Download Energy Transformation on a Roller Coaster

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http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/energy/ce.html (for the Lesson 2 below)
http://www.glenbrook.k12.il.us/gbssci/phys/Class/energy/u5l2a.html
Energy Transformation on a Roller Coaster
A roller coaster ride is a thrilling experience which involves a wealth of physics. Part of the physics of roller coaster is the physics of
work and energy. The ride often begins as a chain and motor (or other mechanical device) exerts a force on the train of cars to lift
the train to the top of a vary tall hill. Once the cars are lifted to the top of the hill, gravity takes over and the remainder of the ride is
an experience of the physics of energy transformation.
At the top of the hill, the cars possess a large quantity of potential energy. Potential energy - the energy of vertical position - is
dependent upon the mass of the object and the height of the object. The car's large quantity of potential energy is due to the fact that
they are elevated to a large height above the ground. As the cars descend the first drop they lose much of this potential energy in
accord with their loss of height. The cars subsequently gain kinetic energy. Kinetic energy - the energy of motion - is dependent
upon the mass of the object and the speed of the object. The train of coaster cars speeds up as they lose height. Thus, their original
potential energy (due to their large height) is transformed into kinetic energy (revealed by their high speeds). As the ride continues,
the train of cars ar continuously losing and gaining height. Each gain in height corresponds to the loss of speed as kinetic energy
(due to speed) is transformed into potential energy (due to height). Each loss in height corresponds to a gain of speed as potential
energy (due to height) is transformed into kinetic energy (due to speed). This transformation of mechanical energy from the form of
potential to the form of kinetic and vice versa is illustrated in the animation below.
A roller coaster ride also illustrates the work-energy theorem. The work-energy theorem states that the work done by external forces
is capable of changing the total amount of mechanical energy from an initial value to some final value. The amount of work done by
the external forces upon the object is equal to the amount of change in the total mechanical energy of the object. The theorem is
often stated in the form of the following mathematical equation.
KEinitial + PEinitial + Wexternal = KEfinal + PEfinal
The left side of the equation includes the total mechanical energy (KE initial + PEinitial) for the initial state of the object plus the work
done on the object by external forces (W external) while the right side of the equation includes the total mechanical energy
(KEfinal + PEfinal) for the final state of the object.
Once a roller coaster has reached its initial summit and begins its descent through loops, turns and smaller hills, the only forces
acting upon the coaster cars are the force of gravity, the normal force and dissipative forces such as air resistance. The force of
gravity is an internal force and thus any work done by it does not change the total mechanical energy of the train of cars. The normal
force of the track pushing up on the cars is an external force; however it is at all times directed perpendicular to the motion of the
cars and thus is unable of ding any work upon the train of cars. Finally, the air resistance force is capable of doing work upon the
cars and thus draining a small amount of energy from the total mechanical energy which the cars possess. However, due to the
complexity of this force and its small contribution to the large quantity of energy possessed by the cars, it is often neglected. By
neglecting the influence of air resistance, it can be said that the total mechanical energy of the train of cars is conserved during the
ride. That is to say, the total amount of mechanical energy (kinetic plus potential) possessed by the cars is the same throughout the
ride. Energy is neither gained nor lost, only transformed from kinetic energy to potential energy and vice versa.
The conservation of mechanical energy by the coaster car in the above animation can be studied using a calculator. At each point in
the ride, the kinetic and potential energies can be calculated using the following equations.
KE = 0.5 * m * (speed)2
PE = m * g * h
If the acceleration of gravity value of 9.8 m/s/s is used along with an estimated mass of the coaster car (say 500 kg), the kinetic
energy and potential energy and total mechanical energy can be determined.
What value do you find for the total mechanical energy of the car at any point along the track?
