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Energy
• Analyzing the motion of an object can often
get to be very complicated and tedious –
requiring detailed knowledge of the path,
frictional forces, etc.
• There has to be an easier way…
• It turns out that there is – it is done by
analyzing the object’s energy.
Energy
The “something” that enables an
object to do work is energy.
Energy is measured in Joules (J).
Forms of Energy:
Mechanical (kinetic and potential)
Thermal (heat)
Electromagnetic (light)
Nuclear
Chemical
Mechanical Energy
Mechanical energy is the form of
energy due to the position or the
movement of a mass.
Kinetic Energy
 Kinetic energy is the energy of motion – it is
associated with the state of motion of an
object.
 The faster an object is moving, the greater it’s
kinetic energy; an object at rest has zero
kinetic energy.
 For an object of mass m, we will define kinetic
energy as:
1
2
K  mv
2
 The SI unit of kinetic energy is the Joule (J).
Kinetic Energy
 If we do positive work on an object by pushing on it
with some force, we can increase the object’s kinetic
energy (and thereby increasing it’s speed).
 We can account for the change in kinetic energy by
saying that the force transferred energy from you to
the object.
 If we do negative work on an object by pushing on it
with some force in the direction opposite to the
direction of motion, we can decrease the object’s
kinetic energy (and decrease it’s speed).
 We can account for the change in kinetic energy by
saying that the force transferred energy from the
object to you.
Kinetic Energy
 Whenever we have a transfer of energy via a force, we
say that work is done on the object by the force.
– Work W is energy transferred to or from an object
by means of a force acting on that object.
– Energy transferred to the object is positive work.
– Energy transferred from the object is negative work.
• Work is nothing more than transferred energy – it
therefore has the same units as energy and is also a
scalar quantity.
• Note that nothing material is transferred.
– Think of it like the balance in two bank accounts: when
money is transferred the number for one account goes down
by some amount and the number for the other account goes
up by the same amount.
Work-Energy Theorem
 Suppose we have an bead which is
constrained to move only along the length of a
frictionless wire.
 We then supply a constant force F on the
bead at some angle  to the wire.
 Because the force is constant, we know that
the acceleration will also be constant.
Work-Energy Theorem
• But because of the constraint, only the force
in the x direction matters, thus Fx = m·ax
where m is the bead’s mass.
• We can relate the bead’s velocity at some
distance down the wire to the acceleration
using:
v  vo  2  a x  d
2
2
Work-Energy Theorem
• Solving for ax, substituting into the Fx equation,
multiplying both sides by d, and distributing the
½·m throughout the equation:
v  vo
ax 
2d
2
Fx  m  a x
2
v  vo
v  vo
Fx  m 
Fx  d  m 
2d
2
1
1
2
2
Fx  d   m  v   m  v o
2
2
2
2
2
2
Work-Energy Theorem
• But we can see that the right side of the
equation is no more than the kinetic energy
after the force has been applied minus the
kinetic energy before the force was applied:
1
1
2
2
Fx  d   m  v   m  v o
2
2
and that – by definition – is the work done
W  Fx  d  Kafter  K before
Work-Energy Theorem
• When calculating the work done on an object
by a force during a displacement, use only
the component of the force that is parallel to
the object’s displacement.
W  Fx  d  F  cos θ  d
1
1
2
2
F  cos θ  d   m  v   m  vo
2
2
– where  is the angle between the force F and the
horizontal.
• The force component perpendicular to the
displacement does no work.
Work-Energy Theorem
Work-Energy Theorem: the net work
done on an object is equal to the
change in kinetic energy of the object.
W = Kf – Ki; F·d = 0.5·m·(vf2 - vi2)
A net force causes an object to change
its kinetic energy because a net force
causes an object to accelerate, and
acceleration means a change in
velocity, and if velocity changes,
kinetic energy changes.
Gravitational Potential Energy
Potential energy (U): an object may store
energy because of its position. Energy that
is stored is called potential energy because
in the stored state it has the potential to do
work.
Work is required to lift objects against
Earth’s gravity.
Potential energy due to elevated positions is
gravitational potential energy.
The amount of gravitational potential energy
possessed by an elevated object is equal to
the work done against gravity in lifting it.
Ug = Fw·h = m·g·h
Work and Potential Energy
• When we throw a tomato up in
the air, negative work is being
done on the tomato which
causes it to slow down during
it’s ascent.
• As a result, the kinetic energy of
the tomato is reduced –
eventually to zero at the highest
point.
• But where did that energy
go???
Work and Potential Energy
• Where it went was into an
increase in the gravitational
potential energy of the tomato.
