Transcript Mechanical Energy

Mechanical Energy
Essential Questions:
Where does all this motion come from?
What is the difference between dynamics and
kinematics?
Is energy real?
What is work?
Why does a gain in potential energy necessitate a
loss in kinetic energy?
How is energy conserved within systems?
When can energy be described as not being
conserved?
Where does all this motion come from?
• Insofar as scientists understand our universe,
there are essentially two “substances” that
provide the make up of our existence:
matter & energy
• Thanks in great part to Albert Einstein,
perhaps the wisest scientist the world has
known to date, we know that matter and
energy are the same thing!
• It never ceases to amaze me how the seeming
complexity of nature is in actuality very
simple...
E=
2
mc
Energy
Speed of light
Mass (matter)
Ever heard this? Bet you have, but I bet you never really
thought about what it meant...
Let’s analyze the units of this quantity before we go on…
So, back to the question: Where does all
this motion come from?
• Motion is just one manifestation of energy
in our lives.
• Other manifestations include heat, pressure,
and electromagnetism, which at the
fundamental level is actually caused by
motion (vibration of charged particles).
• Also, energy can be stored energy via
chemical bonds, springs, gravitational
differentials, and many other sources.
So, energy is movement!
Or, going back and looking at Newton’s
Second Law:
Forces cause accelerations which cause
changes in motion.
This becomes:
Changes in energy result in forces which
change/create motion.
Note: This is glossing over a great deal of complicated physics, but
any physicist will tell you this is the main idea.
Dynamics vs. Kinematics
• The study of motion without regard to the
underlying causes, particularly using time as a
reference in a vector perspective is the study of
kinematics.
– This is what we have been doing all year thus far.
• The study of motion via energy and work is
considered one aspect of dynamics.
– In may cases dynamics mirrors kinematics, but in some
cases dynamics allows for much more simplistic and
intuitive solutions to some VERY complex situations.
– This aspect of our study will focus on one answer to the
question "Why do objects move?“ and our answer to
that question will relate to the physical quantity known
as “work.”
The Work/Energy Theorem
Derivation:
W = KE (= - PE) for later
What does this mean?
• Well, it sounds crazy, but energy is not
really a “real” quantity. It is a derived
product of other physical and chemical
quantities. That does not mean that it is not
real, it just means that it is an abstract
quantity that we study because certain
properties of this “energy” variable help us
figure stuff out about the universe.
What is energy?
• Simply put, energy is the potential to do
work.
• Work: The energy used/used
up/consumed/transferred during a physical
process.
Conservative vs. Non-conservative forces
Dynamics
Conservative Forces
Non-conservative Forces
When work is done, energy is
conserved within a closed system
When work is done, energy is NOT
conserved within a closed system
(i.e., energy leaves the system)
Reversible:
Energy lost can be
regained by reversing process
***Energy is still (always)
conserved globally.***
Non-Reversible:
Energy lost outside the system cannot
be returned by reversing the process
(2nd Law of Thermodyanimcs)
FRICTION
Force and Displacement:
How do vectors play into this?
• Energy is scalar—it doesn’t matter how the
object gets there (in a conservative system),
all that matters is if the state of the object’s
energy before vs. after the event.
• For work to be done by a force, that force
must be applied in the direction of the
displacement.
Fx does work on
the mower.
Fx
Fy
Fy does not. In
fact, Fy makes it
harder to do work
on the mower.
Why?
When force and displacement are in the same direction
𝑊 = 𝐹𝑥
When force and displacement are NOT in the same direction,
but share a unidirectional component:
𝑊 = 𝐹𝑐𝑜𝑠𝜃 𝑥
𝑊 = 𝐹𝑥𝑐𝑜𝑠𝜃
When is work done?
• When a force produces a displacement in
the direction of that force.
• When an object gains or loses kinetic or
potential energy as a result of a force ebing
applied.
Can work be negative?
• YES! This means only that work has been
done against a force.
• Often applied to objects being lifted
“against” gravity:
Applied force (lifting):
W = Fx
W = mgx
Work done by gravity during lifting
Wgrav = -mgx
Negative work can imply that the flow of energy is opposite
the direction of the force being referenced.
Is work done? By what force?
Did F create x? Did the energy of the object change?
• A person lifts a book to place on a high bookshelf.
• A person carries a sack of groceries at a constant
height across a level floor.
• A person carries a sack of groceries at a level
height down a ramp.
• A weight lifter holds a loaded barbell above his
head.
• An airplane comes in for a landing.
