PART FOUR
OSCILLATIONS AND WAVES
4.1. MECHANICAL OSCILLATIONS
• Harmonic motion equation and its solution:
Y-F coax =0,
x= a cos (coot +a),
(4.1a)
where coo is the natural oscillation frequency.
• Damped oscillation equation and its solution:
.X+20;+u)Sx =-. 0,
x = aoe- f
it cos (cot +cc),
(4.1b)
where f3 is the damping coefficient, o) is the frequency of damped oscillations:
(o= 1/(o6-1 82 •
4.1c)
• Logarithmic damping decrement ? and quality factor Q:
X, = f3T , Q =
where T = 2n/co.
• Forced oscillation equation and its steady-state solution:
+213x+ cogx f 0 cos cot, x = a cos (cot — cp),
where
to
• Maximum shift amplitude occurs at
(ores =
2§2.
(4.1g)
4.1. A point oscillates along the x axis according to the law x
a cos (cot — n/4). Draw the approximate plots
(a) of displacement x, velocity projection vx, and acceleration
projection wx as functions of time t;
(b) velocity projection vx and acceleration projection wx as functions of the coordinate x.
4.2. A point moves along the x axis according to the law x
= a sine (cot — n/4). Find:
(a) the amplitude and period of oscillations; draw the plot x (t);
(b) the velocity projection vx as a function of the coordinate x;
draw the plot vx (x).
4.3. A particle performs harmonic oscillations along the x axis
about the equilibrium position x = 0. The oscillation frequency is
= 4.00
At a certain moment of time the particle has a coordinate xo = 25.0 cm and its velocity is equal to vx0 = 100 cm/s.
a—
V (04 — (o2)2 4i32 0)2
tan
2
(pf3ca
4-0)2 •