transcript
- Slide 1
- 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
- Slide 2
- Review of Springs Classic example of periodic motion: Spring
exerts restoring force on block: k = spring constant (a measure of
spring stiffness) Slinky has k = 1 N/m ; auto suspensions have k =
10 5 N/m Movie of vertical spring: Elastic potential energy stored
in spring: U el = 0 when x = 0 (spring relaxed) U el is > 0
always We do not have freedom to pick where x = 0 U el conserves
mechanical energy (Hookes Law)
- Slide 3
- CQ 1: The diagram below shows two different masses hung from
identical Hookes law springs. The Hookes law constant k for the
springs is equal to: A)2 N/cm B)5 N/cm C)10 N/cm D)20 N/cm
- Slide 4
- Example Problem #13.67 Solution (details given in class):
(a)588 N/m (b)0.700 m/s 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.
- Slide 5
- 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
- Slide 6
- 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: Written as a function: x(t) = Acos( t + ) A =
Amplitude of the motion ( t + ) = Phase of the motion = Phase
constant (or phase angle) value depends on the displacement and
velocity of particle at time t = 0 = Angular frequency = 2 /T = 2 f
(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) (Look familiar?? See
previous slide) Position vs Time
- Slide 7
- Simple Harmonic Motion Affect of changes in the amplitude,
period, and phase angle on curves of displacement vs. time: The
velocity of a particle moving with SHM is given by (from
conservation of mechanical energy kA 2 = mv 2 + kx 2 ): From Hookes
Law coupled with Newtons 2 nd Law (kx = ma), the acceleration of a
particle moving with SHM is: Red A > Blue A Red T < Blue T
Red < Blue Fundamentals of Physics, Halliday, Resnick, and
Walker, 6 th ed.
- Slide 8
- 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 viceversa When displacement has its greatest positive value,
acceleration has its greatest negative value, and vice versa When
displacement = 0, acceleration = 0
- Slide 9
- Simple Harmonic Motion From Hookes 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 ( and
period (T) are: used due to strong similarity between SHM and
circular motion
- Slide 10
- 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, 6 th
ed.
- Slide 11
- CQ 2: Interactive Example Problem: Mass on a Spring (Physlet
Physics Problem #16.2, copyright PrenticeHall publishing) Which
animation shows the correct graph of position vs. time for the
ball? A)Animation 1 B)Animation 2 C)Animation 3 D)Animation 4
- Slide 12
- CQ 3: Interactive Example Problem: Measuring Young Tarzans Mass
(ActivPhysics online Problem #9.4, copyright Addison Wesley) What
is Tarzan Jr.s mass? A)14.5 kg B)41.4 kg C)55.7 kg D)130.2 kg
- Slide 13
- 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: = L(mg sin ) = I If is small ( 15 ) then sin : =
(mgL / I ) This equation is the angular equivalent of the condition
for SHM (a = 2 x), so : = (mgL / I ) and T = 2 (I / mgL) Since I =
mL 2 in this case: (independent of mass and amplitude!) Pendulum
vs. Block-Spring (Note that T = period here!)
- Slide 14
- CQ 4: Which of the following would most accurately demonstrate
the kinetic energy of a pendulum? A)Figure A B)Figure B C)Figure C
D)Figure D
- Slide 15
- CQ 5: Interactive Example Problem: Risky Pendulum Walk
(ActivPhysics online Problem #9.11, copyright Addison Wesley) At
what constant speed must the person walk in order to move safely
under the pendulum? A)0.9 m/s B)1.8 m/s C)2.5 m/s D)3.5 m/s
- Slide 16
- 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)
- Slide 17
- 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
- Slide 18
- 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: 1D
snapshots at particular instant in time (transverse)
(longitudinal)
- Slide 19
- 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 (48 km/s P waves) and transverse (25 km/s S
waves) seismic waves Also surface waves which have both
components
- Slide 20
- Properties of Waves Consider traveling waves on a continuous
rope: For the particular case of a transverse wave on a stretched
string (under tension): F = tension (restoring force) = 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 A x y A = wave amplitude = wavelength wave
speed = ( of pattern)
- Slide 21
- 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)
- Slide 22
- 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
- Slide 23
- 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)from 0 cm to 3 cm.
B)from 0 cm to 9 cm. C)from 3 cm to 6 cm. D)from 3 cm to 9 cm.
- Slide 24
- Example Problem #13.59 Partial solution (details given in
class): (a)13.4 m/s 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.