Transcript Conservation of Total Mechanical Energy Standards for WPE

Class notes-- 5.9.11
Conservation of Total Mechanical Energy
Standards for WPE
Reading Reference WPE OUTLINES
TWINS DAY LABS late -10% today
ALL Make up after school M-TH
But not Here Friday…….
On Wednesday the library and Tech Center will be
closed for Guided Study.
BE CAREFUL WITH ΔKE !!!
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ΔKE = KEf - KEi
ΔKE = ½ mvf2 - ½ mvi2
Δ KE = ½ m (vf2 -vi2)
Δ KE = ½ m (Δv2)
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ΔPE = PEf - PEi
ΔPE = mghf - mghi
ΔPE = mg (hf -hi)
ΔPE = mg Δh
not (Δv) 2
Minnesota Standard #2
• 2. When objects change their motion or interact with
other objects in the absence of frictional forces, the
total amount of mechanical energy remains constant.
9P.2.2.2.1
• Explain and calculate the work, power, potential
energy and kinetic energy involved in objects
moving under the influence of gravity and other
mechanical forces.
9P.2.2.2.3
• Use conservation of momentum and
conservation of energy to analyze an elastic
collision of two solid objects in onedimensional motion.
External-internal forces
We categorize a force as
internal or external
because:
• internal forces conserve
Total mechanical energy
(keep same)
• Mechanical Forces keep
all the energy in TME
• external forces either
add or remove
mechanical energy due
to friction or heat
Internal
Forces
External
Forces
Fg
Fsp
FA
Ffr
Fair
FT
F┴
Work  W=Fd
A force and movement parallel to the force
Causes a change in energy from one form to
another.
Energy never disappears.
If the Force is internal, The TME will be
conserved.
W = ΔPE
and
OR
W=ΔE
W = ΔKE
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W = work (in J)
ΔE = change in energy (in J)
PE = gravitational potential energy (in J)
m = mass (in kg)
g = acceleration due to gravity (9.80 m/s2)
h = height above reference level (in m)
KE = kinetic energy (in J)
m = mass (in kg)
v = velocity (in m/s)
Energy
Work is change in ENERGY
PE is ENERGY from POSITION
KE is ENERGY from motion
TME Conserved!!
ΔKE = ΔPE
Use when:
Mechanical Energy is Conserved
so there is no friction or deformation of
Page 6 space 6
objects.
ΔKE = ΔPE
Energy Conservation on an Incline
no friction
TME stays
the same!!
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Mechanical
Energy
changes
between
PE and KE!!
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Consider an ordinary lab cart loaded with bricks and accelerating down an inclined plane. How could work and energy be utilized to analyze the
motion of the loaded cart? Would the total mechanical energy of the cart be altered in the process of rolling down the incline? Or would the total
mechanical energy of the cart merely be conserved?
Of course the answers to these questions begin by determining whether or not there are any external forces doing work upon the loaded cart. If
external forces do work upon the cart, the total mechanical energy of the cart is not conserved; the initial amount of mechanical energy is not the
same as the final amount of mechanical energy. On the other hand, if external forces do not do work upon the loaded cart, then the total
mechanical energy is conserved; that is, mechanical energy is merely transformed from the form of potential energy to the form of kinetic energy
while the total amount of the two forms remains unchanged.
In the case of the cart rolling down the incline, there are three external forces (the normal force, the force of friction and air resistance) and one
internal force (the force of gravity). The normal force does not do work upon the cart because it acts in a direction perpendicular to the direction
of motion. In such instances, the angle between F and d is 90 degrees and the work done by the force is 0 Joules. The force of friction does not
do work upon the cart because it acts upon the wheels of the cart and actually does not serve to displace either the cart nor the wheels. The
friction force only serves to help the wheels turn as the cart rolls down the hill. Friction only does work upon a skidding wheel. Finally, the force
of air resistance does do work upon the cart; air resistance does negative work upon the cart since it acts in a direction opposite the direction of
the cart's motion. Sometimes referred to as a dissipative force, air resistance contributes to a loss in the total amount of mechanical energy
possessed by the cart. Subsequently, it would be expected that there would be a small amount of energy loss as the cart rolls down the hill from
an elevated position to a position just above the ground.
Due to the difficulty in measuring air resistance forces and due to the small amount of existing Fair in situations in which a streamlined object
moves at relatively low speeds, the affect of air resistance is often neglected. If air resistance is neglected, then it would be expected that the
total mechanical energy of the cart would be conserved. The animation below depicts this phenomenon (in the absence of air resistance).
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As the cart rolls down the hill from its elevated position, its mechanical energy is transformed from potential energy to kinetic energy. At a height
just above the ground, the majority of the energy is in the form of kinetic energy. This is to say, energy due to vertical position has been
transformed into energy due to motion. In fact, if air resistance can be neglected, then the amount of potential energy loss equals the amount of
kinetic energy gained. Calculations can be performed to illustrate that the potential energy lost by the 3.0-kg cart in the above animation is equal
to the kinetic energy gained. Assume that the height of the cart changes from 0.40 meters to 0.05 meters and the speed changes from 0 m/s to
2.62 m/s.
TME stays the same!!
KE   PE
Energy Transformation for a Pendulum
no friction
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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. However, 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 direction of motion is 90 degrees. Thus, the force of tension does not do work
upon the bob.
