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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?
 Energy that is stored in elastic materials as a
result of their stretching.
в–є Where







is it found?
Rubber bands
Bungee cords
Trampolines
Springs
Bow and Arrow
Guitar string
Tennis Racquet
Hooke’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).
► Hooke’s Law:
в–є
Felastic = -kx
Where:
(N/m)
k = spring constant = stiffness of spring
x = displacement
Hooke’s Law
в–є What
is the graphical relationship
between the elastic spring force
and displacement?
Felastic = -kx
Slope = k
Displacement
Hooke’s Law
в–є
A force acting on a spring, whether
stretching or compressing, is
always positive.
 Since the spring would prefer to be in a
“relaxed” position, a negative “restoring”
force will exist whenever it is deformed.
 The restoring force will always attempt
to bring the spring and any object
attached to it back to the equilibrium
position.
 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.
 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:
 An oscillation around an equilibrium position will
occur when an object is displaced from its
equilibrium position and released.
 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):
 The Elastic Potential Energy will
be at a maximum
 The force will be at a maximum.
 The acceleration will be at a
maximum.
в–є
At equilibrium (x = 0):
 The Elastic Potential Energy will
be zero
 Velocity will be at a maximum.
 Kinetic Energy will be at a
maximum
 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
пЂ© пЂЅ 0 .14 m
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 – 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.
 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




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.
► Hooke’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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