Analysis of Situations in Which Mechanical Energy is conserved
It has previously been mentioned that there
is a relationship between work and mechanical energy change. Whenever work is
done upon an object by an external force (or nonconservative force), there will be a change in the total
mechanical energy of the object. If only internal
forces are doing work (no work done by external forces), then there is no
change in the total amount of mechanical energy. The total mechanical energy is
said to be conserved. In this part of Lesson 2, we will further
explore the quantitativerelationship between work and mechanical energy in situations in which there
are no external forces doing work.
The quantitative relationship between work and the two forms
of mechanical energy is expressed by the following equation:
KEi + PEi + Wext = KEf + PEf
The equation illustrates that the total mechanical energy (KE
+ PE) of the object is changed as a result of work done by external forces.
There are a host of other situations in which the only forces doing work areinternal or
conservative forces. In such situations, the
total mechanical energy of the object is notchanged. The external work term cancels from the above equation and mechanical
energy is conserved. The previous equation is
simplified to the following form:
KEi + PEi = KEf + PEf
In these situations, the sum of the kinetic and potential
energy is everywhere the same. As the potential energy is increased due to the
stretch/compression of a spring or an increase in its height above the earth,
the kinetic energy is decreased due to the object slowing down. As the
potential energy is decreased due to the return of a spring to its rest
position or a decrease in height above the earth, the kinetic energy is
increased due to the object speeding up. We would say that energy is transformed or changes
its form from kinetic energy to potential energy (or vice versa); yet the total
amount present is conserved - i.e., always the same.
The
Example of Pendulum Motion
The tendency of an object to conserve its mechanical energy is observed
whenever external forces are not doing any overall work. If the influence of friction and air resistance can be ignored (or
assumed to be negligible) and all other external forces are absent or merely
not doing work, then the object is often said to conserve its energy. Consider
a pendulum bob swinging to and fro on the end
of a string. There are only two forces acting upon the pendulum bob. Gravity
(an internal force) acts downward and the tensional force (an external force)
pulls upwards towards the pivot point. The external force does not do work
since at all times it is directed at a 90-degree angle to the motion. (Review a previous page to convince yourself that F•d•cosine angle = 0 J for the force of tension.)
As the pendulum bob swings to and fro, its height above the tabletop (and in turn its speed) is constantly
changing. As the height decreases, potential energy is lost;
and simultaneously the kinetic energy is gained.
Yet at all times, the sum of the potential and kinetic energies of the bob
remains constant. The total mechanical energy is 6 J. There is no loss or gain
of mechanical energy, only a transformation from kinetic energy to potential
energy (and vice versa). This is depicted in the diagram below.
As the 2.0-kg pendulum bob in the above diagram swings to and fro, its
height and speed change. Use energy equations and the above data to determine
the blanks in the above diagram. Click the button to view answers.
A: h = 0.306 m (6 J = 2 kg *9.8 m/s/s * h)
B: h = 0.153 m (3 J = 2 kg *9.8 m/s/s * h)
C: v = 1.73 m/s (3 J = 0.5 * 2 kg * v2)
D: h = 0 m (0 J = 2 kg * 9.8 m/s/s*h)
E: v = 2.45 m/s (6 J = 0.5 * 2 kg * v2)
F: h = 0.306 m (6 J = 2 kg * 9.8
m/s/s * h)
Learning From Lab
A common Physics lab involves the analysis of a pendulum in
its back and forth motion. The transformation and conservation of mechanical
energy is the focus of the lab. 
A 0.200-kg (200 gram) pendulum is typically released from rest at
location A. The bob passes through a photogate at location B and another
photogate at location C. The speed of the pendulum bob can be determined from
the width of the bob and the photogate times. The speed and mass can be used to
determine the kinetic energy of the bob at each of the three locations. The
heights of the bob above the tabletop at each of the three locations can be
measured and used to determine the potential energy of the bob. The data should
reflect that the mechanical energy changes
its form as the bob passes from location A to B to C. Yet the total mechanical
energy should remain relativity constant. Sample data for such a lab are shown
below.
|
Loc'n |
Height |
Speed |
PE |
KE |
TME |
|
A |
0.400 m |
0 m/s |
0.784 J |
0 J |
0.784 J |
|
B |
0.248 m |
1.70 m/s |
0.486 J |
0.289 J |
0.775 J |
|
C |
0.096 m |
2.47 m/s |
0.188 J |
0.610 J |
0.798 J |
The sample data show that the pendulum bob loses potential
energy as it swings from the more elevated location at A to the lower location
at B and at C. As this loss of potential energy occurs, the pendulum bob gains kinetic
energy. Yet the total mechanical energy remains
approximately 0.785 Joules. We would say that total mechanical energy is
conserved as the potential energy is transformed into kinetic energy.
The
Example of a Roller Coaster
A roller coaster operates on this same
principle of energy transformation. Work is
initially done on a roller coaster car to lift to its initial summit. Once
lifted to the top of the summit, the roller coaster car has a large quantity of potential
energy and virtually no kinetic energy (the car
is almost at rest). If it can be assumed that no external forces are
doing work upon the car as it travels from the initial summit to the end of the
track (where finally an external braking system is employed), then the total
mechanical energy of the roller coaster car is conserved. As the
car descends hills and loops, its potential energy is transformed into kinetic
energy as the car speeds up. As the car climbs up hills and loops, its kinetic
energy is transformed into potential energy as the car slows down. Yet in the
absence of external forces doing work, the total mechanical energy of the
car is conserved.
Conservation of energy on a roller coaster ride means that
the total amount of mechanical energy is the same at every location along the
track. The amount of kinetic energy and the amount of potential energy is
constantly changing. Yet the sum of the kinetic and potential energies is
everywhere the same. This is illustrated in the diagram below. The total
mechanical energy of the roller coaster car is a constant value
of 40 000 Joules.
The
Example of a Ski Jumper
The motion of a ski jumper is also governed by the transformation
of energy. As a ski jumper glides down the hill towards
the jump ramp and off the jump ramp towards the ground, potential
energy is transformed into kinetic energy. If it can be
assumed that no external forces are doing work upon the ski jumper as it
travels from the top of the hill to the completion of the jump, then the total
mechanical energy of the ski jumper is conserved. Consider Li
Ping Phar, the esteemed Chinese ski jumper. She starts at rest on top of a
100-meter hill, skis down the 45-degree incline and makes a world record
setting jump. Assuming that friction and air resistance have a negligible affect upon
Li's motion and assuming that Li never uses her poles for propulsion, her total
mechanical energy would never change.
Of course it should be noted that the original assumption
that was made for both the roller coaster car and the ski jumper is that there
were no external forces doing work. In actuality, there are external forces doing
work. Both the roller coaster car and the ski jumper experience the force of
friction and the force of air resistance during the course of their motion.
Friction and air resistance are both external forces and would do work upon the
moving object. In fact, the presence of friction and air resistance would do
negative work and cause the total mechanical energy to decrease during the
course of the motion. While the assumption that mechanical energy is conserved
is an invalid assumption, it is a useful approximation that assists in the
analysis of an otherwise complex motion.