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# Simple Harmonic Motion and Elasticity

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```Simple Harmonic
Motion & Elasticity
Chapter 10
Elastic Potential Energy
в–є What
is it?
п‚§ Energy that is stored in elastic materials as a
result of their stretching.
в–є Where
п‚§
п‚§
п‚§
п‚§
п‚§
п‚§
п‚§
is it found?
Rubber bands
Bungee cords
Trampolines
Springs
Bow and Arrow
Guitar string
Tennis Racquet
HookeвЂ™s Law
A spring can be stretched or compressed with
a force.
в–є The force by which a spring is compressed or
stretched is proportional to the magnitude of
the displacement (F п‚џ x).
в–є HookeвЂ™s Law:
в–є
Felastic = -kx
Where:
(N/m)
k = spring constant = stiffness of spring
x = displacement
HookeвЂ™s Law
в–є What
is the graphical relationship
between the elastic spring force
and displacement?
Felastic = -kx
Slope = k
Displacement
HookeвЂ™s Law
в–є
A force acting on a spring, whether
stretching or compressing, is
always positive.
п‚§ Since the spring would prefer to be in a
вЂњrelaxedвЂќ position, a negative вЂњrestoringвЂќ
force will exist whenever it is deformed.
п‚§ The restoring force will always attempt
to bring the spring and any object
attached to it back to the equilibrium
position.
п‚§ Hence, the restoring force is always negative.
Example 1:
в–є
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?
в–є
Known:
m = 0.55 kg
x = -2.0 cm
g = 9.81 m/s2
в–є
Felastic
Equations:
Fnet = 0 = Felastic + Fg
(1)
Felastic = -kx
(2)
Fg = -mg
(3)
Substituting 2 and 3 into 1 yields:
k = -mg/x
k = -(0.55 kg)(9.81 m/s2)/-(0.020 m)
k = 270 N/m
Fg
Elastic Potential Energy in a
Spring
в–є The
force exerted to put a spring in
tension or compression can be used to
do work. Hence the spring will have
Elastic Potential Energy.
в–є Analogous to kinetic energy:
PEelastic = ВЅ kx2
Example 2:
в–єWhat
is the
difference
in the
potential
в–є A
0.55 kg
mass
is attached
to aelastic
vertical
spring with
energy
the system
when
the If
deflection
is is
a
springofconstant
of 270
N/m.
the spring
maximum4.0
in either
theits
positive
orposition,
negativewhat is
stretched
cm from
original
direction?
the
Elastic Potential Energy?
в–є
Known:
m = 0.55 kg
x = -4.0 cm
k = 270 N/m
g = 9.81 m/s2
в–є
Felastic
Equations:
PEelastic = ВЅ kx2
PEelastic = ВЅ (270 N/m)(0.04 m)2
PEelastic = 0.22 J
Fg
Elastic Potential Energy
в–є What
is area under the curve?
Displacement
A = ВЅ bпѓ—h
A = ВЅ xпѓ—F
A = ВЅ xпѓ—kпѓ—x
A = ВЅ kпѓ—x2
Which you should see
equals the elastic
potential energy
What is Simple Harmonic Motion?
в–єSimple
harmonic motion exists whenever
there is a restoring force acting on an object.
п‚§ The restoring force acts to bring the object back to
an equilibrium position where the potential energy
of the system is at a minimum.
Simple Harmonic Motion &
Springs
в–є Simple
Harmonic Motion:
п‚§ An oscillation around an equilibrium position will
occur when an object is displaced from its
equilibrium position and released.
п‚§ For a spring, the restoring force F = -kx.
в–є The
spring is at equilibrium
when it is at its relaxed length.
(no restoring force)
в–є Otherwise, when in tension or
compression, a restoring
force will exist.
Simple Harmonic Motion &
Springs
в–є
At maximum displacement (+ x):
п‚§ The Elastic Potential Energy will
be at a maximum
п‚§ The force will be at a maximum.
п‚§ The acceleration will be at a
maximum.
в–є
At equilibrium (x = 0):
п‚§ The Elastic Potential Energy will
be zero
п‚§ Velocity will be at a maximum.
п‚§ Kinetic Energy will be at a
maximum
п‚§ The acceleration will be zero, as
will the unbalanced restoring
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
E f пЂЅ Eo
1
2
mv
2
f
пЂ«
1
2
1
2
ho пЂЅ
пЂЅ
I пЃ· f пЂ« mgh
2
f
пЂ«
1
2
kh f пЂЅ
2
kh o пЂЅ mgh o
2
2 mg
k
пЂЁ
2 пЂЁ0 . 20 kg пЂ© 9 . 8 m s
28 N m
2
1
2
mv o пЂ«
2
1
2
I пЃ· o пЂ« mgh o пЂ«
2
1
2
2
kh o
Simple Harmonic Motion of Springs
в–є Oscillating
systems such as that of a spring follow
a sinusoidal wave pattern.
Harmonic Motion of Springs вЂ“ 1
в–є Harmonic Motion of Springs (Concept Simulator)
в–є
Frequency of Oscillation
в–є For
a spring oscillating system, the frequency
and period of oscillation can be represented by
the following equations:
f пЂЅ
1
k
2пЃ°
m
and T пЂЅ 2пЃ°
m
k
в–є Therefore,
if the mass of the spring and the spring constant
are known, we can find the frequency and period at which
the spring will oscillate.
п‚§ Large k and small mass equals high frequency of
oscillation (A small stiff spring).
Harmonic Motion & Simple The
Pendulum
в–є
в–є
Simple Pendulum: Consists of a massive object called
a bob suspended by a string.
Like a spring, pendulums go through
simple harmonic motion as follows.
T = 2ПЂв€љl/g
Where:
T = period
l = length of pendulum string
g = acceleration of gravity
в–є
Note:
1.
2.
This formula is true for only small angles of Оё.
The period of a pendulum is independent of its mass.
Conservation of ME & The
Pendulum
In a pendulum, Potential Energy is converted into
Kinetic Energy and vise-versa in a continuous
repeating pattern.
в–є
PE = mgh
KE = ВЅ mv2
MET = PE + KE
MET = Constant
п‚§
п‚§
п‚§
п‚§
в–є
Note:
1.
2.
3.
Maximum kinetic energy is achieved at the lowest point
of the pendulum swing.
The maximum potential energy is achieved at the top of
the swing.
When PE is max, KE = 0, and when KE is max, PE = 0.
Key Ideas
в–є Elastic
Potential Energy is the energy stored
in a spring or other elastic material.
в–є HookeвЂ™s Law: The displacement of a spring
from its unstretched position is proportional
the force applied.
в–є The slope of a force vs. displacement graph
is equal to the spring constant.
в–є The area under a force vs. displacement
graph is equal to the work done to compress
or stretch a spring.
Key Ideas
в–є Springs
and pendulums will go through
oscillatory motion when displaced
from an equilibrium position.
в–є The period of oscillation of a simple
pendulum is independent of its angle
of displacement (small angles) and
mass.
в–є Conservation
of energy: Energy can be
converted from one form to another, but it is
always conserved.
```
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