Transcript Vibrations and Waves

Vibrations
and
Waves
Chapter 11
• Most object oscillate (vibrate)
because solids are elastic and they
will vibrate when given an impulse
• Tuning forks, violin strings, even a
spider’s web
• The spider uses the vibrations to
detect prey
11-1 Simple Harmonic Motion
• An object that vibrates or oscillates
moves back and forth over the same
path in the same time period
• This is said to be periodic motion
• Equilibrium position the position
where the net force equals zero.
Spring Constant
(Spring stiffness constant)
• The magnitude of the restoring force,
F of a spring is directly proportional
to the displacement, x
• F=-kx
• Negative because the restoring force
is always in the opposite direction of
the displacement
• k, spring stiffness constant, depends
upon the spring
• The force is not a constant force and
therefore the acceleration is not
constant either. The linear equations
developed earlier cannot be used.
• However, acceleration at a certain
point can be found by using
Newton’s 2nd law
• Displacement, x, (m) is measured
from the equilibrium position
• Amplitude, A, (m) is the maximum
displacement (should be equal on
either side of equilibrium)
• Cycle a complete vibration
• Period, T, (s) the time to complete
one cycle
• Frequency, f, (Hz) the number of
complete cycles per second
• Periods and
frequencies are
reciprocals of each
other
1
f 
T
1
T
f
• Any vibrating system for which F=-kx
is said to exhibit simple harmonic
motion (SHM)
• Such a system is often called a
simple harmonic oscillator (SHO)
11-2 Energy in the SHO
• Work has to be done to stretch the
spring and W=½kx2, this equals the
gain in PE (sec. 6-4)
• So, PE=½kx2
• The total energy must include KE
• E=½mv2 + ½kx2
• V is velocity of mass
• X is displacement from equilibrium
At maximum amplitude
• At the extreme points v=0 and x = A
• E = ½kA2
• So the total mechanical energy of
SHO is proportional to the square of
the amplitude
At equilibrium
• The x value is zero so there is 0 PE
and the velocity is maximum
• E = ½m2max
At any point
• Energy is part KE and part PE
• ½mv2 + ½kx2 = ½kA2
• Rearranging

k 2
2
v 
A x
m
2

11-3 The Period
• The period depends upon the stiffness of
a spring, but also on the mass that is
oscillating
• Oddly though, period doesn’t depend
upon the amplitude
• So
1
f 
2
k
m