• An airplane takes off.
• A pitcher throws a baseball.
Work, Energy, Force and Graphs
Remember how
we looked at the
“area under” the
velocity vs. time
graph to
calculate
something?
What did we calculate? Does above/below axis matter?
How much work is
done by this force
as it displaces the
object 0.5m?
Does it matter if the axis went “under” the horizontal axis? What
would that mean mathematically? What would that mean physically?
Another example:
What work is done by
this force during the
first 5m of
displacement? During
all 10m?
Types of Energy:
• Kinetic Energy
– Anything moving has kinetic energy.
• Potential Energy
– Stored energy
• For our purposes: PE’s are energies of position
• In general: PE’s are energies of state
• e.g., springs, chemical bonds, all matter, objects at
heights, etc…
Kinetic Energy
(from the work energy theorem)
1 2
KE  mv
2
Derivation of Gravitational PE
Falling object
0
Gain and loss of energy
Throwing a ball upwards
Changes in energy are reversed from
falling body.
Falling:
Gains KE, loses GPE
Thrown up:
Loses KE, gains GPE
Conclusion:
A gain in KE = loss in PE and
A loss of KE = gain in PE
+KE = -PE
W = KE = -PE
Complete work/energy theorem
• The scenario in the prior problem assumes a
closed, conservative system. Why is this
premise only partly valid in real life (a.k.a.
outside of magical physics land)?
• Consider a bouncing ball.
– Why? What does that have to do with anything?
– Figure out where I’m going with this…
Potential Energy
• Gravitational PE:
– GPE is understood as the energy difference
caused by the relative heights of different
objects above the ground (at h=0m, GPE = 0 J)
GPE  m gh
• Spring PE:
– Spring PE is the energy stored in a compressed
or stretched spring. It describes the spring’s
tendency to snap back into its original shape.
1 2 Where k, is the spring constant or stiffness of
SPE  kx the spring and x is the displacement of the
2
spring from its neutral position.
The Law of Conservation of
Energy
The total amount of energy in any system is
constant.
Energy can neither be created no destroyed, only
changed to a new form.
This principle is ALWAYS true. There are NO
EXCEPTIONS. (just erroneous assumptions)
Energybefore = Energyafter
Work
Work is the amount of energy used up in a mechanical
process.
W = Fx = KE = - PE
Work-Energy Theorem
Works nicely with the impulse-momentum theorem to
help define a system.
Impulse/Momentum is Newtonian Kinematics (VECTORS)
Work/Energy is Dynamics (SCALAR)
Power
𝑊 ∆𝐸
𝑃=
=
𝑡
𝑡
• Power is the RATE at which energy is used.
It has many mathematical forms, but
anytime you see any energy variable
described over a period of time, you are
dealing with power.
• More power does not necessarily mean
more energy…
Units for Energy, etc.
• The SI unit for energy is the Newton-Meter,
also called the Joule (J). It is a
(kg)(m2)/(s2).
• Power is energy over time, so J/s which is
also called a Watt (W) (like a 40 W light
bulb…)
– Be CAREFUL! Capital W is the unit for
power, but as a variable is means work or
energy. Know your context.
AP-1: We will do these in a later unit.
Using conservation of energy to
help solve collision problems:
• New definitions!
– Elastic Collision: A collision where kinetic
energy is conserved.
– Inelastic Collision: A collision where kinetic
energy is not conserved.
– The TOTAL energy, sometimes called the
“mechanical energy” in any system is
conserved in any collision.
Problem:
• How do you solve an elastic collision
system where both objects bounce off of
one another elastically and both objects
continue to move?
• Example:
– Two objects of mass m = 2.4 kg collide in an
elastic collision. The first object is traveling
3.9 m/s to the right and the second is traveling
1.5 m/s also to the right. Find the final
velocities of both objects after an elastic
collision.
How do you solve this?
• Set up your conservation of momentum
equation (m1v1i + m2v2i = …)
– Notice that you have two unknown variables,
v1f and v2f.
• Now set up an equation that indicates
conservation of kinetic energy.
– Notice that you have the same two unknowns.
• Solve the system of two equations either by
substitution, or by elimination.
REMEMBER!
• You may only conserve kinetic energy in an
elastic collision!
• Certain systems in which objects collide
inelastically may be solved using
conservation of energy (all types of E) if
more information is known about what
happens either before or after the collision.
You will not have to evaluate these systems
in this course, but you will be responsible
for using Cons. Of Energy to solve elastic
collisions.