Download 13.1-4 Spring force and elastic energy revisited. (Hooke’s law)

Ch13. Vibration and waves
First, we want to know the speed and position
x  A cos(2ft )
of an object undergoing simple harmonic
v  2 Af sin(2ft )
motion.
a   A( 2f ) 2 cos(2ft )
We can use the law of conservation of
mechanical energy to determine the speed of We can find an important equation from these
an object.
equations.
1 2 1 2 1 2
Fs  kx
1. Fs   kx
kA  mv  kx
2
2
2
negative sign means that force is always pushes
 mA(2f ) 2 cos(2ft )  kxAcos(2ft )
k 2
or pulls the object toward the equilibrium
2
v
(A  x )
m(2f ) 2  k
position.
m
The elastic potential energy is
1 k
f 
In
order
to
calculate
the
complete
motion
of
an
1
2 m
PE s  kx2
object undergoing simple harmonic motion, we
2
m
need to use calculus. However, we can use the
T  2
k
In this chapter, we will use this equation of
excel spreadsheet and equations of motion
motion to study how an object moves under
under a constant acceleration to calculate the 2. you can also see that the velocity equation
elastic force.
motion approximately.
and the position equation satisfy the law of
The motion under the elastic force is common Here is the graphs we obtained from the
conservation of energy.
in nature, so this motion has a name, simple
calculation.
1 2 1 2 1 2
Simple Harmonic Motion
harmonic motion.
kA  mv  kx
2
2
2
6
2
2
2
kA  m A(2f ) sin(2ft )   k  A cos(2ft ) 
Simple harmonic motion occurs when the net
4
2
force is proportional to the displacement from
2


k
2
2
x(m)
the equilibrium point and is always toward the
kA  m A(
) sin(2ft )   k  A cos(2ft ) 
0
m
a(m/s^2)
equilibrium point.(Serway&Vuille)


0
5
10
15
x(m)
13.1-4 Spring force and elastic energy
revisited. (Hooke’s law)
We studied spring force and spring elastic
energy in chapter 5.
The elastic force is
-2
-4
As we always do before we study new chapters,
-6
we will define new measurement quantities.
t(s)
Amplitude(A): The maximum distance of the
object from its equilibrium position.
The position, acceleration and velocity graphs
Period(T): The time it takes the object to move are sinusoidal functions. We can write down
through one complete cycle of motion.
equations for position, velocity and
Frequency(f): The number of complete cycles or acceleration.
vibrations per unit of time. f =1/T.
v(m/s)

kA2  kA2 sin(2ft )   cos(2ft ) 
2
2

kA2  kA2
3. 2πf is called angular frequency. ω=2πf.
The following video shows how the position and
the acceleration are changing when an object
undergoes simple harmonic motion
https://youtu.be/eeYRkW8V7Vg
13.5 Motion of pendulum.
Any object with a restoring force can undergo
simple harmonic motion. A simple pendulum is
a good example.
acceleration.
 F
N
13.7-8 Waves


kg / m
 
Wave is the motion of a disturbance which
propagates through a medium or vacuum(light
kgm / s 2

propagates through vacuum).
kg / m
We need to determine the force constant of a Types of waves
simple pendulum.
m2
Transverse wave

From a free-body diagram, we can get the
s2
Particles of the disturbed medium move in a
following equation.
 m/s
direction perpendicular to the wave motion.
13.10
Interference
of
waves
Ft   mg sin 
Example) string wave, light, ocean wave.
Two traveling waves can meet and pass through
x
Ft   mg
each other without being destroyed or even
L
Longitudinal wave
altered unlike collisions between particles.
mg
Particles of the disturbed medium move in a
k
Constructive interference
L
direction parallel to the wave motion.
When two waves are in phase, the
Where m is the mass of an object suspended by Example) sound, ocean wave(ocean wave is a displacements are added together.
a light string, L is the length of the light string. combination of transverse and longitudinal
waves).
Destructive interference

L
When two waves are out of phase the final
We define following new physical quantities.
displacement is difference between the two
Wavelength(λ)
waves.
m
Distance
between
two
successive
points
that
x
https://youtu.be/uKrvTA4SKVU
behave identically.
Then from the frequency equation,
T  2
m
k
 2
m
mg / L
 2
L
g
Wave speed(v)
v  f
13.9 the speed of waves on string.
The speed of waves on string is,
v
F

The period does not depend on the mass but We can see that this equation has a correct unit
only on the length of the string and gravitational by dimensional analysis.
13.11 reflection of waves
When a pulse on string is reflected, the pulse is
inverted if the reflection point is fixed. If the
end is free to move up and down, the pulse is
reflected without inversion.
(Please see the diagrams in the chapter 13.11)
A uniform sting has a mass M of 0.030 kg and a
length L of 6.00 m. Tension is maintained in A 0.500 kg object connected to a ling spring
the string by suspending a block of mass m
with a spring constant of 20.0 N/m oscillates
= 2.00 kg from one end. Find the speed of a
on a frictionless horizontal surface. (a)
transverse wave pulse on this string.
Calculate the total energy of the system and
A wave traveling in the positive x-direction is
the maximum speed of the object if the
pictured in Figure below. Find the
amplitude of the motion is 0.0300m.(b)
amplitude, wave length, speed and period
What is the velocity of the object when the
of the wave if it has a frequency of 8.00 Hz.
displacement is 0.0200m? (c) compute the
kinetic energy and potential energy of the
In the figure, ∆x =0.400m and ∆y=0.150m.
system when the displacement is 0.0200m.
Using a small pendulum of length 0.171 m, a
geophysicist counts 72.0 complete swings in
a time of 60.0 s. What is the value of g in
this location?