The total mechanical energy is
A final feature of the above animation is the use of work-energy bar charts. Work-energy bar charts are a conceptual tool which
depict the amount of each form of energy possessed by an object as it an undergoes a particular motion. The work-energy bar charts
for the coaster car illustrate that the car's energy is transformed from potential to kinetic and vice versa; yet the total amount of
mechanical energy remains the same during the course of the motion.
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http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/energy/pe.html
Energy Transformation for a Pendulum
The motion of a pendulum is a classic example of mechanical energy conservation. A pendulum consists of a mass (known as a bob)
attached by a string to a pivot point. As the pendulum moves it sweeps out a circular arc, moving back and forth in a periodic
fashion. Neglecting air resistance (which would indeed be small for an aerodynamically shaped bob), there are only two forces
acting upon the pendulum bob. One force is gravity; the force of gravity acts in a downward direction and does work upon the
pendulum bob. However, gravity is an internal force (or conservative force) and thus does not serve to change the total amount of
mechanical energy of the bob. The other force acting upon the bob is the force of tension. Tension is an external force and if it did
do work upon the pendulum bob it would indeed serve to change the total mechanical energy of the bob. The force of tension does
not do work since it always acts in a direction perpendicular to the motion of the bob. At all points in the trajectory of the pendulum
bob, the angle between the force of tension and its instantaneous displacement is 90 degrees. Thus, the force of tension does not do
work upon the bob [F*d*cosine(90 degrees) is 0 Joules].
Since there are no external forces doing work, the total mechanical energy of the pendulum bob is conserved. The conservation of
mechanical energy is demonstrated in the animation below. Observe the KE and PE bars of the bar chart; their sum is a constant
value.
Observe that the falling motion of the bob is accompanied by an increase in speed. As the bob loses height and PE, it gains speed
and KE; yet the total of the two forms of mechanical energy is conserved.
To test you understanding of the conservation principle, use the heights and the speeds given in the table below to fill in the
remaining cells at the various locations in a 0.200-kg bob's trajectory. (Note: mass = 0.200 kg)
h
(m)
2.000
1.490
1.128
0.897
0.357
Speed
(m/s)
0.0 m/s
PE
(J)
KE
(J)
TME
(J)
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http://www.glenbrook.k12.il.us/gbssci/phys/Class/energy/u5l2bc.html
Application and Practice Questions
Lesson 2 has thus far focused on how to analyze motion situations using the work-energy theorem. The theorem could be
summarized by the following statements:
There is a relationship between work and mechanical energy change. Whenever work is done upon an object by an
external 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), there is no change in total mechanical energy; the total mechanical energy is said to
be "conserved." The quantitative relationship between work and the two forms of mechanical energy is expressed by
the following equation:
KEi + PEi + Wext = KEf + PEf
Now an effort will be made to apply the theorem to a variety of motion scenarios in order to test our understanding.
Check Your Understanding
Use your understanding of the work-energy theorem to answer the following questions. Then depress mouse on the "pop-up menu"
to view the answers.
1. Consider the falling motion of the ball in the following two frictionless situations. For each situation, indicate what type of forces
are doing work upon the ball. Indicate whether the energy of the ball is conserved and explain why. Finally, indicate the kinetic
energy and the velocity of the 2-kg ball just prior to striking the ground.
The only force doing w ork is gravity. Since
The only force doing w ork is gravity. Since
2. If frictional forces and air resistance were acting upon the falling ball in #1 would the kinetic energy of the ball just prior to
striking the ground be more, less, or equal to the value predicted in #1?
The kinetic energy w ould be less in a situation
3. The cartoon strip below depicts a pile-driver falling from a high elevation (diagram A) to a low elevation (diagram B) before it
encounters the force of a spike which ultimately brings it to rest (diagram C). Assume that there is no air resistance and that the
spike moves only slightly. Fill in the blanks in the cartoon strip.
In diagram B, the KE is 10 000 Joules (since
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4. A worker pushes a 50.0-kg cylinder up a frictionless incline at constant speed to a height of 3-meters. A diagram of the situation
and a free-body diagram is shown below. Note that the force of gravity has two components (parallel and perpendicular component);
the parallel component balances the applied force and the perpendicular component balances the normal force.
