We went over the homework problems posted last week. We discussed frequency, its realtionship to
angular frequency (see last week's page), as well as its relationship to the
period.
We returned to theory for the pendulum started last Thursday. We examined the forces acting
on the pendulum "bob" and broke them into radial and tangential components. The tangential
component was -mgsin(q) where q is the angle
between the pendulum and the vertical (equilibrium). The minus sign indicates that this is a
restoring force pushing the bob towrd the equilibrium position. Some discussion led
us to an expression for the tangential acceleration given by
L d2q/dt2, where L is the length of the
pendulum. We further examined the so-called "small angle" approximation
sin(q) » q
provided the angle is measured in radians. The tangential component of F=ma together with the
small-angle approximation gives an equation very much like we encoutered for the mass on a spring.
q(t) = qm
cos(w t + f0)
Thursday:
We started discussing waves using sound as our example. In this case, the air serves as a
medium. We established that it is not the medium "as a whole" that moves, but a disturbance
of the medium from its equilibrium that travels through the medium. Again, in the case of
sound, the equilibrium was uniform distribution of air density and air pressure. A source
of sound (such as one's vocal chords) caused regions of more dense or less dense air.
We mentioned the term "longitudinal" to refer to the situation in which the displacements from
equilibrium were in the same direction (along the same line) as direction in which the
wave travelled as well as the trem "transverse" to refer to the situation in which the displacements
occurred at right angles to the direction in which the wave moved. The sound wave we had been
considering was an example of a longitudinal wave, and we introduced the example of a wave on
a string as an example of a tranverse wave.
We discussed how if we looked at one point, the wave phenomena looked like vibration (reapeating
behavior in time) and that if we looked at it as a function of position we also expected
repeated behavior. We started examining a particular function
y(x,t) = ym cos ( k x ± w t +
f0)
where k is known as the wave number and is equal to 2 p /
l, where l is the wavelength, which the the distance
one travels before the above wave function begins repeating its behavior. The symbol
± accounts for the fact that the wave may be travelling
in the negative x direction (usually the left) corresponding to the + or may be travelling
in the positive x direction (usually the right) corresponding to the -. The speed of the wave
v is given by
v = l / T = l f = w / k
We started solving a travelling wave problem but ran out of time.
