The Joule is the
unit of work.
1 Joule = 1 Newton *
1 meter
1J = 1 N *
m
In fact, any unit of force times any unit of
displacement is equivalent to a unit of work. Some
nonstandard units for work are shown below. Notice that
when analyzed, each set of units is equivalent to a force
unit times a displacement unit.
In summary, work is a force acting upon an object to
cause a displacement. When a force acts to cause an
object to be displaced, three quantities must be known in
order to calculate the amount of work. Those three
quantities are force, displacement and the angle between
the force and the displacement.
