Transcript PHYS 342: Modern Physics

Periodic Motion
• We are surrounded by oscillations – motions that
repeat themselves (periodic motion)
– Grandfather clock pendulum, boats bobbing at anchor,
oscillating guitar strings, pistons in car engines
• Understanding periodic motion is essential for the
study of waves, sound, alternating electric currents,
light, etc.
• An object in periodic motion experiences restoring
forces or torques that bring it back toward an
equilibrium position
• Those same forces cause the object to “overshoot”
the equilibrium position
• Think of a block oscillating on a spring or a
pendulum swinging back and forth past its
equilibrium position
Review of Springs
• Classic example of periodic motion:
– Spring exerts restoring force on block:
Fs  kx
(Hooke’s Law)
– k = spring constant (a measure of
spring stiffness)
– “Slinky” has k = 1 N/m; auto
suspensions have k = 105 N/m
– Movie of vertical spring:
1 2
• Elastic potential energy stored in spring: U el  kx
2
–
–
–
–
Uel = 0 when x = 0 (spring relaxed)
Uel is > 0 always
We do not have freedom to pick where x = 0
Uel conserves mechanical energy
CQ 1: The diagram below shows two different masses
hung from identical Hooke’s law springs. The Hooke’s
law constant k for the springs is equal to:
A)
B)
C)
D)
2 N/cm
5 N/cm
10 N/cm
20 N/cm
Example Problem #13.67
A 3.00-kg object is fastened to a light spring,
with the intervening cord passing over a pulley
(see figure). The pulley is frictionless, and its
inertia may be neglected. The object is
released from rest when the spring is
unstretched. If the object drops 10.0 cm before
stopping, find (a) the spring constant of the
spring and (b) the speed of the object when it is
5.00 cm below its starting point.
Solution (details given in class):
(a) 588 N/m
(b) 0.700 m/s
Periodic Motion
• Sequence of “snapshots” of
a simple oscillating system:
• Frequency ( f ) = number of oscillations that are
completed each second
– Units of frequency = Hertz 1 Hz = 1 oscillation per
second = 1 s–1
• Period = time for one complete oscillation (or cycle) =
T = 1/f
Simple Harmonic Motion
• For the motion shown in the previous slide, a graph
of the displacement x as a function of time looks like
the following:
Position vs Time
(Look familiar??
See previous slide)
• Written as a function: x(t) = Acos(wt + f)
– A = Amplitude of the motion
– (wt + f) = Phase of the motion
– f = Phase constant (or phase angle)  value depends on
the displacement and velocity of particle at time t = 0
– w = Angular frequency = 2p/T = 2pf (measures rate of
change of an angular quantity in rad/s)
• Simple harmonic motion (SHM) = periodic motion is
a sinusoidal function of time (represented by sine or
cosine function)
Simple Harmonic Motion
• Affect of changes in the amplitude, period, and phase
Fundamentals of
Halliday,
angle on curves of displacement vs. time: Physics,
Resnick, and Walker,
6th ed.
Red A > Blue A
Red T < Blue T
Red f < Blue f
• The velocity of a particle moving with SHM is given
by (from conservation of mechanical energy  ½ kA2
= ½ mv2 + ½ kx2):
k 2 2
v

A
m
x

• From Hooke’s Law coupled with Newton’s 2nd Law
(–kx = ma), the acceleration of a particle moving with
SHM is:
a  kx / m
Simple Harmonic Motion
• Since v and a both depend on x,
they also are sinusoidal functions
of time (see figure at right)
• The relationship between
displacement, velocity, and
acceleration in SHM is
demonstrated by the following:
– When the magnitude of the
displacement is greatest, the magnitude of the velocity is
least and vice–versa
– When displacement has its greatest positive value,
acceleration has its greatest negative value, and vice–
versa
– When displacement = 0, acceleration = 0
Simple Harmonic Motion
• From Hooke’s Law, we have another definition of
SHM:
– Motion executed by a particle of mass m subject to a force
that is proportional to the displacement of the particle but
opposite in sign
• We can further analyze SHM by comparing it to
uniform circular motion
– For example, when a ball is attached to a turntable
rotating with constant angular speed, the shadow of the
ball moves back and forth with SHM
• The angular frequency (w and period (T) are:
– w used due to strong similarity
k
w