Of the forces acting upon the cylinder, which one(s) do work upon it?
Both gravity and applied forces do w ork.
Based upon the types of forces acting upon the system and their classification as internal or external forces, is energy conserved?
Explain.
No! The applied force is an external force,
Calculate the amount of work which the man does upon the cylinder.
Start w ith TMEi + Wext = TMEf
Use the following diagram to answer questions #5 - #7. Neglect the effect of friction and air resistance.
5. As the object moves from point A to point D across the frictionless surface, the sum of its gravitational potential and kinetic
energies
a. decreases, only.
b. decreases and then increases.
c. increases and then decreases.
d. remains the same.
The answ er is D. The total mechanical energy
6. The object will have a minimum gravitational potential energy at point
a. A.
b. B.
c. C.
d. D.
e. E.
The answ er is B. Gravitaional potential energy
7. The object's kinetic energy at point C is less than its kinetic energy at point
a. A only.
b. A, D, and E.
c. B only.
d. D and E.
The answ er is C. Since the total mechanical energy is
8. Many drivers education books provide tables which relate a car's braking distance to the speed of the car (see table below). Utilize
what you have learned about the stopping distance-velocity relationship to complete the table.
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a = 108 ft (compare 20 mph to 40 mph - a
9. Some driver's license exams have the following question. A car moving 50 km/hr skids 15 meters with locked brakes.
How far will the car skid with locked brakes if it is moving at 150 km/hr?
The car skids 135 m.
10. Two arrows are fired into a bale of hay. If one has twice the speed of the other, how much farther does the faster arrow
penetrate? (Assume that the force of the haystack on the arrows are constant).
The arrow penetrates four times as far.
11. Use the law of conservation of energy (assume no friction) to fill in the blanks at the various marked positions for a 1000-kg
roller coaster car.
A: h = 45 m (from 450 000=1000*10*h)
12. If the angle of the initial drop in the roller coaster diagram above were 60 degrees (and all other factors were kept constant),
would the speed at the bottom of the hill be any different? Explain.
The angle does not effect the speed at the
13. Determine Li Ping Phar's (m=50 kg) speed at locations B, C, D and E.
B: 20 000 = 0.5*50*v^2
14. An object which weighs 10 N is dropped from rest from a height of 4 meters above the ground. When it has free-fallen 1 meter
its total mechanical energy with respect to the ground is
a. 2.5 J b. 10 J c. 30 J d. 40 J
The answ er is D. Energy is conserved in free-fall
15. During a certain time interval, a 20-N object free-falls 10 meters. The object gains _____ Joules of kinetic energy during this
interval.
a. 10
b. 20
c. 200 d. 2000
The answ er is C. Energy is conserved in free-fall
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http://www.glenbrook.k12.il.us/gbssci/phys/Class/energy/u5l2c.html#Q1
Bar Chart Illustrations
One tool which can be utilized to express an understanding of the work-energy theorem is a bar chart. A work-energy bar chart
represents the amount of energy possessed by an object by means of a vertical bar. The length of the bar is representative of the
amount of energy present - a longer bar representing a greater amount of energy. In a work-energy bar chart, a bar is constructed for
each form of energy. According to the work-energy theorem, the sum of all forms of initial energy plus the work done on the object
by external forces equals the sum of all forms of final energy.
KEi + PEi + Wext = KEf + PEf
In a work-energy bar chart, a bar is used to represent the amount of each term in the above equation. Consequently, the sum of the
bar heights for the initial condition (initial energy + work) must equal the sum of the bar heights for the final condition (final
energy). Since the potential energy comes in two forms - the elastic potential energy stored in springs (PEs) and the gravitational
potential energy (PEg) - the above equation is rewritten as
KEi + PEig + PEis + Wext = KEf + PEfg + PEfs
In this portion of Lesson 2, we will investigate the use and meaning of work-energy bar charts and make an effort to apply this
understanding to a variety of motions involving energy changes or energy transformations.
