Transcript Simple Harmonic Motion & Elasticity

Simple Harmonic
Motion & Elasticity
Chapter 10
Elastic Potential Energy
► What
is it?
 Energy that is
a result of their
► Where







is it found?
in
.
materials as
Law
►A
spring can be
or
with a
.
► The
by which a spring is
compressed or stretched is
the magnitude of the
(
).
► Hooke’s Law:
Where:
spring (
to
Felastic =
= spring constant =
)
= displacement
of
Hooke’s Law
► What
is the graphical relationship
between the elastic spring force
and displacement?
Felastic = -kx
Displacement
Hooke’s Law
►
A force acting on a spring, whether
stretching or compressing, is
always
.
 Since the spring would prefer to be in a
“relaxed” position, a negative “
”
force will exist whenever it is deformed.
 The
force will always
attempt to bring the spring and any
object attached to it back to the
position.
 Hence, the restoring force is always
.
Example 1:
►
A 0.55 kg mass is attached to a vertical spring. If
the spring is stretched 2.0 cm from its original
position, what is the spring constant?
►
Known:
m=
x=
g=
►
Equations:
Fnet =
=
+
(1)
=
(2)
=
(3)
Substituting 2 and 3 into 1 yields:
k=
k=
k=
Elastic
Spring
► The
in a
exerted to put a spring
in tension or compression can be used
to do
. Hence the spring will
have Elastic
.
► Analogous to kinetic energy:
=
Example 2:
►What
is the
maximum
value
elastic
► A
0.55 kg
mass
is attached
to aofvertical
potential
spring
with
energy
the system
when
the If
spring
is allowed
to
a
springofconstant
of 270
N/m.
the spring
is
oscillate from
its from
relaxed
with no weight
stretched
4.0 cm
its position
original position,
what is
on it?
the
Elastic Potential Energy?
►
Known:
m = 0.55 kg
x = -4.0 cm
k = 270 N/m
g = 9.81 m/s2
►
Felastic
Equations:
PEelastic =
PEelastic =
PEelastic =
Fg
Elastic Potential Energy
► What
is area under the curve?
A=
A=
A=
A=
Which you should see
equals the
Displacement
Simple Harmonic Motion &
Springs
► Simple
Harmonic Motion:
 An
around an
will occur when an object is
from its equilibrium position and
 For a spring, the restoring force F = -kx.
► The
spring is at equilibrium
when it is at its relaxed length.
(
)
► Otherwise, when in tension or
compression, a restoring
force
exist.
.
Simple Harmonic Motion &
Springs
►
At
displacement (+ ):
 The Elastic Potential Energy will
be at a
 The force will be at a
.
 The acceleration will be at a
.
►
At
(x =
):
 The Elastic Potential Energy will
be
 Velocity will be at a
.
 Kinetic Energy will be at a
 The acceleration will be
as will the
force.
,
10.3 Energy and Simple Harmonic Motion
Example 3 Changing the Mass of a Simple
Harmonic Oscilator
A 0.20-kg ball is attached to a
vertical spring. The spring
constant is 28 N/m. When
released from rest, how far
does the ball fall before being
brought to a momentary stop by
the spring?
10.3 Energy and Simple Harmonic Motion
Eo  E f






Simple Harmonic Motion of Springs
► Oscillating
a
►
►
systems such as that of a spring follow
pattern.
Harmonic Motion of Springs – 1
Harmonic Motion of Springs (Concept Simulator)
Frequency of Oscillation
► For
a spring oscillating system, the frequency and
period of oscillation can be represented by the
following equations:
f 
► Therefore,

and T 
if the
of the spring and the
are known, we can find the
at which the spring will oscillate.
k and
frequency of oscillation (A
mass equals
and
spring).
Harmonic Motion & The Simple
Pendulum
►
►
Simple Pendulum: Consists of a massive object
called a
suspended by a string.
Like a spring, pendulums go through
as follows.
T
Where:
=
=
=
►
Note:
1.
2.
This formula is true for only
The period of a pendulum is
of .
of its mass.
Conservation of ME & The
Pendulum
In a pendulum,
is converted
into
and vise-versa in a
continuous repeating pattern.
►
PE = mgh
KE = ½ mv2
MET = PE + KE
MET =




►
Note:
1.
2.
3.
kinetic energy is achieved at the
point of the pendulum swing.
The
potential energy is achieved at the
of the swing.
When
is
,
= , and when
is
,
=
.
Key Ideas
► Elastic
Potential Energy is the
in a spring or other elastic material.
► Hooke’s Law: The
of a spring
from its
is
the
applied.
► The
of a
vs.
is equal to the
.
► The
under a
vs.
is equal to the
done to
compress or stretch a spring.
Key Ideas
► Springs
and pendulums will go through
oscillatory motion when
from an
position.
► The
of
of a
simple pendulum is
of
its
of displacement (small
angles) and
.
► Conservation
of energy: Energy can be
converted from one form to another, but it is
.