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.
h
(m)
2.000
Speed
(m/s)
0.0 m/s
PE
(J)
KE
(J)
TME
(J)
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. Use the pull down menus to check your work.
1.490
3.920 J
(Note: mass = 0.200 kg)
1.128
0.897
0.357
Energy Transformation on a Roller Coaster
no fricton
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A roller coaster ride is a thrilling experience which involves a wealth of physics. Part of the physics of a roller coaster i s 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 in 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 are 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 corres ponds 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.
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A roller coaster ride also illustrates the work and energy relationship. 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 relationship 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 (KEinitial + PEinitial) for the initial state of the object plus the work done on the object by external forces (Wexternal) 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 incapable of doing 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 * mass * (speed)^2
PE = mass * g * height
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 k inetic 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?
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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.
TME stays the same!!
KE = 0.5 * mass * (speed)^2
PE = mass * g * height
How High Will It Go?
Without friction
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The animation below depicts the motion of a young child sliding across the snow on a sled (combined mass of 50 kg). The child comes to a small decline to a valley below which is immediately followed by a steep incline. The child begins the descent down the decline with a speed of 8 m/s. If the snow is assumed to be friction free, then how high along the opposite incline will the child slide before eventually comi ng to a stop? And what variables will influence
the answer? Would a change in the child's mass affect the height achieved on the opposite incline? Would a change in the angl es of the decline and the incline affect the height achieved on the the opposite incline? Would a change in the child's speed affect the height achieved on the opposite incline?
The motion of the sledder in the animation below is similar to the motion of a roller coaster car on a roller coaster track. The similarity of the two motions is associated with the work-energy relationship. The energy bar charts accompanying the animation depict this relationship. Energy bar charts are a conceptual tool amount the amount of each form of energy possessed by an object as it an undergoes a particular motion. Observe from the animation that the
total mechanical energy (TME) of the sledder remains constant throughout the motion. The total mechanical energy is the sum of the two forms of mechanical energy - kinetic energy (KE) and potential energy (PE). While the individual amounts of kinetic and potential energy are undergoing change, their sum will always be the same amount. As on a roller coaster, energy is transformed from potential energy to kinetic energy and vice versa. Provided that
external forces (such as friction forces and applied forces) do not do work, the total amount of mechanical energy will be held constant.
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In the initial state on top of the hill, the sledder has both kinetic (energy of motion) and potential energy (energy of vertical position). The amount of kinetic energy is dependent upon mass and speed, and computed from the equation KE=0.5*m*v2. For t he 50-kg sledder (includes the sled's mass) moving at 8.0 m/s, the kinetic energy is 1600 Joules. The amount of potential energy is dependent upon mass and height and is found using the equation PE=m*g*h
where g is the acceleration of gravity (approximated here to be 10 m/s/s). For a 50-kg sledder on top of a 4.0-meter high hill, the potential energy is 2000 Joules. The total amount of mechanical energy (kinetic plus potential) is then 3600 J. Since thi s quantity of mechanical energy will be conserved, the sledder must still have 3600 Joules of mechanical energy by the time he/s he comes to a stop on the opposite incline. At that instant, all 3600 Joules of
mechanical energy will be in the form of potential energy. The height of the sledder can then be calculated using the equation PE = m*g*h where PE = 3600 J, m = 50 kg, and g = 10 m/s/s (the same approximation as used earlier). The substitution and algebra yields an answer of 7.2 meters.
Note that the solution to the above problem does not take into account the angles of the decline and incline. The angles woul d only have an affect upon the acceleration of the sledder along the hills. A steeper incline would be consistent with a greater acceleration value; yet this greater acceleration would occur for less time, thus yielding the same speed at the bottom of the hill and the same final height on the opposite incline. Thus the angles of the decline and
incline have no affect upon the final height achieved by the sledder.
But perhaps the mass of the sledder would have an affect upon the final height which the sledder would achieve? After all, the mass of the sledder does enter into the equations. To investigate the affect of mass upon the scenario, perform the same calculations as above for an 80-kg sledder with the same initial height of 4.0 meters and the same initial speed of 8.0 m/s. Use the pull down menu to check each individual answer.
1. Determine the initial kinetic energy of the 80-kg sledder.
2. Determine the initial potential energy of the 80-kg sledder.
3. Determine the initial mechanical energy of the 80-kg sledder.
4. Determine the mechanical energy possessed by the sledder when he/she comes to a stop at the final height.
5. When finally stopped on the opposite incline, how much potential energy will the sledder have?
6. Determine the final height achieved by this more massive (80 kg) sledder?
7. Does the mass of the sledder have an affect upon the height achieved by the sledder?
• Write about what you learned about this benchmark in
this class. Include concepts learned and how it would
apply to everyday life examples. Also describe how
you would use the equations and do the calculations.
9P.2.2.2.1
1=I was here
2=kind of.
3=know it
4= know well
5=EXCELLENT
Rank 1-5________
9P.2.2.2.3
1=I was here
2=kind of.
3=know it
4= know well
5=EXCELLENT
Rank 1-5________
Explain and calculate the work,
power, potential energy and kinetic
energy involved in objects moving
under the influence of gravity and
other mechanical forces.
Use conservation of momentum and
conservation of energy to analyze an
elastic collision of two solid objects
in one-dimensional motion.
Due tomorrow…..