Irodov – Problems in General Physics

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- fit cos (cot +cc), (4.1b)

where f3 is the damping coefficient, o) is the frequency of damped oscillations:

(o= 1/(o6- 182 • 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 func-
tions 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 coor-
dinate xo = 25.0 cm and its velocity is equal to vx0 = 100 cm/s.

a—

V ( 0 4 — (o^2 )2 4i32 0)^2

tan (p-^2 f3ca^

4 - 0) 2 •