The following procedure might be useful for constructing work-energy bar charts:

analyze the initial and final states of the object in order to make decisions about the presence or absence of the different
forms of energy

analyze the forces acting upon the object during the motion to determine if external forces are doing work and whether the
work (if present) is positive or negative

construct bars on the chart to illustrate the presence and absence of the various forms of energy for the initial and final state
of the object; the height of the individual bars is not important; what is important is that the sum of the heights on the left of
the chart is balanced by the sum of the heights on the right of the chart
The above procedure is illustrated below.
Consider a ball falling from the top of a pillar to the ground below; ignore air
resistance. The initial state is the ball at rest on top of the pillar and the final state is
the ball just prior to striking the ground. Given this motion and the identification of
the initial and final state of the ball, decisions can be made about the presence and
absence of each form of energy. Since there is no motion at the top of the pillar, there
is no initial kinetic energy. Since the ball is elevated above the ground while on top
of the pillar, there is an initial gravitational potential energy (PEg). There are no
springs involved; thus, there is neither initial nor final elastic potential energy (PEs).
In the final condition (just prior to striking the ground), the ball is moving; thus, there
is a final kinetic energy. And finally, the ball is no longer elevated above the ground
so there is no final gravitational potential energy. The ball falls under the influence of
gravity (an internal force) alone; thus there are no external forces present and none
doing work. The diagram at the right summarizes this analysis.
To complete the bar chart, an arbitrarily decided height for each bar is decided upon and a bar is constructed for each form of
energy. As mentioned before, it is not important how high each bar is!! It is only important that the sum of the bar heights on the left
balance the sum of the bar heights on the right.
Observe that the work-energy bar chart reveals that
 there is no kinetic and elastic potential energy in the initial state
 there is no gravitational and elastic potential energy in the final state
 there is no work done by external forces
 the sum of the heights on the right (5 units) equals the sum of the heights on the left (5 units)
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It is not important as to how high the two bars are in the above bar chart. If the bars were 4 units high instead of 5 units high, then
that would not be of importance. The decision about bar height is entirely arbitrary; what is important is that the bar height is the
same on the left as it is on the right.
Now we will repeat the process for a car which skids from a high speed to a stop across level ground with its brakes applied. The
initial state is the car traveling at a high speed and the final state is the car at rest. Initially, the car has kinetic energy (since it is
moving) but does not have gravitational potential energy (since the height is zero) nor elastic potential energy (since there are no
springs). In the final state of the car, there is neither kinetic energy (since the car is at rest) nor potential energy (since there is no
height nor springs). The force of friction between the tires of the skidding car and the road does work on the car. Friction is an
external force and does negative work since its direction is opposite the direction of the car's motion.
Now that the analysis is complete, the bar charts can be constructed. The charts must be consistent with the above analysis.
Observe that the bar for work is a downward bar. This is consistent with the fact that the work done by friction is negative work.
Whenever negative work is done by external forces, the Wext bar will be a downward bar. Note also that the sum of the bar height
on the left side (+5 plus -5) is the same as the sum of the bar heights on the right side of the chart. One final comment is in order:
even though the score is balanced (i.e., the height of all bars on the left equals the height of all bars on the right), energy is not
conserved. The bar chart includes both energy and work on the left side of the chart; if work is done by external forces then the only
reason that the score is balanced is that the Wext makes up for the difference in total mechanical energy.