• The reverse happens when the
tomato begins to fall down.
• Now the positive work done by
the gravitational force causes
the gravitational potential
energy to be reduced and the
tomato’s kinetic energy to
increase.
Work and Potential Energy
• From this we can see that for either the rise or
fall of the tomato, the change ΔU in the
gravitational potential energy is the negative of
the work done on the tomato by the
gravitational force
• In equation form we get: U  W
 Only changes in potential
energy have meaning; it is
important that all heights be
measured from the same
origin.
 In many problems, the
ground is chosen as the zero
level for the determination of
the height.
 As the ball falls from A to B,
the potential energy at A is
converted to kinetic energy
at B.
 The amount of potential
energy of the ball at point A
will equal the amount of
kinetic energy of the ball at
point B.
Elastic Potential Energy
 Stretching or compressing an elastic object
requires energy and this energy is stored in
the elastic object as elastic potential energy.
 The work required to stretch or compress a
spring is dependent on the force constant k.
 The force constant will not change for a
particular spring as long as the spring is
not permanently distorted (which occurs
when the elastic limit of the spring is
exceeded).
 F = k·x
Elastic Potential Energy
 The force required to stretch/compress an
elastic object is not a constant force. The
work needed varies with the amount of
stretch/compression.
W = 0.5·k·x2
 The potential energy of an elastic object is
equal to the work done on the elastic object.
Ue = 0.5·k·x2
 In actual practice, a small fraction of the work
in stretching/compressing an elastic object is
converted into heat energy in the spring.
Work and Elastic Potential Energy
• If we give the block a shove to
the right, the kinetic energy of
the block is transferred into
elastic potential energy as the
spring compresses.
• The work done in compressing
the spring is the negative of the
change in the block’s kinetic
energy.
• And of course the reverse
happens when the spring
stretches back out – potential
energy gets transformed back
into kinetic energy.
Conservation of Mechanical Energy
 Conservative forces: all the work done is
stored as energy and is available to do work
later. Example: gravitational forces, elastic
forces.
 Nonconservative (dissipative) forces: the
force generally produces a form of energy
that is not mechanical.
Friction is a nonconservative (dissipative)
force because it produces heat (thermal
energy, not mechanical).
 The total amount of energy in any closed
system remains constant.
Conservation of Mechanical Energy
The sum of the potential and kinetic
energy of a system remains constant
when no dissipative forces (like
friction) act on the system.
Law of Conservation of Energy:
energy cannot be created or destroyed;
it may be changed from one form to
another or transferred from one object
to another, but the total amount of
energy never changes.
Conservation of Mechanical Energy
In a closed system in which
gravitational potential energy and
kinetic energy are involved, the
potential energy at the highest point is
equal to the kinetic energy at the
lowest point.
m·g·h = 0.5·m·v2
Notice that the sum of the
potential energy (PE) and kinetic
energy (KE) at every point is
40000 J. Energy is conserved.
The velocity at the lowest point
can be determined by the height:
v  2g h
Conservation of Mechanical Energy
The change in velocity due to a change
in height can also be determined:
v  2  g  h  2  g  hf  hi
The height can be determined by the
initial velocity (vf = 0 m/s):
2
v
h
2g
Conservation of Mechanical Energy
In a closed system in which elastic
potential energy and kinetic energy
are involved, the potential energy at
the maximum distance of
stretch/compression is equal to the
kinetic energy at the equilibrium (rest)
position.
2
2
0.5  k  x  0.5  m  v
k  x 2  m  v2
v
k  x2
m
k
m  v2
x2
Conservation of Mechanical Energy
Conservation of Energy Equation:
W done (by applied force) + Ugravitational
before + Uelastic before + K before =
Ugravitational after + Uelastic after + K after
+ W done (usually by friction)
Fapplied·d + m ·g ·hi + 0.5 ·k ·xi2 + 0.5 ·m ·vi2
= m ·g ·hf + 0.5 ·k ·xf2 + 0.5 ·m ·vf2 + FF·d
• If there is no change in height: m ·g ·hi and
m ·g ·hf drop out of the equation.
• If there is no spring or elastic object:
0.5 ·k ·xi2 and 0.5 ·k ·xf2 drop out of the
equation.
• If there is no change in velocity: 0.5 ·m ·vi2
and 0.5 ·m ·vf2 drop out of the equation.
• If there is no applied force (a push/pull that
you supply): Fapplied·d drops out of the
equation.
• If there is no friction: FF·d drops out of the
equation.
Helpful Online Links
Work – Energy Theorem:
The Work-Energy Theorem
Elastic Constant k:
Hooke’s Law Applet