2
p
f

between SHM and circular motion
m
m
T  2p
k
Energy in Simple Harmonic Motion
• In Chapter 5 we saw that the total mechanical energy
of a linear oscillator (mass on a spring) was
conserved if the motion proceeded without friction
• We can now see directly how both the kinetic and
potential energies vary with time, yet the total
mechanical energy remains constant in time
E vs. t
Fundamentals of Physics, Halliday,
Resnick, and Walker, 6th ed.
CQ 2: Interactive Example Problem:
Mass on a Spring
Which animation shows the correct graph of
position vs. time for the ball?
A)
B)
C)
D)
Animation 1
Animation 2
Animation 3
Animation 4
(Physlet Physics Problem #16.2, copyright Prentice–Hall publishing)
CQ 3: Interactive Example Problem:
Measuring Young Tarzan’s Mass
What is Tarzan Jr.’s mass?
A)
B)
C)
D)
14.5 kg
41.4 kg
55.7 kg
130.2 kg
(ActivPhysics online Problem #9.4, copyright Addison Wesley)
The Simple Pendulum
• A simple pendulum consists of a particle
of mass m (bob) suspended from one
end of an unstretchable, massless string
of length L fixed at the other end
• The component of gravity tangent to the
path of the bob provides a restoring torque about the
pivot point:
Pendulum vs. Block-Spring
t = –L(mg sinq) = Ia
• If q is small ( 15°) then sinq  q: a = –(mgL / I )q
• This equation is the angular equivalent of the
condition for SHM (a = –w2 x), so: (Note that T = period here!)
w = (mgL / I )½ and
T = 2p(I / mgL)½
• Since I = mL2 in this case:
T  2p
L
g
(independent of
mass and
amplitude!)
CQ 4: Which of the following would most
accurately demonstrate the kinetic energy of a
pendulum?
A)
B)
C)
D)
Figure A
Figure B
Figure C
Figure D
CQ 5: Interactive Example Problem: Risky
Pendulum Walk
At what constant speed must the person walk in
order to move safely under the pendulum?
A)
B)
C)
D)
0.9 m/s
1.8 m/s
2.5 m/s
3.5 m/s
(ActivPhysics online Problem #9.11, copyright Addison Wesley)
Shock Absorbers
• Shock absorbers provide a
damping of the oscillations
– A piston moves through a viscous
fluid like oil
– The piston has holes in it, which
creates a (reduced) viscous force on the piston, regardless
of the direction it moves (up or down)
– Viscous force reduces amplitude of oscillations smoothly
after car hits bump in road
– When oil leaks out of the shock absorber, the damping is
insufficient to prevent oscillations
• Shock absorber is example of an
underdamped oscillator (see also
critically damped and overdamped)
Wave Motion
• The wave is another basic model used to describe
the physical world (the particle is another example)
• Any wave is characterized as some sort of
“disturbance” that travels away from its source
• In many cases, waves are result of oscillations
– For example, sound waves produced by vibrating string
• For now, we will concentrate on mechanical waves
traveling through a material medium
– For example: water, sound, seismic waves
– The wave is the propagation of the disturbance: they do
not carry the medium with it
• Electromagnetic waves do not require a medium
• All waves carry momentum and energy
Types of Waves
• A traveling wave is a disturbance (pulse) that travels
along the medium with a definite speed
• A transverse wave produces particles in the
medium that move perpendicular to the motion of
the wave pulse
• A longitudinal wave produces particles that move
parallel to the motion of the wave pulse
• Both transverse and longitudinal waves can be
represented by waveforms: 1–D snapshots at
particular instant in time
(transverse)
(longitudinal)
Types of Waves
• In solids, both transverse and longitudinal waves can
exist
– Transverse waves result from shear disturbance
– Longitudinal waves result from compressional disturbance
• Only longitudinal waves propagate in fluids (they can
be compressed but do not sustain shear stresses)
– Transverse waves can travel along surface of liquid,
though (due to gravity or surface tension)
• Sound waves are longitudinal
– Each small volume of air vibrates back and forth along
direction of travel of the wave
• Earthquakes generate both longitudinal (4–8 km/s P
waves) and transverse (2–5 km/s S waves) seismic
waves
– Also surface waves which have both components
Properties of Waves
• Consider traveling waves on a continuous rope:
y
A = wave amplitude
A
x
l = wavelength
wave speed = v  fl (of pattern)
• For the particular case of a transverse wave on a
stretched string (under tension):
F
v
– F = tension (restoring force)
m
– m = mass per unit length (property of medium)
• Multiple traveling waves can meet and pass through
each other without being destroyed or altered
– We can hear multiple voices in a crowded room
• When multiple waves overlap, the wave in the
overlap region is determined by the superposition
principle
Properties of Waves
• Superposition principle: The overlap of 2 or more
waves (having small amplitude) results in a wave
that is a point-by-point summation of each individual
wave
(constructive interference)
(destructive interference)
Properties of Waves
• Traveling waves can both reflect and transmit
across a boundary between 2 media
– Reflected wave pulse is inverted (not inverted) if wave
reaches a boundary that is fixed (free to move)
Reflection of Waves
Wave Pulse One End Fixed
Wave Pulse Sliding Support
CQ 6: Waves A and B, pictured below, may or may not
be in phase. If wave A and wave B are superimposed,
the range of possible amplitudes for the resulting wave
will be:
A)
B)
C)
D)
from 0 cm to 3 cm.
from 0 cm to 9 cm.
from 3 cm to 6 cm.
from 3 cm to 9 cm.
Example Problem #13.59
A 2.65-kg power line running between two towers has a
length of 38.0 m and is under a tension of 12.5 N.
(a)What is the speed of a transverse pulse set up on
the line?
(b)If the tension in the line was unknown, describe a
procedure a worker on the ground might use to
estimate the tension.
Partial solution (details given in class):
(a) 13.4 m/s