As a final example consider a skier that starts from rest on top of hill A and skis into the valley and back up onto hill B. The skier
utilizes her poles to propel herself across the snow, thus doing work to change her total mechanical energy. The initial state is on top
of hill A and the final state is on top of hill B. Suppose that friction and air resistance have a negligible effect on the motion. In the
initial state, the skier has no kinetic energy (the skier is said to be "at rest"). There is no elastic potential energy in both the initial and
the final states (since there are no springs). The skier has gravitational potential energy in both the initial and the final states (since
the skier is at an elevated position). Finally, work is being done by external forces since the skier is said to be using "her poles to
propel herself across the snow." This work is positive work since the force of the snow on her poles is in the same direction as her
displacement.
Now that the analysis is complete, the bar charts can be constructed. The charts must be consistent with the above analysis.
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Observe that the bar for work is upward bar. This is consistent with the fact that the work done by the poles is positive work.
Whenever positive work is done by external forces, the Wext bar will be an upward bar. Note also that the sum of the bar height on
the left side (+5 plus +2) is the same as the sum of the bar heights on the right side of the chart. The score is balanced. As mentioned
earlier, the exact heights of the individual bars is not important. it is only important that the bars exist, that they are in the correct
direction (upward) and that their sum on the left is the same as the sum on the right. On occasions it is customary to utilize a
different type of work-energy bar chart that looks like the diagram at the right. If external forces are not doing work, then the total
mechanical energy is conserved. The Wext term cancels from the work-energy equation leaving the equation
KEi + PEi = KEf + PEf
This equation shows that the total mechanical energy (potential energy plus kinetic energy) is the same in the initial state as it is in
the final state. In fact, if external forces are not doing work, then the total mechanical energy will be the same throughout the entire
motion. For such situations it is customary to use the different style of bar chart to depict energy conservation. A series of bars are
shown for a variety of positions throughout the motion; each set of bars reveals that the total mechanical energy (TME) is always the
same while the potential energy (PE) and the kinetic energy (KE) are constantly changing.
As an example consider a roller coaster car in the ideal situation in which the force air resistance is assumed negligible (indeed, an
idealized situation). Since the normal force acts at right angles to the motion at all times, it does not do work. The only force doing
work on the roller coaster car is gravity; and since the force of gravity is an internal force, the total mechanical energy is conserved
(i.e., not changing). The energy may change forms - transforming from potential to kinetic and vice versa; yet the total amount will
never change. The diagram below depicts the conservation of total mechanical energy and the transformation of potential and kinetic
energy for a roller coaster car at five positions along a track.
In conclusion, bar charts are a useful tool for depicting the influence of external forces (if present) upon the total mechanical energy.
It is a conceptual means of representing one's understanding of the work-energy theorem.
Check Your Understanding
Construct analyses and work-energy bar charts for the following motions. Then check the answers by depressing mouse on the "popup menu" and by clicking on the link.
1. A ball is dropped from rest from a bridge. As the ball falls through the air, it encounters a small amount of air resistance. The final
state of the ball is the instant before it strikes the water.
PEg - present (ball is elevated)
2. A volleyball player spikes the ball at just above net level and drives it over the net. The initial state is the ball just prior to the
spike. The final state of the ball is the instant before it strikes the ground.
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PEg - present (ball is elevated)
3. A spring gun is used to project a sponge dart into the air. The gun is held at a height of 1-meter before the trigger is pulled. The
loaded spring gun is the initial state and the sponge dart at its peak is the final state.
PEg - present (dart is elevated)
4. A baseball is caught by a catcher after passing over home plate. The initial state is the baseball moving at high speed just prior to
hitting the catcher's mitt. The final state is the baseball just after the catcher has applied the force to stop the ball. Assume that the
ball does not change height as the catcher is catching it.
PEg - present (ball is elevated a little)
5. In the Stopping Distance vs. Velocity lab, a Hot Wheels car starts at an elevated position, moves down an incline to the level
ground, strikes a box and skids to a stop. Consider three states for the car: state A is the top of the incline; state B is the bottom of
the incline before striking the box; state C is after the car has been brought to a stop. Use the diagram at the right and your
understanding of the work-energy theorem to fill in the bar charts for the motion from A to B and from B to C.